A101321 -id:A101321 - cp's OEIS frontend

A060544 Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1.

1, 10, 28, 55, 91, 136, 190, 253, 325, 406, 496, 595, 703, 820, 946, 1081, 1225, 1378, 1540, 1711, 1891, 2080, 2278, 2485, 2701, 2926, 3160, 3403, 3655, 3916, 4186, 4465, 4753, 5050, 5356, 5671, 5995, 6328, 6670, 7021, 7381, 7750, 8128, 8515, 8911, 9316

Offset: 1

Henry Bottomley, Apr 02 2001

Triangular numbers not == 0 (mod 3). - Amarnath Murthy, Nov 13 2005
Shallow diagonal of triangular spiral in A051682. - Paul Barry, Mar 15 2003
Equals the triangular numbers convolved with [1, 7, 1, 0, 0, 0, ...]. - Gary W. Adamson & Alexander R. Povolotsky, May 29 2009
a(n) is congruent to 1 (mod 9) for all n. The sequence of digital roots of the a(n) is A000012(n). The sequence of units' digits of the a(n) is period 20: repeat [1, 0, 8, 5, 1, 6, 0, 3, 5, 6, 6, 5, 3, 0, 6, 1, 5, 8, 0, 1]. - Ant King, Jun 18 2012
Divide each side of any triangle ABC with area (ABC) into 2n + 1 equal segments by 2n points: A_1, A_2, ..., A_(2n) on side a, and similarly for sides b and c. If the hexagon with area (Hex(n)) delimited by AA_n, AA_(n+1), BB_n, BB_(n+1), CC_n and CC_(n+1) cevians, we have a(n+1) = (ABC)/(Hex(n)) for n >= 1, (see link with java applet). - Ignacio Larrosa Cañestro, Jan 02 2015; edited by Wolfdieter Lang, Jan 30 2015
For the case n = 1 see the link for Marion's Theorem (actually Marion Walter's Theorem, see the Cugo et al, reference). Also, the generalization considered here has been called there (Ryan) Morgan's Theorem.  - Wolfdieter Lang, Jan 30 2015
Pollock states that every number is the sum of at most 11 terms of this sequence, but note that "1, 10, 28, 35, &c." has a typo (35 should be 55). - Michel Marcus, Nov 04 2017
a(n) is also the number of (nontrivial) paths as well as the Wiener sum index of the (n-1)-alkane graph. - Eric W. Weisstein, Jul 15 2021

T. D. Noe, Table of n, a(n) for n = 1..1000
Ignacio Larrosa Cañestro, Hexágono y estrella determinados por tres pares de cevianas simétricas, (java applet).
Al Cugo et al., Marion's theorem, The Mathematics Teacher 86 (1993) p. 619.
John Elias, Illustration of Initial Terms
F. Pollock, On the Extension of the Principle of Fermat's Theorem of the Polygonal Numbers to the Higher Orders of Series Whose Ultimate Differences Are Constant. With a New Theorem Proposed, Applicable to All the Orders, Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.
Eric Weisstein's World of Mathematics, Alkane Graph
Eric Weisstein's World of Mathematics, Graph Path
Eric Weisstein's World of Mathematics, Marion's Theorem
Eric Weisstein's World of Mathematics, Wiener Sum Index
Index entries for two-way infinite sequences
Index entries for sequences related to centered polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001263, A027468, A081266, A190152.

List([1..50],n->(2*n-1)^2+(n-1)*n/2); # Muniru A Asiru, Mar 01 2019

[(2*n-1)^2+(n-1)*n/2: n in [1..50]]; // Vincenzo Librandi, Nov 18 2015

H := n -> simplify(1/hypergeom([-3*n,3*n+3,1],[3/2,2],3/4)); A060544 := n -> H(n-1); seq(A060544(i),i=1..19); # Peter Luschny, Jan 09 2012

Take[Accumulate[Range[150]], {1, -1, 3}] (* Harvey P. Dale, Mar 11 2013 *)
LinearRecurrence[{3, -3, 1}, {1, 10, 28}, 50] (* Harvey P. Dale, Mar 11 2013 *)
FoldList[#1 + #2 &, 1, 9 Range @ 50] (* Robert G. Wilson v, Feb 02 2011 *)
Table[(3 n - 1) (3 n - 2)/2, {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)
Table[Binomial[3 n - 1, 2], {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)
Table[PolygonalNumber[3 n - 2], {n, 20}] (* Eric W. Weisstein, Jul 15 2021 *)

PARI
```
a(n)=(3*n-1)*(3*n-2)/2
```

[(3*n-1)*(3*n-2)/2 for n in (1..50)] # G. C. Greubel, Mar 02 2019

a(n) = C(3*n, 3)/n = (3*n-1)*(3*n-2)/2 = A001504(n-1)/2.
a(n) = a(n-1) + 9*(n-1) = A060543(n, 3) = A006566(n)/n.
a(n) = A025035(n)/A025035(n-1) = A027468(n-1) + 1 = A000217(3*n-2).
a(1-n) = a(n).
From Paul Barry, Mar 15 2003: (Start)
a(n) = C(n-1, 0) + 9*C(n-1, 1) + 9*C(n-1, 2); binomial transform of (1, 9, 9, 0, 0, 0, ...).
a(n) = 9*A000217(n-1) + 1.
G.f.: x*(1 + 7*x + x^2)/(1-x)^3. (End)
Narayana transform (A001263) of [1, 9, 0, 0, 0, ...]. - Gary W. Adamson, Dec 29 2007
a(n-1) = Pochhammer(4,3*n)/(Pochhammer(2,n)*Pochhammer(n+1,2*n)).
a(n-1) = 1/Hypergeometric([-3*n,3*n+3,1],[3/2,2],3/4). - Peter Luschny, Jan 09 2012
From Ant King, Jun 18 2012: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 2*a(n-1) - a(n-2) + 9.
a(n) = A000217(n) + 7*A000217(n-1) + A000217(n-2).
Sum_{n>=1} 1/a(n) = 2*Pi/(3*sqrt(3)) = A248897.
(End)
a(n) = (2*n-1)^2 + (n-1)*n/2. - Ivan N. Ianakiev, Nov 18 2015
a(n) = A101321(9,n-1). - R. J. Mathar, Jul 28 2016
E.g.f.: (2 + 9*x^2)*exp(x)/2 - 1. - G. C. Greubel, Mar 02 2019
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=1} a(n)/n! = 11*e/2 - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 11/(2*e) - 1. (End)
a(n) = A000567(n) + A005449(n-1) (see illustration in links). - John Elias, Nov 10 2020
a(n) = P(2*n,4)*P(3*n,3)/24 for n>=2, where P(s,k) = ((s - 2)*k^2 - (s - 4)*k)/2 is the k-th s-gonal number. - Lechoslaw Ratajczak, Jul 18 2021

Additional description from Terrel Trotter, Jr., Apr 06 2002

Formulas by Paul Berry corrected for offset 1 by Wolfdieter Lang, Jan 30 2015

A062786 Centered 10-gonal numbers.

Original entry on oeis.org

1, 11, 31, 61, 101, 151, 211, 281, 361, 451, 551, 661, 781, 911, 1051, 1201, 1361, 1531, 1711, 1901, 2101, 2311, 2531, 2761, 3001, 3251, 3511, 3781, 4061, 4351, 4651, 4961, 5281, 5611, 5951, 6301, 6661, 7031, 7411, 7801, 8201, 8611, 9031, 9461, 9901, 10351, 10811

Offset: 1

Jason Earls, Jul 19 2001

Deleting the least significant digit yields the (n-1)-st triangular number: a(n) = 10*A000217(n-1) + 1. - Amarnath Murthy, Dec 11 2003
All divisors of a(n) are congruent to 1 or -1, modulo 10; that is, they end in the decimal digit 1 or 9. Proof: If p is an odd prime different from 5 then 5n^2 - 5n + 1 == 0 (mod p) implies 25(2n - 1)^2 == 5 (mod p), whence p == 1 or -1 (mod 10). - Nick Hobson, Nov 13 2006
Centered decagonal numbers. - Omar E. Pol, Oct 03 2011
The partial sums of this sequence give A004466. - Leo Tavares, Oct 04 2021
The continued fraction expansion of sqrt(5*a(n)) is [5n-3; {2, 2n-2, 2, 10n-6}]. For n=1, this collapses to [2; {4}]. - Magus K. Chu, Sep 12 2022
Numbers m such that 20*m + 5 is a square. Also values of the Fibonacci polynomial y^2 - x*y - x^2 for x = n and y = 3*n - 1. This is a subsequence of A089270. - Klaus Purath, Oct 30 2022
All terms can be written as a difference of two consecutive squares a(n) = A005891(n-1)^2 - A028895(n-1)^2, and they can be represented by the forms (x^2 + 2mxy + (m^2-1)y^2) and (3x^2 + (6m-2)xy + (3m^2-2m)y^2), both of discriminant 4. - Klaus Purath, Oct 17 2023

T. D. Noe, Table of n, a(n) for n = 1..1000
Leo Tavares, Illustration: Pentagonal Stars.
Leo Tavares, Illustration: Mid-section Stars.
Leo Tavares, Illustration: Mid-section Star Pillars.
Leo Tavares, Illustration: Trapezoidal Rays.
R. Yin, J. Mu, and T. Komatsu, The p-Frobenius Number for the Triple of the Generalized Star Numbers, Preprints 2024, 2024072280. See p. 2.
Index entries for sequences related to centered polygonal numbers.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001263, A124080, A101321, A028387, A016861, A003154, A005891, A000217, A004466, A144390, A000326, A060544.

Cf. also A016754, A002378, A069099, A045943, A003215, A046092, A001844, A028896, A005448, A024966, A082970.

List([1..50], n-> 1+5*n*(n-1)); # G. C. Greubel, Mar 30 2019

[1+5*n*(n-1): n in [1..50]]; // G. C. Greubel, Mar 30 2019

FoldList[#1+#2 &, 1, 10Range@ 45] (* Robert G. Wilson v, Feb 02 2011 *)
1+5*Pochhammer[Range[50]-1, 2] (* G. C. Greubel, Mar 30 2019 *)

j=[]; for(n=1,75,j=concat(j,(5*n*(n-1)+1))); j

for (n=1, 1000, write("b062786.txt", n, " ", 5*n*(n - 1) + 1) ) \\ Harry J. Smith, Aug 11 2009

def a(n): return(5*n**2-5*n+1) # Torlach Rush, May 10 2024

[1+5*rising_factorial(n-1, 2) for n in (1..50)] # G. C. Greubel, Mar 30 2019

a(n) = 5*n*(n-1) + 1.
From  Gary W. Adamson, Dec 29 2007: (Start)
Binomial transform of [1, 10, 10, 0, 0, 0, ...];
Narayana transform (A001263) of [1, 10, 0, 0, 0, ...]. (End)
G.f.: x*(1+8*x+x^2) / (1-x)^3. - R. J. Mathar, Feb 04 2011
a(n) = A124080(n-1) + 1. - Omar E. Pol, Oct 03 2011
a(n) = A101321(10,n-1). - R. J. Mathar, Jul 28 2016
a(n) = A028387(A016861(n-1))/5 for n > 0. - Art Baker, Mar 28 2019
E.g.f.: (1+5*x^2)*exp(x) - 1. - G. C. Greubel, Mar 30 2019
Sum_{n>=1} 1/a(n) = Pi * tan(Pi/(2*sqrt(5))) / sqrt(5). - Vaclav Kotesovec, Jul 23 2019
From Amiram Eldar, Jun 20 2020: (Start)
Sum_{n>=1} a(n)/n! = 6*e - 1.
Sum_{n>=1} (-1)^n * a(n)/n! = 6/e - 1. (End)
a(n) = A005891(n-1) + 5*A000217(n-1). - Leo Tavares, Jul 14 2021
a(n) = A003154(n) - 2*A000217(n-1). See Mid-section Stars illustration. - Leo Tavares, Sep 06 2021
From Leo Tavares, Oct 06 2021: (Start)
a(n) = A144390(n-1) + 2*A028387(n-1). See Mid-section Star Pillars illustration.
a(n) = A000326(n) + A000217(n) + 3*A000217(n-1). See Trapezoidal Rays illustration.
a(n) = A060544(n) + A000217(n-1). (End)
From Leo Tavares, Oct 31 2021: (Start)
a(n) = A016754(n-1) + 2*A000217(n-1).
a(n) = A016754(n-1) + A002378(n-1).
a(n) = A069099(n) + 3*A000217(n-1).
a(n) = A069099(n) + A045943(n-1).
a(n) = A003215(n-1) + 4*A000217(n-1).
a(n) = A003215(n-1) + A046092(n-1).
a(n) = A001844(n-1) + 6*A000217(n-1).
a(n) = A001844(n-1) + A028896(n-1).
a(n) = A005448(n) + 7*A000217(n).
a(n) = A005448(n) + A024966(n). (End)
From Klaus Purath, Oct 30 2022: (Start)
a(n) = a(n-2) + 10*(2*n-3).
a(n) = 2*a(n-1) - a(n-2) + 10.
a(n) = A135705(n-1) + n.
a(n) = A190816(n) - n.
a(n) = 2*A005891(n-1) - 1. (End)

Better description from Terrel Trotter, Jr., Apr 06 2002

A100119 a(n) = n-th centered n-gonal number.

Original entry on oeis.org

1, 2, 7, 19, 41, 76, 127, 197, 289, 406, 551, 727, 937, 1184, 1471, 1801, 2177, 2602, 3079, 3611, 4201, 4852, 5567, 6349, 7201, 8126, 9127, 10207, 11369, 12616, 13951, 15377, 16897, 18514, 20231, 22051, 23977, 26012, 28159, 30421, 32801, 35302

Offset: 0

Jonathan Vos Post, Dec 26 2004

a(n) is n times the n-th triangular number plus 1. - Thomas M. Green, Nov 16 2009
From Gary W. Adamson, Jul 31 2010: (Start)
Equals (1, 2, 3, 4, ...) convolved with (1, 0, 4, 7, 10, 13, ...).
Example: a(5) = 76 = (6, 5, 4, 3, 2, 1) dot (1, 0, 4, 7, 10, 13) = (6 + 0 + 16 + 21 + 20 + 13). (End)

			a(2) = 2*3 + 1 = 7, a(3) = 3*6 + 1 = 19, a(4) = 4*10 + 1 = 41. - _Thomas M. Green_, Nov 16 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

See also A101357 (Cumulative sums of the n-th n-gonal numbers).
A diagonal of A101321.
Cf. A060354, A098547, A101357.
Cf. A006003, A005920.

I:=[1, 2, 7, 19]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 25 2012

Table[(n^3+n^2)/2+1,{n,0,6!}] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2010 *)
LinearRecurrence[{4,-6,4,-1},{1,2,7,19},40] (* Vincenzo Librandi, Jun 25 2012 *)

a(n) = n^2*(n+1)/2+1; \\ Altug Alkan, Sep 21 2018

a(n) = 1 + n*(n + n^2)/2 = 1 + (1/2)*n^2 + (1/2) * n^3 = 1 + mean(n^2, n^3). - Joshua Zucker, May 03 2006
Equals A002411(n) + 1. - Olivier Gérard, Jun 20 2007
G.f.: (1 - 2*x + 5*x^2 - x^3) / (x-1)^4. - R. J. Mathar, Apr 04 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 25 2012
a(n) = (A098547(n)+1)/2. - Richard Turk, Jul 18 2017
a(n) = A060354(n+2) - A000290(n+1) = A006003(n+1) - A005563(n) and for n>0 A005920(n) - A068601(n+1). - Bruce J. Nicholson, Jun 23 2018

Corrected and extended by Joshua Zucker, May 03 2006

A350209 a(n) is the smallest positive integer which can be represented as the sum of distinct centered n-gonal numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

65, 151, 330, 316, 515, 645, 888, 824, 1017, 1348, 1474, 1614, 1655, 2006, 2233, 2255, 2779, 2805, 2882, 2865, 3364, 3653, 3930, 4035, 4325, 4765, 4675, 5016, 4904, 5446, 5913, 5990, 6061, 6954, 6850, 7681, 8313, 7846, 7591, 9035, 8736

Offset: 3

Ilya Gutkovskiy, Dec 19 2021

nonn

Eric Weisstein's World of Mathematics, Centered Polygonal Number

Cf. A101321, A350195, A350203, A350204, A350207, A350210.

A352349 a(n) is the largest number that is not the sum of distinct centered n-gonal numbers.

Original entry on oeis.org

158, 238, 492, 860, 1318, 1922, 2648, 3602, 4996, 6782, 9232, 12042, 14747

Offset: 3

Ilya Gutkovskiy, Mar 12 2022

Eric Weisstein's World of Mathematics, Centered Polygonal Number

Cf. A007419, A101321, A352350.

A365445 a(n) is the index of the least prime that is also a centered n-gonal number, or -1 if none exists.

Original entry on oeis.org

8, 3, 11, 4, 14, -1, -1, 5, 19, 6, 22, 14, 36, 7, 27, 8, 43, 18, 31, 9, 34, 21, 36, 22, 38, 10, 795, 11, 64, 25, 46, 27, 47, 12, 48, 50, 183, 13, 394, 14, 83, 121, 58, 15, 61, 169, 94, 36, 63, 16, 489, 38, 67, 68, 105, 17, 623, 18, 73, 74, 75, 44, 347

Offset: 3

Ilya Gutkovskiy, Sep 25 2023

sign

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000
Eric Weisstein's World of Mathematics, Centered Polygonal Number

Centered k-gonal primes listed in A276261.

Cf. A000040, A000720, A101321, A365815.

Table[PrimePi[SelectFirst[n # (# + 1)/2 + 1 & /@ Range[100], PrimeQ]], {n, 3, 65}] /. (PrimePi[Missing["NotFound"]] -> -1)

a(n) = if ((n==8) || (n==9), return(-1)); my(k=0, p); while (!isprime(p=1+n*k*(k-1)/2), k++); primepi(p); \\ Michel Marcus, Sep 27 2023

a(n) = A000720(A365815(n)). - Michel Marcus, Sep 27 2023

A365815 a(n) is the least centered n-gonal prime, or -1 if none exists.

Original entry on oeis.org

19, 5, 31, 7, 43, -1, -1, 11, 67, 13, 79, 43, 151, 17, 103, 19, 191, 61, 127, 23, 139, 73, 151, 79, 163, 29, 6091, 31, 311, 97, 199, 103, 211, 37, 223, 229, 1093, 41, 2707, 43, 431, 661, 271, 47, 283, 1009, 491, 151, 307, 53, 3499, 163, 331, 337, 571, 59, 4603, 61

Offset: 3

Ilya Gutkovskiy, Sep 25 2023

sign

Ilya Gutkovskiy, Scatterplot of a(n) up to n=10000
Eric Weisstein's World of Mathematics, Centered Polygonal Number

Centered k-gonal primes listed in A276261.

Cf. A000040, A101321, A365445.

Table[SelectFirst[n # (# + 1)/2 + 1 & /@ Range[100], PrimeQ], {n, 3, 60}] /. (Missing["NotFound"] -> -1)

a(n) = if ((n==8) || (n==9), return(-1)); my(k=0, p); while (!isprime(p=1+n*k*(k-1)/2), k++); p; \\ Michel Marcus, Sep 27 2023

A279830 a(n) = the least integer that is centered polygonal in exactly n ways.

Original entry on oeis.org

4, 7, 37, 31, 91, 181, 211, 421, 631, 1891, 1261, 2521, 6931, 18481, 20791, 13861, 27721, 41581, 83161, 138601, 245701, 235621, 180181, 556921, 360361, 540541, 1670761, 1081081, 1413721, 2702701, 2162161, 6486481, 3063061, 8288281, 13430341, 6846841, 10270261, 6126121

Offset: 1

Daniel Sterman, Dec 20 2016

nonn

a(n) has exactly n representations as a centered r-gonal number P(r,m) = 1 + r*m*(m+1)/2, with m > 1, r > 0.
a(n) appears n+1 times in A101321, due to the second column containing every positive integer.
a(n)-1 is the first appearance of n+1 in A007862.

			a(4)=31, because 31 is a centered triangular number (A005448), a centered pentagonal number (A005891), a centered decagonal number (A062786), and a central polygonal number (A002061). No number less than 31 has 4 representations.

Lars Blomberg, Table of n, a(n) for n = 1..143

Cf. A007862 (see alternative definition: the number of ways to represent n+1 as a centered polygonal number).
Cf. A063778 (the equivalent for polygonal numbers).
Subset of A275340 (the list of nontrivial centered polygonal numbers).
Subset of A101321 (centered polygonal numbers read by antidiagonals).

f[n_] := Length@Select[Divisors[2 n - 2], IntegerQ@Sqrt[1 + 4 #] &] - 1;
Do[If[IntegerQ[A279830[f[i]]], , A279830[f[i]] = i], {i, 10000}];
A279830 /@ Range[13]
(* Davin Park, Dec 28 2016 *)

Corrected and extended by Davin Park, Dec 27 2016

A279831 Numbers that are centered k-gonal numbers for two or more values of k.

Original entry on oeis.org

1, 7, 13, 16, 19, 22, 25, 31, 37, 43, 46, 49, 55, 61, 64, 67, 73, 76, 79, 85, 91, 97, 103, 106, 109, 111, 115, 121, 127, 133, 136, 139, 141, 145, 148, 151, 154, 157, 163, 166, 169, 172, 175, 181, 187, 190, 191, 193, 196, 199, 205, 211, 217, 221, 223, 226, 229, 232, 235, 241, 247, 253, 256

Offset: 1

Daniel Sterman, Dec 20 2016

nonn

Numbers satisfying 1 + n*m*(m+1)/2 for two or more values of (n,m), where n>=0 m>1.

Numbers in this sequence appear in A101321 at least three times (because the second column contains every positive integer).

			19 is in the sequence because 19 is both a centered triangular number and a centered hexagonal number.

Daniel Sterman, Table of n, a(n) for a(n)<1000

Cf. A090428 (rough equivalent for polygonal numbers).
Cf. A101321 (table of all centered polygonal numbers).
Cf. A275340 (list of nontrivial centered polygonal numbers).

A279846 Numbers that are centered k-gonal numbers for three or more values of k.

Original entry on oeis.org

1, 31, 37, 43, 46, 61, 67, 73, 79, 85, 91, 106, 109, 121, 127, 133, 136, 145, 151, 157, 166, 169, 181, 199, 211, 217, 226, 232, 235, 241, 253, 265, 271, 274, 277, 289, 295, 301, 307, 313, 316, 325, 331, 337, 343, 361, 379, 391, 397, 406, 409, 421, 433, 451, 463, 469, 481, 496, 505, 511

Offset: 1

Daniel Sterman, Dec 20 2016

nonn

Numbers satisfying 1 + n*m*(m+1)/2 for three or more values of (n,m), where n >= 0 m > 1.

Numbers in this sequence appear in A101321 at least four times (because the second column contains every positive integer).

			109 is in the sequence because 109 is a centered triangular number, a centered 18-gonal number, and a centered 36-gonal number.

Daniel Sterman, Table of n, a(n) for a(n)<1000

Cf. A062712 (rough equivalent for polygonal numbers).
Cf. A101321 (table of all centered polygonal numbers).
Cf. A275340 (list of nontrivial centered polygonal numbers).

cp's OEIS Frontend

A060544 Centered 9-gonal (also known as nonagonal or enneagonal) numbers. Every third triangular number, starting with a(1)=1.

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A062786 Centered 10-gonal numbers.

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A100119 a(n) = n-th centered n-gonal number.

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A350209 a(n) is the smallest positive integer which can be represented as the sum of distinct centered n-gonal numbers in exactly n ways, or 0 if no such integer exists.

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A352349 a(n) is the largest number that is not the sum of distinct centered n-gonal numbers.

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A365445 a(n) is the index of the least prime that is also a centered n-gonal number, or -1 if none exists.

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A365815 a(n) is the least centered n-gonal prime, or -1 if none exists.

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A279830 a(n) = the least integer that is centered polygonal in exactly n ways.