A000537 -id:A000537 - cp's OEIS frontend

A316458 Expansion of 60x(1 + 4*x + x^2) / (1 - x)^5.

60, 540, 2160, 6000, 13500, 26460, 47040, 77760, 121500, 181500, 261360, 365040, 496860, 661500, 864000, 1109760, 1404540, 1754460, 2166000, 2646000, 3201660, 3840540, 4570560, 5400000, 6337500, 7392060, 8573040, 9890160, 11353500, 12973500, 14760960

Offset: 1

Colin Barker, Aug 12 2018

Seems to be the negative of the second column of A316349.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000537, A316349, A316457, A316459.

Vec(60*x*(1 + 4*x + x^2) / (1 - x)^5 + O(x^40))

PARI
```
a(n) = 15*n^4 + 30*n^3 + 15*n^2
```

G.f.: 60*x*(1 + 4*x + x^2) / (1 - x)^5.
a(n) = 60 * A000537(n).
a(n) = 15*n^4 + 30*n^3 + 15*n^2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

A339483 Number of regular polygons that can be drawn with vertices on a centered hexagonal grid with side length n.

Original entry on oeis.org

0, 9, 75, 294, 810, 1815, 3549, 6300, 10404, 16245, 24255, 34914, 48750, 66339, 88305, 115320, 148104, 187425, 234099, 288990, 353010, 427119, 512325, 609684, 720300, 845325, 985959, 1143450, 1319094, 1514235, 1730265, 1968624, 2230800, 2518329, 2832795

Offset: 0

Peter Kagey, Dec 06 2020

The only regular polygons that can be drawn with vertices on the centered hexagonal grid are equilateral triangles and regular hexagons.

			There are a(2) = 75 regular polygons that can be drawn on the centered hexagonal grid with side length 2: A000537(2) = 9 regular hexagons and A008893(n) = 66 equilateral triangles.
The nine hexagons are:
    * . *       . * .       * * .
   . . . .     * . . *     * . * .
  * . . . *   . . . . .   . * * . .
   . . . .     * . . *     . . . .
    * . *       . * .       . . .
      1           1           7
which are marked with the number of ways to draw the hexagons up to translation.
The 66 equilateral triangles are:
    * . .       * . .       * . .       * . *       * . .       . . .
   * * . .     . . * .     . . . .     . . . .     . . . .     * . . *
  . . . . .   . * . . .   . . . * .   . . * . .   . . . . *   . . . . .
   . . . .     . . . .     * . . .     . . . .     . . . .     . . . .
    . . .       . . .       . . .       . . .       * . .       . * .
     24          14          12          12           2           2
which are marked with the number of ways to draw the triangles up to translation and dihedral action of the hexagon.

Peter Kagey, Table of n, a(n) for n = 0..10000
Burkard Polster, What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented, Mathologer video (2020).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000537 (regular hexagons), A008893 (equilateral triangles).
Cf. A338323 (cubic grid).
Cf. A003215, A180324, A185096, A322677, A348670.

a[n_] := n*(n+1)*(2*n+1)^2/2; Array[a, 35, 0] (* Amiram Eldar, Jun 20 2025 *)

a(n) = A000537(n) + A008893(n).
a(n) = (1/2)*(n+1)*n*(2*n+1)^2.
a(n) = 3*A180324(n).
Sum_{n>=1} 1/a(n) = 10 - Pi^2 (A348670). - Amiram Eldar, Jun 20 2025
From Elmo R. Oliveira, Aug 20 2025: (Start)
G.f.: -3*x*(x + 3)*(3*x + 1)/(x - 1)^5.
E.g.f.: exp(x)*x*(2 + x)*(9 + 24*x + 4*x^2)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A185096(n)/4 = A322677(n)/32. (End)

A004282 a(n) = n(n+1)^2(n+2)^2/12.

Original entry on oeis.org

0, 3, 24, 100, 300, 735, 1568, 3024, 5400, 9075, 14520, 22308, 33124, 47775, 67200, 92480, 124848, 165699, 216600, 279300, 355740, 448063, 558624, 690000, 845000, 1026675, 1238328, 1483524, 1766100, 2090175

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A006542, A107891, A114242.

[n*(n+1)^2*(n+2)^2/12: n in [0..50]]; // Vincenzo Librandi, Feb 09 2012

a:= n-> binomial(2+n, 2)*binomial(2+n, 3): seq(a(n), n=0..31); # Zerinvary Lajos, Apr 26 2007

Table[n*(n+1)^2*(n+2)^2/12,{n,0,40}] (* Vincenzo Librandi, Feb 09 2012 *)

a(n) = binomial(n+2,2)*binomial(n+2,3); \\ Charles R Greathouse IV, Feb 09 2012

a(n) = C(2+n, 2)*C(2+n, 3) = A000217(n+1)*A000292(n). - Zerinvary Lajos, Jan 10 2006
a(n-1) = Sum_{1 <= x_1, x_2 <= n} x_1*(det V(x_1,x_2))^2 = Sum_{1 <= i,j <= n} i*(i-j)^2, where V(x_1,x_2) is the Vandermonde matrix of order 2. - Peter Bala, Sep 21 2007
G.f.: x*(3+6*x+x^2)/(1-x)^6. - Colin Barker, Feb 09 2012
a(n) = Sum_{k=0..n} Sum_{i=0..n} (n-i+1) * C(k+1,k-1). - Wesley Ivan Hurt, Sep 21 2017
a(n) = A004302(n+1) - A000537(n+1). - J. M. Bergot, Mar 28 2018
From Amiram Eldar, May 29 2022: (Start)
Sum_{n>=1} 1/a(n) = 30 - 3*Pi^2.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/2 - 24*log(2) + 12. (End)

A133820 Triangle whose rows are sequences of increasing cubes: 1; 1,8; 1,8,27; ... .

Original entry on oeis.org

1, 1, 8, 1, 8, 27, 1, 8, 27, 64, 1, 8, 27, 64, 125, 1, 8, 27, 64, 125, 216, 1, 8, 27, 64, 125, 216, 343, 1, 8, 27, 64, 125, 216, 343, 512, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Offset: 1

Peter Bala, Sep 25 2007

Reading the triangle by rows produces the sequence 1,1,8,1,8,27,1,8,27,64,..., analogous to A002260.

			Triangle starts
1;
1, 8;
1, 8, 27;
1, 8, 27, 64;
1, 8, 27, 64, 125;

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Cf. A000537 (row sums), A002260, A019522, A133819, A133821, A133823.

a133820 n k = a133820_tabl !! (n-1) !! (k-1)
a133820_row n = a133820_tabl !! (n-1)
a133820_tabl = map (`take` (tail a000578_list)) [1..]
-- Reinhard Zumkeller, Nov 11 2012

Module[{nn=10,c},c=Range[nn]^3;Flatten[Table[Take[c,n],{n,10}]]] (* Harvey P. Dale, Mar 05 2014 *)

O.g.f.: (1+4qx+q^2x^2)/((1-x)(1-qx)^4) = 1 + x(1 + 8q) + x^2(1 + 8q + 27q^2) + ... .

Offset changed by Reinhard Zumkeller, Nov 11 2012

A140144 a(1)=1, a(n)=a(n-1)+n^1 if n odd, a(n)=a(n-1)+ n^0 if n is even.

Original entry on oeis.org

1, 2, 5, 6, 11, 12, 19, 20, 29, 30, 41, 42, 55, 56, 71, 72, 89, 90, 109, 110, 131, 132, 155, 156, 181, 182, 209, 210, 239, 240, 271, 272, 305, 306, 341, 342, 379, 380, 419, 420, 461, 462, 505, 506, 551, 552, 599, 600, 649, 650, 701, 702, 755, 756, 811, 812, 869

Offset: 1

Artur Jasinski, May 12 2008

Equals triangle A177990 * [1,2,3,...]. - Gary W. Adamson, May 16 2010

Girtrude Hamm, Classification of lattice triangles by their two smallest widths, arXiv:2304.03007 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000027, A000217, A000330, A000537, A000538, A000539, A136047, A140113.
Cf. A177990. - Gary W. Adamson, May 16 2010
Cf. A002378 (even bisection), A028387 (odd bisection).

a = {}; r = 1; s = 0; Do[k = 0; Do[k = k + (Sin[Pi m/2]^2) m^r + (Cos[Pi m/2]^2) m^s, {m, 1, n}]; AppendTo[a, k], {n, 1, 100}]; a

From R. J. Mathar, Feb 22 2009: (Start)
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
G.f.: x*(-1-x-x^2+x^3)/ ((1+x)^2*(x-1)^3). (End)
a(n) = Sum_{k=1..n} k^(k mod 2). - Wesley Ivan Hurt, Nov 20 2021

A164307 Primes in A081175.

Original entry on oeis.org

3, 5, 17, 257, 65537

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 12 2009

nonn

The 6th term is too large to include in the data section (see Example section or the b-file).
Primes of the form sum_{j=1..u} j^x for some x>0, u>1. (Since the case of x=1 leads to the triangular numbers with no additional primes, this is equivalent to the definition.)
Primes in A000330 (x=2), or in A000537 (x=3), or in A000538 (x=4), or in A000539 (x=5) etc. See A164312 for the corresponding x values.

			a(1) = 1^1 + 2^1 = 3.
a(2) = 1^2 + 2^2 = 5.
a(3) = 1^4 + 2^4 = 17.
a(4) = 1^8 + 2^8 = 257.
a(5) = 1^16 + 2^16 = 65537.
a(6) = 1^1440 + 2^1440 + 3^1440 + 4^1440 + 5^1440 = 3.287049497374559048967261852*10^1006 = 3287049497374559048967261852 ... 458593539025033893379.

N. J. A. Sloane, Table of n, a(n) for n = 1..6

Cf. A000215, A070592, A019434, A092506, A093179, A100270, A123599.

lst={};Do[s=0;Do[If[PrimeQ[s+=n^x],AppendTo[lst,s];Print[Date[],s]],{n, 4!}],{x,7!}];lst

Edited by R. J. Mathar, Aug 22 2009

Corrected by N. J. A. Sloane, Nov 23 2015 at the suggestion of Jaroslav Krizek.

A253169 Smallest m such that A256188(m) = n.

Original entry on oeis.org

1, 2, 9, 5, 6, 36, 10, 11, 12, 100, 17, 18, 19, 20, 225, 26, 27, 28, 29, 30, 441, 37, 38, 39, 40, 41, 42, 784, 50, 51, 52, 53, 54, 55, 56, 1296, 65, 66, 67, 68, 69, 70, 71, 72, 2025, 82, 83, 84, 85, 86, 87, 88, 89, 90, 3025, 101, 102, 103, 104, 105, 106, 107

Offset: 1

Reinhard Zumkeller, Mar 26 2015

nonn

A256188(a(n)) = n and A256188(m) != n for m < a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A256188, A000537, A010054, A000290.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a253169 = (+ 1) . fromJust . (`elemIndex` a256188_list)

a(n) = if n triangular then n^2 else A004202(n + 1 - A002024(n)).

A257448 a(n) = 13(2^n - 1) - 3n^2 - 9*n.

Original entry on oeis.org

1, 9, 37, 111, 283, 657, 1441, 3051, 6319, 12909, 26149, 52695, 105859, 212265, 425161, 851043, 1702903, 3406725, 6814477, 13630095, 27261451, 54524289, 109050097, 218101851, 436205503, 872412957, 1744828021, 3489658311, 6979319059, 13958640729

Offset: 1

Luciano Ancora, Apr 23 2015

These numbers belong to a family of sequences obtained as follows:
. A000225:   1*(2^n-1);
. A050488:   3*(2^n-1) - 2*n;
.    a(n):  13*(2^n-1) - 3*n^2 -  9*n;
. A257449:  75*(2^n-1) - 4*n^3 - 18*n^2 -  52*n;
. A257450: 541*(2^n-1) - 5*n^4 - 30*n^3 - 130*n^2 - 375*n,
where the sequence 1, 3, 13, 75, 541, ... is A000670 (after the first term), and A208744 gives the triangle of coefficients:
2;
3, 9;
4, 18, 52;
5, 30, 130, 375;
6, 45, 260, 1125, 3246;
7, 63, 455, 2625, 11361, 32781, etc.
Also, the antidiagonal sums in the array are given by the formula (6*n^2 + 6*k*n + (k-1)*k)*(k+n)!/((k+3)!*(n-1)!) for k = 0, 1, 2, 3, 4, ... (see Example field).

			By the second comment, the array begins (antidiagonals in A046902):
k=0: 1,  8, 27,  64,  125,  216, ...  A000578
k=1: 1,  9, 36, 100,  225,  441, ...  A000537
k=2: 1, 10, 46, 146,  371,  812, ...  A024166
k=3: 1, 11, 57, 203,  574, 1386, ...  A101094
k=4: 1, 12, 69, 272,  846, 2232, ...  A101097
k=5: 1, 13, 82, 354, 1200, 3432, ...  A101102
k=6: 1, 14, 96, 450, 1650, 5082, ...  A254469
...
See also A254469 (Example field).

Index entries for linear recurrences with constant coefficients, signature (5, -9, 7, -2).

Cf. A000225, A000670, A050488, A208744, A257449, A257450.

[13*(2^n-1)-3*n^2-9*n: n in [1..30]]; // Vincenzo Librandi, Apr 24 2015

Table[13 (2^n - 1) - 3 n^2 - 9n, {n, 30}]
CoefficientList[Series[x (1 + 4 x + x^2)/((1 - x)^3*(1 - 2 x)), {x, 0, 30}], x] (* Michael De Vlieger, Nov 14 2016 *)

G.f.: x*(1+4*x+x^2)/((1-x)^3*(1-2*x)).

a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>4. - Ray Chandler, Jul 25 2015

Edited by Bruno Berselli, Apr 28 2015

A264854 a(n) = n(n + 1)(11n^2 + 11n - 10)/24.

Original entry on oeis.org

0, 1, 14, 61, 175, 400, 791, 1414, 2346, 3675, 5500, 7931, 11089, 15106, 20125, 26300, 33796, 42789, 53466, 66025, 80675, 97636, 117139, 139426, 164750, 193375, 225576, 261639, 301861, 346550, 396025, 450616, 510664, 576521, 648550, 727125, 812631, 905464, 1006031

Offset: 0

Ilya Gutkovskiy, Nov 26 2015

Partial sums of centered 11-gonal (or hendecagonal) pyramidal numbers.

OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Pyramidal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A004467.

Cf. similar sequences provided by the partial sums of centered k-gonal pyramidal numbers: A006522 (k=1), A006007 (k=2), A002817 (k=3), A006325 (k=4), A006322 (k=5), A000537 (k=6), A006323 (k=7), A006324 (k=8), A236770 (k=9), A264853 (k=10), this sequence (k=11), A062392 (k=12), A264888 (k=13).

[n*(n+1)*(11*n^2+11*n-10)/24: n in [0..50]]; // Vincenzo Librandi, Nov 27 2015

Table[n (n + 1) (11 n^2 + 11 n - 10)/24, {n, 0, 50}]

a(n)=n*(n+1)*(11*n^2+11*n-10)/24 \\ Charles R Greathouse IV, Jul 26 2016

G.f.: x*(1 + 9*x + x^2)/(1 - x)^5.
a(n) = Sum_{k = 0..n} A004467(k).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Nov 27 2015

A265377 Sums of two or more consecutive positive cubes.

Original entry on oeis.org

9, 35, 36, 91, 99, 100, 189, 216, 224, 225, 341, 405, 432, 440, 441, 559, 684, 748, 775, 783, 784, 855, 1071, 1196, 1241, 1260, 1287, 1295, 1296, 1584, 1729, 1800, 1925, 1989, 2016, 2024, 2025, 2241, 2331, 2584, 2800, 2925, 2989, 3016, 3024, 3025, 3059, 3060

Offset: 1

Robert Israel, Dec 07 2015

nonn

All numbers of the form A000537(b) - A000537(a) for 0 <= a <= b-2.
A217843 minus (A000578 minus A131643).
n is in the sequence iff n = s*t where (s+t)/2 = A000217(u) and (s-t)/2 = A000217(v) with u-v >= 2.
If a(k(n)) = A000537(n+1), k(n) >= A000217(n) for n > 0. - Altug Alkan, Dec 07 2015
See A062682 for sums of two or more consecutive positive cubes in more than one way. - Reinhard Zumkeller, Dec 16 2015

			a(1) = 1^3 + 2^3 = 9.
a(2) = 2^3 + 3^3 = 35.
a(3) = 1^3 + 2^3 + 3^3 = 36.

Robert Israel, Table of n, a(n) for n = 1..10000

Subset of A217843.
Cf. A000217, A000537, A000578, A131643.
Cf. A062682 (subsequence).

import Data.Set (singleton, deleteFindMin, insert, Set)
a265377 n = a265377_list !! (n-1)
a265377_list = f (singleton (1 + 2^3, (1, 2))) (-1) where
   f s z = if y /= z then y : f s'' y else f s'' y
              where s'' = (insert (y', (i, j')) $
                           insert (y' - i ^ 3 , (i + 1, j')) s')
                    y' = y + j' ^ 3; j' = j + 1
                    ((y, (i, j)), s') = deleteFindMin s
-- Reinhard Zumkeller, Dec 17 2015

amin:= proc(b,N) local r;
  r:= b^2*(b+1)^2 - 4*N; if r > 0 then iroot(r,4) else 1 fi
end proc:
A265377:= proc(N) # to get all terms <= N
  local  a,b;
  sort(convert(select(`<=`,{seq(seq(b^2*(b+1)^2/4 - a^2*(a-1)^2/4,
       a = amin(b,N) .. b-1), b=2..1+iroot(floor(N/2),3))},N),list))
end proc:
A265377(10000);

With[{nn=12},Select[Sort[Flatten[Table[Total/@Partition[Range[nn]^3,n,1],{n,2,nn}]]],#<=((nn(nn+1))/2)^3&]] (* Harvey P. Dale, Dec 25 2015 *)

cp's OEIS Frontend

A316458 Expansion of 60*x*(1 + 4*x + x^2) / (1 - x)^5.

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A339483 Number of regular polygons that can be drawn with vertices on a centered hexagonal grid with side length n.

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Mathematica

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A004282 a(n) = n*(n+1)^2*(n+2)^2/12.

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A133820 Triangle whose rows are sequences of increasing cubes: 1; 1,8; 1,8,27; ... .

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A140144 a(1)=1, a(n)=a(n-1)+n^1 if n odd, a(n)=a(n-1)+ n^0 if n is even.

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A164307 Primes in A081175.

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Mathematica

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A253169 Smallest m such that A256188(m) = n.

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A257448 a(n) = 13*(2^n - 1) - 3*n^2 - 9*n.

A316458 Expansion of 60x(1 + 4*x + x^2) / (1 - x)^5.

A004282 a(n) = n(n+1)^2(n+2)^2/12.

A257448 a(n) = 13(2^n - 1) - 3n^2 - 9*n.

A264854 a(n) = n(n + 1)(11n^2 + 11n - 10)/24.