Кіріспе
Фигуралы сан – үшбұрышты және шаршы сандардың ұғымын бесбұрышты санға дейін кеңейтетін фигуралы сан, бірақ алғашқы екеуінен айырмашылығы, бесбұрышты сандардың құрылысына қатысы бар үлгілер айналымдық симметриялы емес. n-ші бесбұрышты сан pn – бұл n нүктеге дейінгі жақтары бар тұрақты бесбұрыштардың контурларынан тұратын нүктелер үлгісіндегі ерекше нүктелердің саны, егер бесбұрыштар бір нүктеге бөлінетін етіп үстінен жабысады. Мысалы, үшіншісі 1, 5 және 10 нүктеден тұратын контурлардан құрылады, бірақ 5-тің 1 және 3 нүктелері 10-ның 3 нүктесімен сәйкес келеді, нәтижесінде 12 ерекше нүкте қалады, олардың 10-ы бесбұрыш түрінде, ал 2-сі ішінде орналасқан. pn формуламен беріледі:
Figurate number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
n ≥ 1 үшін. Алғашқы бесбұрышты сандар:
Figurate number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
n-ші бесбұрышты сан – n-ден басталатын n бүтін санның қосындысы (яғни n-ден 2n-1-ге дейін). Сонымен қатар, келесі қатынастар орындалады:
Figurate number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
Бесбұрышты сандар үшбұрышты сандармен тығыз байланысты. n-ші бесбұрышты сан – (3n − 1)-ші үшбұрышты санның үштен біріне тең. Сонымен қатар, Tn – n-ші үшбұрышты сан:
Figurate number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
Жалпыланған бесбұрышты сандар жоғарыда келтірілген формуладан алынады, бірақ n 0, 1, −1, 2, −2, 3, −3, 4 тізбегіндегі мәндерді қабылдайды, нәтижесінде келесі тізбек пайда болады:
Figurate number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Жалпыланған бесбұрышты сандар Эйлердің бүтін сандарды бөлу теориясында маңызды, оның бесбұрышты сан теоремасында көрсетілгендей. Бесбұрышты санды құрайтын өрнектің ең сыртқы бесбұрышының ішіндегі нүктелер саны өзі жалпыланған бесбұрышты сан болып табылады.
Figurate number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The nth pentagonal number pn is the number of distinct dots in a pattern of dots consisting of the outlines of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. pn is given by the formula:
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187
The nth pentagonal number is the sum of n integers starting from n (i. e. from n to 2n 1). The following relationships also hold:
Pentagonal numbers are closely related to triangular numbers. The nth pentagonal number is one third of the (3n − 1)th triangular number. In addition, where Tn is the nth triangular number:
Generalized pentagonal numbers are obtained from the formula given above, but with n taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4 , producing the sequence:
0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots inside the outermost pentagon of a pattern forming a pentagonal number is itself a generalized pentagonal number.
Басқа қасиеттері
n>0 үшін, 2 немесе 3 сандары кірмейтін n бөлікке бөлінетін сандардың әр түрлі комбинацияларының саны. – 3-ке бөлінгенде 1 қалдық беретін алғашқы n табиғи санның қосындысы.
for n>0 is the number of different compositions of into n parts that don't include 2 or 3.
is the sum of the first n natural numbers congruent to 1 mod 3.
is the sum of the first n natural numbers congruent to 1 mod 3.