A050492 -id:A050492 - cp's OEIS frontend

A152725 a(n) = n(n+1)(n^4 + 2n^3 - 2n^2 - 3*n + 3)/2.

0, 1, 63, 666, 3430, 12195, 34461, 83188, 178956, 352485, 647515, 1124046, 1861938, 2964871, 4564665, 6825960, 9951256, 14186313, 19825911, 27219970, 36780030, 48986091, 64393813, 83642076, 107460900, 136679725, 172236051, 215184438

Offset: 0

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A062392, A062393 (for 5th powers), A011934, A152726 (for 7th powers).

[n*(n+1)*(n^4+2*n^3-2*n^2-3*n+3)/2: n in [0..50]]; // G. C. Greubel, Sep 01 2018

k=0;lst={k};Do[k=n^6-k;AppendTo[lst,k],{n,1,5!}];lst
LinearRecurrence[{7,-21,35,-35,21,-7,1}, {0,1,63,666,3430,12195,34461}, 50] (* G. C. Greubel, Sep 01 2018 *)
CoefficientList[Series[-((x (1+56 x+246 x^2+56 x^3+x^4))/(-1+x)^7),{x,0,30}],x] (* Harvey P. Dale, Aug 03 2024 *)

a(n)=n*(n+1)*(n^4+2*n^3-2*n^2-3*n+3)/2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = n^6 - (n-1)^6 + (n-2)^6 - ... + ((-1)^n)*0^6.
G.f.: x*(1 + 56*x + 246*x^2 + 56*x^3 + x^4) / (1-x)^7. - R. J. Mathar, Jul 08 2013
a(n) = A050492(A000217(n)). - Kelvin Voskuijl, Jun 18 2025
E.g.f.: exp(x)*x*(2 + 61*x + 160*x^2 + 95*x^3 + 18*x^4 + x^5)/2. - Stefano Spezia, Jun 19 2025

Offset set to 0 by R. J. Mathar, Aug 15 2010

A175113 a(n) = ((2*n + 1)^6 + 1)/2.

Original entry on oeis.org

1, 365, 7813, 58825, 265721, 885781, 2413405, 5695313, 12068785, 23522941, 42883061, 74017945, 122070313, 193710245, 297411661, 443751841, 645733985, 919132813, 1282863205, 1759371881, 2375052121, 3160681525, 4151882813

Offset: 0

R. J. Mathar, Feb 13 2010

Convolution of the finite sequence 1, 358, 5279, 11764, 5279, 358, 1 with A000579. Partial sums of A175114.
Subsequence of A001844 because a(n)=(A050492(n+1)-1)^2+A050492(n+1)^2. - Bruno Berselli, Dec 28 2010
a(n) is also the first integer in a sum of (2*n + 1)^6 consecutive integers that equals (2*n + 1)^12. - Patrick J. McNab, Dec 26 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
R. J. Mathar, Point counts of D_k and some A_k and E_k integer lattices inside hypercubes, arXiv:1002.3844, Variable V_6^(g)(n).
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

I:=[1, 365, 7813, 58825, 265721, 885781, 2413405]; [n le 7 select I[n] else 7*Self(n-1) - 21*Self(n-2) + 35*Self(n-3) - 35*Self(n-4) + 21*Self(n-5) - 7*Self(n-6) + Self(n-7): n in [1..40]]; // Vincenzo Librandi, Dec 20 2012

CoefficientList[Series[(1 + 358*x + 5279*x^2 + 11764*x^3 + 5279*x^4 + 358*x^5 + x^6)/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Dec 20 2012 *)

a(n)= 7*a(n-1) -21*a(n-2) +35*a(n-3) -35*a(n-4) +21*a(n-5) -7*a(n-6) +a(n-7).
G.f.: (1+358*x+5279*x^2+11764*x^3+5279*x^4+358*x^5+x^6)/(1-x)^7.
a(n) = (2*n^2+2*n+1)*(16*n^4+32*n^3+20*n^2+4*n+1). - Bruno Berselli, Dec 27 2010

A050533 Thickened pyramidal numbers: a(n) = 2(n+1)n + Sum_{i=1..n} (4i(i-1) + 1).

Original entry on oeis.org

0, 5, 22, 59, 124, 225, 370, 567, 824, 1149, 1550, 2035, 2612, 3289, 4074, 4975, 6000, 7157, 8454, 9899, 11500, 13265, 15202, 17319, 19624, 22125, 24830, 27747, 30884, 34249, 37850, 41695, 45792, 50149, 54774, 59675, 64860, 70337, 76114, 82199, 88600, 95325

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 29 1999

This sequence is the partial sums of A053755. - J. M. Bergot, May 31 2012

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

Cf. A000292, A000447, A000217, A050492, A053755.

I:=[0, 5, 22, 59]; [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..40]]; // Vincenzo Librandi, Apr 27 2012

CoefficientList[Series[x*(5+2*x+x^2)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Apr 27 2012 *)
LinearRecurrence[{4,-6,4,-1},{0,5,22,59},40] (* Harvey P. Dale, May 08 2012 *)

a(n)=n*(4*n^2+6*n+5)/3 \\ Charles R Greathouse IV, Apr 16 2012

a(n) = (1/3)*n*(5 + 6*n + 4*n^2) = binomial(2*n+1, 3) + 2*(n+1)*n = A000447(n) + 4*A000217(n).
G.f.: x*(5+2*x+x^2)/(1-x)^4. - Colin Barker, Apr 16 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Apr 27 2012
E.g.f.: exp(x)*x*(15 + 18*x + 4*x^2)/3. - Elmo R. Oliveira, Aug 08 2025

A268201 a(n) = 4n^3 - 6n^2 + 3*n - 1.

Original entry on oeis.org

0, 13, 62, 171, 364, 665, 1098, 1687, 2456, 3429, 4630, 6083, 7812, 9841, 12194, 14895, 17968, 21437, 25326, 29659, 34460, 39753, 45562, 51911, 58824, 66325, 74438, 83187, 92596, 102689, 113490, 125023, 137312, 150381, 164254, 178955, 194508, 210937, 228266, 246519, 265720

Offset: 1

Juri-Stepan Gerasimov, Apr 16 2016

Nonnegative numbers n such that 2*n+1 is a cube.

Or, (y^k-1)/2 for k odd. - N. J. A. Sloane, Mar 05 2022

			a(1) = 0 because 4*1^3 - 6*1^2 + 3*1 - 1 = 0.
a(2) = 13 because 4*2^3 - 6*2^2 + 3*2 - 1 = 13.

H. Brocard, #2158, L'Intermédiaire des Mathématiciens, 10 (1903), 282-283

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. nonnegative numbers n such that 2*n + k is a cube: A271828 (k=-3), A050492 (k=-1), this sequence (k=1).

Magma
```
[((2*n-1)^3-1)/2: n in [0..41]];
```

A268201:=n->4*n^3 - 6*n^2 + 3*n - 1: seq(A268201(n), n=1..80); # Wesley Ivan Hurt, Apr 17 2016

Table[((2 n - 1)^3 - 1)/2, {n, 41}] (* or *)
CoefficientList[Series[(13*x + 10*x^2 + x^3)/(-1 + x)^4, {x, 0, 40}],
   x] (* Michael De Vlieger, Apr 16 2016 *)

lista(nn) = for(n=1, nn, print1(4*n^3-6*n^2+3*n-1, ", ")); \\ Altug Alkan, Apr 17 2016

G.f.: (13*x + 10*x^2 + x^3)/(-1 + x)^4. - Michael De Vlieger, Apr 16 2016

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4. - Wesley Ivan Hurt, Apr 17 2016

A274324 Number of partitions of n^3 into at most two parts.

Original entry on oeis.org

1, 1, 5, 14, 33, 63, 109, 172, 257, 365, 501, 666, 865, 1099, 1373, 1688, 2049, 2457, 2917, 3430, 4001, 4631, 5325, 6084, 6913, 7813, 8789, 9842, 10977, 12195, 13501, 14896, 16385, 17969, 19653, 21438, 23329, 25327, 27437, 29660, 32001, 34461, 37045, 39754

Offset: 0

Colin Barker, Jun 18 2016

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

A subsequence of A008619.
Cf. A099392 (n^2), A274325 (n^5).
Cf. also A050492.

[(3+(-1)^n+2*n^3)/4 : n in [0..50]]; // Wesley Ivan Hurt, Jun 25 2016

A274324:=n->(3+(-1)^n+2*n^3)/4: seq(A274324(n), n=0..50); # Wesley Ivan Hurt, Jun 25 2016

Table[(3+(-1)^n+2*n^3)/4, {n, 0, 50}] (* Wesley Ivan Hurt, Jun 25 2016 *)

\\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)).
b(n) = (3+(-1)^n+2*n)/4
vector(50, n, n--; b(n^3))

Coefficient of x^(n^3) in 1/((1-x)*(1-x^2)).
a(n) = A008619(n^3).
a(n) = (3+(-1)^n+2*n^3)/4.
a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>4.
G.f.: (1-2*x+4*x^2+3*x^3) / ((1-x)^4*(1+x)).
From Stefano Spezia, Sep 28 2022: (Start)
a(n) = A050492((n+1)/2) for n odd.
E.g.f.: ((2 + x + 3*x^2 + x^3)*cosh(x) + (1 + x + 3*x^2 + x^3)*sinh(x))/2. (End)

A359844 a(n) = ((2*n+1)^8 + 1)/2.

Original entry on oeis.org

1, 3281, 195313, 2882401, 21523361, 107179441, 407865361, 1281445313, 3487878721, 8491781521, 18911429681, 39155492641, 76293945313, 141214768241, 250123206481, 426445518721, 703204309121, 1125937695313, 1756239726961, 2676004630241, 3992462614561, 5844100138801

Offset: 0

Jianing Song, Jan 15 2023

Jianing Song, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. {((2*n+1)^k + 1)/2}: A000012 (k=0), A001477 (k=1), A219086 (k=2), A050492 (k=3), A175110 (k=4), A175113 (k=6), this sequence (k=8).

((2*Range[0, 25] + 1)^8 + 1)/2 (* Paolo Xausa, Jan 23 2025 *)

PARI
```
a(n) = ((2*n+1)^8 + 1)/2
```

def A359844(n): return ((n<<1)+1)**8+1>>1 # Chai Wah Wu, Jan 15 2023

a(n) = A359499(n)/A359498(n) = 16 * A359498(n) + 1.

A271828 a(n) = 4n^3 - 18n^2 + 27*n - 12.

Original entry on oeis.org

1, 2, 15, 64, 173, 366, 667, 1100, 1689, 2458, 3431, 4632, 6085, 7814, 9843, 12196, 14897, 17970, 21439, 25328, 29661, 34462, 39755, 45564, 51913, 58826, 66327, 74440, 83189, 92598, 102691, 113492, 125025, 137314, 150383, 164256, 178957, 194510, 210939, 228268, 246521, 265722

Offset: 1

Juri-Stepan Gerasimov, Apr 15 2016

This sequence lists all positive integers n such that 2*n - 3 is a cube. Only for first term 2*n - 3 generates a negative cube that is -1. - Altug Alkan, Apr 15 2016

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. positive integers n such that 2*n + k is a cube: this sequence (k=-3), A050492 (k=-1), A268201 (k=1).

Magma
```
[((2*n-1)^3+3)/2: n in [0..40]];
```

Table[((2 n - 1)^3 + 3)/2, {n, 0, 41}] (* or *)
Rest@ CoefficientList[Series[x (1 - 2 x + 13 x^2 + 12 x^3)/(1 - x)^4, {x, 0, 42}], x] (* Michael De Vlieger, Apr 16 2016 *)
LinearRecurrence[{4,-6,4,-1},{1,2,15,64},70] (* Harvey P. Dale, Jun 06 2022 *)

lista(nn) = for(n=0, nn, print1(((2*n-1)^3+3)/2, ", ")); \\ Altug Alkan, Apr 15 2016

a(n+1) = A050492(n)+1.

G.f.: x*(1 - 2*x + 13*x^2 + 12*x^3)/(1 - x)^4. - Ilya Gutkovskiy, Apr 15 2016

cp's OEIS Frontend

A152725 a(n) = n*(n+1)*(n^4 + 2*n^3 - 2*n^2 - 3*n + 3)/2.

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A175113 a(n) = ((2*n + 1)^6 + 1)/2.

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A050533 Thickened pyramidal numbers: a(n) = 2*(n+1)*n + Sum_{i=1..n} (4*i*(i-1) + 1).

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A268201 a(n) = 4*n^3 - 6*n^2 + 3*n - 1.

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A274324 Number of partitions of n^3 into at most two parts.

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Magma

Maple

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A359844 a(n) = ((2*n+1)^8 + 1)/2.

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A271828 a(n) = 4*n^3 - 18*n^2 + 27*n - 12.

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A152725 a(n) = n(n+1)(n^4 + 2n^3 - 2n^2 - 3*n + 3)/2.

A050533 Thickened pyramidal numbers: a(n) = 2(n+1)n + Sum_{i=1..n} (4i(i-1) + 1).

A268201 a(n) = 4n^3 - 6n^2 + 3*n - 1.

A271828 a(n) = 4n^3 - 18n^2 + 27*n - 12.