A062786 -id:A062786 - cp's OEIS frontend

A244911 Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 7, 7, 1, 1, 5, 10, 13, 11, 1, 1, 6, 13, 19, 21, 16, 1, 1, 7, 16, 25, 31, 31, 22, 1, 1, 8, 19, 31, 41, 46, 43, 29, 1, 1, 9, 22, 37, 51, 61, 64, 57, 37, 1, 1, 10, 25, 43, 61, 76, 85, 85, 73, 46, 1, 1, 11, 28, 49, 71, 91, 106, 113, 109, 91

Offset: 0

Kival Ngaokrajang, Jul 07 2014

T(n,k) is the total number of boxes, when we start with 1 center box (n = 0) then expand 1 box on k-arms for each n iteration. See illustration in links.
It seems that column C(k) = centered k-gonal numbers, and row R(n) = A000217(n)*k + 1.
The triangle under the main diagonal is A121722.
Column N (CN) is the Narayana transform (A001263) of (1, N, 0, 0, 0, ...). Example: C2 (1, 3, 7, 13, ...) is the Narayana transform of (1, 2, 0, 0, 0, ...). - Gary W. Adamson, Oct 01 2015

			Table begins:
       C0  C1  C2  C3  C4  C5
  n/k  0   1   2   3   4   5   ...
R0 0   1   1   1   1   1   1   ...
R1 1   1   2   3   4   5   6   ...
R2 2   1   4   7   10  13  16  ...
R3 3   1   7   13  19  25  31  ...
R4 4   1   11  21  31  41  51  ...
R5 5   1   16  31  46  61  76  ...
R6 6   1   22  43  64  85  106 ...
R7 7   1   29  57  85  113 141 ...
R8 8   1   37  73  109 145 181 ...
R9 9   1   46  91  136 181 226 ...
  ...  ... ... ... ... ... ... ...
C1 = A000124, C2 = A002061, C3 = A005448, C4 = A001844, C5 = A005891, C6 = A003215, C7 = A069099, C8 = A016754, C9 = A060544, C10 = A062786, C11 = A069125, C12  =  A003154.
R1 = A000027, R2 = A016777, R3 = A016921, R4 = A017281, R5 = 15*k + 1, R6 = A215146, R7 = A161714.

Kival Ngaokrajang, Illustration for n = 0..3, k = 1..4

Cf. A000217, A121722, A000124, A002061, A005448, A001844, A005891, A003215, A069099, A016754, A060544, A062786, A069125, A003154, A000027, A016777, A016921, A017281, A215146, A161714.

T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

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

A330082 a(n) = 5*A064038(n+1).

Original entry on oeis.org

0, 5, 15, 15, 25, 75, 105, 70, 90, 225, 275, 165, 195, 455, 525, 300, 340, 765, 855, 475, 525, 1155, 1265, 690, 750, 1625, 1755, 945, 1015, 2175, 2325, 1240, 1320, 2805, 2975, 1575, 1665, 3515, 3705, 1950, 2050, 4305, 4515, 2365, 2475, 5175, 5405, 2820, 2940

Offset: 0

Paul Curtz, Dec 01 2019

Main column of a pentagonal spiral for A026741:
                       (25)
                    49 (15) 31
                24  29 (15)  8  16
            47  14   7 ( 5)  3  17  33
        23  27  13   2 ( 0)  1   7   9  17
          45  13   6   3   1   4  19  35
            22  25  11   5   9  10  18
              43  12  23  11  21  37
                21  41  20  39  19
a(n) = 5 * A064038(n+1) from a pentagonal spiral.
Compare to A319127 =  6 * A002620 in the hexagonal spiral:
                22  23  23  22 (24)
              20  12  13  13 (12) 25
            21  10   5   4 ( 6) 14  25
          21  11   5   1 ( 0)  7  15  24
        20  11   4   1 ( 0)  2   7  15  26
          18  10   2   3   3   6  14  27
            19   8   9   9   8  16  27
              19  18  16  17  17  26
                30  28  29  29  28

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-6,10,-12,12,-10,6,-3,1).

Cf. A026741, A028895, A064038. A033429, A062786, A087348, A147874, A158447, A168668 are in the spiral.

A330082[n_]:=5Numerator[n(n+1)/4];Array[A330082,100,0] (* Paolo Xausa, Dec 04 2023 *)

concat(0, Vec(5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3) + O(x^50))) \\ Colin Barker, Dec 08 2019

a(n) = A026741(A028895(n)).
From Colin Barker, Dec 08 2019: (Start)
G.f.: 5*x*(1 + 4*x^3 + x^6) / ((1 - x)^3*(1 + x^2)^3).
a(n) = 3*a(n-1) - 6*a(n-2) + 10*a(n-3) - 12*a(n-4) + 12*a(n-5) - 10*a(n-6) + 6*a(n-7) - 3*a(n-8) + a(n-9) for n>8.
a(n) = (-5/16 + (5*i)/16)*(((-3-3*i) + (-i)^n + i^(1+n))*n*(1+n)) where i=sqrt(-1).
(End)

More terms from Colin Barker, Dec 22 2019

Name corrected by Paolo Xausa, Dec 04 2023

A332495 a(n-2) = a(n-6) + 5(1+2n) with a(0)=0, a(1)=2, a(2)=7, a(3)=15 for n>=4.

Original entry on oeis.org

0, 2, 7, 15, 25, 37, 52, 70, 90, 112, 137, 165, 195, 227, 262, 300, 340, 382, 427, 475, 525, 577, 632, 690, 750, 812, 877, 945, 1015, 1087, 1162, 1240, 1320, 1402, 1487, 1575, 1665, 1757, 1852, 1950, 2050, 2152, 2257

Offset: 0

Paul Curtz, Feb 14 2020

a(-2)=2, a(-1)=0. 4 evens followed by 4 odds.
Last digit is only 0, 2, 5, 7.
The vertical spoke S-N of the pentagonal spiral for A004526.
                          37
                      37  25  25
                  36  24  15  15  26
              36  24  14   7   8  16  26
          35  23  14   7   2   3   8  16  27
      35  23  13   6   2   0   0   3   9  17  27
        34  22  13   6   1   1   4   9  17  28
          34  22  12   5   5   4  10  18  28
            33  21  12  11  11  10  18  29
              33  21  20  20  19  19  29
                32  32  31  31  30  30
Rank of multiples of 10: 0, 7, 8, 15, 16, ... = A047521. Compare to A154260 in the formula.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).

Cf. A004526, A033429, A062786, A168668, A135706, A147874, 2*A147875 (all in the spiral).

Cf. A005408, A047521, A154260

CoefficientList[Series[x (2 + x + 2 x^2)/((1 - x)^3*(1 + x^2)), {x, 0, 42}], x] (* Michael De Vlieger, Feb 14 2020 *)

concat(0, Vec(x*(2 + x + 2*x^2) / ((1 - x)^3*(1 + x^2)) + O(x^40))) \\ Colin Barker, Feb 14 2020

a(-1-n) = a(n).
a(2*n) + a(1+2*n) = 2, 22, 62, ... = A273366(n).
Second differences give the sequence of period 4: repeat [3, 3, 2, 2].
From Colin Barker, Feb 14 2020: (Start)
G.f.: x*(2 + x + 2*x^2) / ((1 - x)^3*(1 + x^2)).
a(n) = 3*a(n-1) - 4*a(n-2) + 4*a(n-3) - 3*a(n-4) + a(n-5) for n>4.
(End)
Multiples of 10: 10*(0, 7, 9, 30, 34, ... = A154260).
4*a(n) = A087960(n) +5*n -1 +5*n^2. - R. J. Mathar, Feb 28 2020

A350760 Decimal expansion of Pi/(2sqrt(5)) tan(Pi/(2*sqrt(5))).

Original entry on oeis.org

5, 9, 4, 6, 7, 8, 1, 2, 3, 5, 3, 5, 2, 7, 8, 5, 1, 9, 1, 6, 8, 1, 1, 4, 2, 6, 9, 7, 6, 0, 5, 5, 4, 9, 3, 7, 6, 0, 3, 6, 3, 9, 4, 6, 9, 6, 1, 0, 2, 4, 2, 4, 3, 7, 9, 0, 5, 1, 1, 2, 5, 6, 8, 9, 5, 7, 9, 4, 5, 2, 5, 6, 3, 2, 6, 6, 1, 9, 0, 1, 5, 8, 8, 8, 4, 5, 2, 7, 3, 8, 9, 2, 6, 1, 2, 6, 0, 2, 5, 5, 2, 4, 3, 1, 0

Offset: 0

Amiram Eldar, Jan 14 2022

			0.59467812353527851916811426976055493760363946961024...

Robert Frontczak, Problem B-1267, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 58, No. 2 (2020), p. 179; The Zeta Riemann Function and the Cotangent Function, Solution to Problem B-1267 by Brian Bradie, ibid., Vol. 59, No. 2 (2021), pp. 179-180.

Cf. A000032, A000045, A062786, A244979.

RealDigits[(Pi/(2*Sqrt[5]))*Tan[Pi/(2*Sqrt[5])], 10, 100][[1]]

Equals Sum_{n>=1} zeta(2*n)*Fibonacci(2*n)/5^n (Frontczak, 2020).
Equals -1 + Sum_{n>=1} zeta(2*n)*Lucas(2*n)/5^n (Frontczak, 2020).
Equals Sum_{n>=0} 1/A273366(n).

A368426 Centered 10-gonal numbers which are sphenic numbers.

Original entry on oeis.org

20801, 22781, 37411, 47531, 55651, 75031, 80011, 100111, 120901, 133661, 161101, 177661, 191101, 199001, 201001, 230051, 236531, 240901, 245311, 263351, 279661, 289201, 313751, 323851, 326401, 368561, 376751, 436601, 439561, 445511, 472781, 475861, 488281, 507211, 539561

Offset: 1

Massimo Kofler, Dec 24 2023

nonn

			A062786(65) = 20801 = 5 * 65 * (65-1) + 1 = 11 * 31 * 61.
A062786(68) = 22781 = 5 * 68 * (68-1) + 1 = 11 * 19 * 109.

Robert Israel, Table of n, a(n) for n = 1..5080

Intersection of A007304 and A062786.

isc10:= proc(n) issqr(20*n+5) end proc:
P:= select(isprime, [seq(seq(10*i+j,j=[-1,1]),i=1..1000)]): nP:= nops(P):
M:= P[1]*P[2]*P[-1]:
A:= NULL:
for i from 1 to nP-2 while P[i]^3 < M do
  for j from i+1 to nP-1 while P[i]*P[j]^2 < M do
    for k from j+1 to nP do
      q:= P[i]*P[j]*P[k];
      if q > M then break fi;
      if isc10(q) then A:= A,q fi
od od od:
sort([A]); # Robert Israel, Dec 24 2023

Select[Table[5*n*(n+1) + 1, {n, 1, 330}], FactorInteger[#][[;; , 2]] == {1, 1, 1} &] (* Amiram Eldar, Dec 24 2023 *)

A386485 a(0) = 1; thereafter a(n) = 5n^2 - 5n + 2.

Original entry on oeis.org

1, 2, 12, 32, 62, 102, 152, 212, 282, 362, 452, 552, 662, 782, 912, 1052, 1202, 1362, 1532, 1712, 1902, 2102, 2312, 2532, 2762, 3002, 3252, 3512, 3782, 4062, 4352, 4652, 4962, 5282, 5612, 5952, 6302, 6662, 7032, 7412, 7802, 8202, 8612, 9032, 9462, 9902, 10352, 10812, 11282, 11762, 12252, 12752, 13262, 13782, 14312

Offset: 0

N. J. A. Sloane, Aug 18 2025

Maximum number of regions that can be formed in the plane by drawing n regular pentagons (of any size). Differs from A062786 and A124080 by a small constant shift, but is included here because of its geometrical applications.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A051890, A069894, A077588, A386480.

A062788 Squares of the form 5n*(n-1)+1.

Original entry on oeis.org

1, 361, 116281, 37442161, 12056259601, 3882078149401, 1250017107847561, 402501626648765281, 129604273763794572961, 41732173650315203728201, 13437630311127731805907801, 4326875228009479326298583761, 1393240385788741215336338063281

Offset: 1

Jason Earls, Jul 19 2001

Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Bisection of A049683.

Cf. A062786.

for(n=1,10^8,x=(5*n*(n-1)+1); if(issquare(x),print(x)))

a(n) = A049683(2*n-1) = (1/16) * (Lucas(12*n-6) - 2). - Joerg Arndt, Apr 09 2023
From Chai Wah Wu, Apr 17 2024: (Start)
a(n) = 323*a(n-1) - 323*a(n-2) + a(n-3) for n > 3.
G.f.: x*(-x^2 - 38*x - 1)/((x - 1)*(x^2 - 322*x + 1)). (End)

More terms from Sean A. Irvine, Apr 08 2023

A109119 a(n) = 2(5n^2 + 5n + 1)^3.

Original entry on oeis.org

2, 2662, 59582, 453962, 2060602, 6885902, 18787862, 44376082, 94091762, 183467702, 334568302, 577609562, 952759082, 1512116062, 2321871302, 3464647202, 5042017762, 7177208582, 10017976862, 13739671402, 18548472602

Offset: 0

Emeric Deutsch, Jun 19 2005

Kekulé numbers for certain benzenoids.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 310).

a:=n->2*(5*n^2+5*n+1)^3: seq(a(n),n=0..28);

G.f.: 2(1 + 1324z + 20495z^2 + 46360z^3 + 20495z^4 + 1324z^5 + z^6)/(1-z)^7.

a(n) = 2*A062786(n)^3. - R. J. Mathar, Jul 22 2022

A298760 Numbers k such that there is a record number of consecutive prime centered k-gonal numbers after 1.

Original entry on oeis.org

1, 2, 6, 10, 46, 102, 7186, 6382932

Offset: 1

Amiram Eldar, Jan 26 2018

The number of consecutive primes is 1, 3, 4, 7, 8, 9, 10, 11.

			The first 8 centered 10-gonal numbers (A062786) are 1, 11, 31, 61, 101, 151, 211, 281, and all of them except for 1 are primes (A090562). The previous record is 4 primes, for centered hexagonal numbers 7, 19, 37, 61 (A003215), therefore 6 and 10 are in the sequence.
From _Michel Marcus_, Feb 12 2018: (Start)
  Number of primes after the 1
1: 1  2  4  7  11  16 ...  : 1   <- record
2: 1  3  7 13  21  31 ...  : 3   <- record
3: 1  4 10 19  31  46 ...  : 0
4: 1  5 13 25  41  61 ...  : 2
5: 1  6 16 31  51  76 ...  : 0
6: 1  7 19 37  61  91 ...  : 4   <- record
....
(End)

Eric Weisstein's World of Mathematics, Centered Polygonal Number
Wikipedia, Centered polygonal number

Cf. A000124, A002061, A003215, A062786, A090562.

f[n_, k_] := k*n (n - 1)/2 + 1; a[k_] := Module[{n = 2}, While[PrimeQ[f[n, k]], n++]; n - 2]; am = 0; seq={}; Do[a1 = a[n]; If[a1 > am, AppendTo[seq, n]; am = a1], {n,1,10^7}]; seq

cp's OEIS Frontend

A244911 Table read by antidiagonals: T(n,k) = n*k + T(n-1,k) for n >=1, T(0,k) = 1.

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

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A330082 a(n) = 5*A064038(n+1).

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A332495 a(n-2) = a(n-6) + 5*(1+2*n) with a(0)=0, a(1)=2, a(2)=7, a(3)=15 for n>=4.

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A350760 Decimal expansion of Pi/(2*sqrt(5)) * tan(Pi/(2*sqrt(5))).

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A368426 Centered 10-gonal numbers which are sphenic numbers.

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A386485 a(0) = 1; thereafter a(n) = 5*n^2 - 5*n + 2.

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A062788 Squares of the form 5n*(n-1)+1.

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A332495 a(n-2) = a(n-6) + 5(1+2n) with a(0)=0, a(1)=2, a(2)=7, a(3)=15 for n>=4.

A350760 Decimal expansion of Pi/(2sqrt(5)) tan(Pi/(2*sqrt(5))).

A386485 a(0) = 1; thereafter a(n) = 5n^2 - 5n + 2.