The centered polygonal numbers are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers of dots with a constant number of sides. Each side of a polygonal layer contains one more dot than each side in the previous layer; so starting from the second polygonal layer, each layer of a centered k-gonal number contains k more dots than the previous layer.
Each centered k-gonal number in the series is k times the previous triangular number, plus 1. This can be formalized by the expression , where n is the series rank, starting with 0 for the initial 1. For example, each centered square number in the series is four times the previous triangular number, plus 1. This can be formalized by the expression .
The following diagrams show a few examples of centered polygonal numbers and their geometric construction. Compare these diagrams with the diagrams in Polygonal number.
centered triangular number
centered square number
centered pentagonal number
centered hexagonal number
Centered square numbers
1
5
13
25
Centered hexagonal numbers
1
7
19
37
Formulas
As can be seen in the above diagrams, the nth centered k-gonal number can be obtained by placing k copies of the (n−1)th triangular number around a central point; therefore, the nth centered k-gonal number can be mathematically represented by
The difference of the n-th and the (n+1)-th consecutive centered k-gonal numbers is k(2n+1).
The n-th centered k-gonal number is equal to the n-th regular k-gonal number plus (n-1)2.
Just as is the case with regular polygonal numbers, the first centered k-gonal number is 1. Thus, for any k, 1 is both k-gonal and centered k-gonal. The next number to be both k-gonal and centered k-gonal can be found using the formula:
which tells us that 10 is both triangular and centered triangular, 25 is both square and centered square, etc.
F. Tapson (1999). The Oxford Mathematics Study Dictionary (2nd ed.). Oxford University Press. pp. 88–89. ISBN0-19-914-567-9.
January 30, 2023
centered, polygonal, number, centered, polygonal, numbers, class, series, figurate, numbers, each, formed, central, surrounded, polygonal, layers, dots, with, constant, number, sides, each, side, polygonal, layer, contains, more, than, each, side, previous, la. The centered polygonal numbers are a class of series of figurate numbers each formed by a central dot surrounded by polygonal layers of dots with a constant number of sides Each side of a polygonal layer contains one more dot than each side in the previous layer so starting from the second polygonal layer each layer of a centered k gonal number contains k more dots than the previous layer Contents 1 Examples 1 1 Centered square numbers 1 2 Centered hexagonal numbers 2 Formulas 3 Sum of reciprocals 4 ReferencesExamples EditEach centered k gonal number in the series is k times the previous triangular number plus 1 This can be formalized by the expression k n n 1 2 1 displaystyle frac kn n 1 2 1 where n is the series rank starting with 0 for the initial 1 For example each centered square number in the series is four times the previous triangular number plus 1 This can be formalized by the expression 4 n n 1 2 1 displaystyle frac 4n n 1 2 1 These series consist of the centered triangular numbers 1 4 10 19 31 46 64 85 109 136 166 199 OEIS A005448 centered square numbers 1 5 13 25 41 61 85 113 145 181 221 265 OEIS A001844 centered pentagonal numbers 1 6 16 31 51 76 106 141 181 226 276 331 OEIS A005891 centered hexagonal numbers 1 7 19 37 61 91 127 169 217 271 331 397 OEIS A003215 which are exactly the difference of consecutive cubes i e n3 n 1 3 centered heptagonal numbers 1 8 22 43 71 106 148 197 253 316 386 463 OEIS A069099 centered octagonal numbers 1 9 25 49 81 121 169 225 289 361 441 529 OEIS A016754 which are exactly the odd squares centered nonagonal numbers 1 10 28 55 91 136 190 253 325 406 496 595 OEIS A060544 which include all even perfect numbers except 6 centered decagonal numbers 1 11 31 61 101 151 211 281 361 451 551 661 OEIS A062786 centered hendecagonal numbers 1 12 34 67 111 166 232 309 397 496 606 727 OEIS A069125 centered dodecagonal numbers 1 13 37 73 121 181 253 337 433 541 661 793 OEIS A003154 which are also the star numbers and so on The following diagrams show a few examples of centered polygonal numbers and their geometric construction Compare these diagrams with the diagrams in Polygonal number centeredtriangularnumber centeredsquarenumber centeredpentagonalnumber centeredhexagonalnumber Centered square numbers Edit 1 5 13 25 Centered hexagonal numbers Edit 1 7 19 37 Formulas EditAs can be seen in the above diagrams the nth centered k gonal number can be obtained by placing k copies of the n 1 th triangular number around a central point therefore the nth centered k gonal number can be mathematically represented by C k n k n 2 n 1 1 displaystyle C k n frac kn 2 n 1 1 The difference of the n th and the n 1 th consecutive centered k gonal numbers is k 2n 1 The n th centered k gonal number is equal to the n th regular k gonal number plus n 1 2 Just as is the case with regular polygonal numbers the first centered k gonal number is 1 Thus for any k 1 is both k gonal and centered k gonal The next number to be both k gonal and centered k gonal can be found using the formula k 2 2 k 1 1 displaystyle frac k 2 2 k 1 1 which tells us that 10 is both triangular and centered triangular 25 is both square and centered square etc Whereas a prime number p cannot be a polygonal number except the trivial case i e each p is the second p gonal number many centered polygonal numbers are primes In fact if k 3 k 8 k 9 then there are infinitely many centered k gonal numbers which are primes assuming the Bunyakovsky conjecture Since all centered octagonal numbers are also square numbers and all centered nonagonal numbers are also triangular numbers and not equal to 3 thus both of them cannot be prime numbers Sum of reciprocals EditThe sum of reciprocals for the centered k gonal numbers is 1 2 p k 1 8 k tan p 2 1 8 k displaystyle frac 2 pi k sqrt 1 frac 8 k tan left frac pi 2 sqrt 1 frac 8 k right if k 8p 2 8 displaystyle frac pi 2 8 if k 8References Edit centered polygonal numbers in OEIS wiki content Table of related formulae and values Neil Sloane amp Simon Plouffe 1995 The Encyclopedia of Integer Sequences San Diego Academic Press Fig M3826 Weisstein Eric W Centered polygonal number MathWorld F Tapson 1999 The Oxford Mathematics Study Dictionary 2nd ed Oxford University Press pp 88 89 ISBN 0 19 914 567 9 Retrieved from https en wikipedia org w index php title Centered polygonal number amp oldid 1100585348, wikipedia, wiki, book, books, library,
