A000537 -id:A000537 - cp's OEIS frontend

A097988 a(n) = Sum_{d dividing n} tau(d)^3 = (Sum_{d dividing n} tau(d))^2.

1, 9, 9, 36, 9, 81, 9, 100, 36, 81, 9, 324, 9, 81, 81, 225, 9, 324, 9, 324, 81, 81, 9, 900, 36, 81, 100, 324, 9, 729, 9, 441, 81, 81, 81, 1296, 9, 81, 81, 900, 9, 729, 9, 324, 324, 81, 9, 2025, 36, 324, 81, 324, 9, 900, 81, 900, 81, 81, 9, 2916, 9, 81, 324

Offset: 1

Lekraj Beedassy, Sep 07 2004

When n = p^e is a prime power, we have the corollary a(n) = Sum_{r=1..e+1} r^3 = (Sum_{r=1..e+1} r)^2, i.e. A000537(n) = (A000217(n))^2.

3^A001221(n) always divides a(n) except if n > 1 and included in A000578. - Enrique Pérez Herrero, Jul 12 2010

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 47.
Jean-Marie De Koninck and Armel Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 562, pp. 75, 265; Ellipses Paris 2004.
William J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 85, Problem 2.
William J. LeVeque, Fundamentals of Number Theory, Dover Publications Inc, 1977, p. 125.
Joe Roberts, The Lure of Integers, MAA, 1992, Integer 3, pages 8-9.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 84.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Enrique Pérez Herrero)

Cf. A000005, A000217, A000537, A007425.

with(numtheory); f:=proc(n) local t1; t1:=divisors(n); add(sigma[0](i), i in t1)^2; end;

tau[1,n_Integer] := 1; SetAttributes[tau, Listable]; tau[k_Integer,n_Integer] := Plus@@(tau[k-1,Divisors[n]]); A097988[n_] := tau[3,n]^2; Table[A097988[n], {n, 100}] (* Enrique Pérez Herrero, Jul 12 2010 *)
f[n_]:=Total[DivisorSigma[0,Divisors[n]]]^2;f/@Range[100] (* Ivan N. Ianakiev, Mar 05 2015 *)
a[n_] := DivisorSum[n, DivisorSigma[0, #]&]^2; Array[a, 100] (* Jean-François Alcover, Dec 02 2015 *)
f[p_, e_] := ((e+1)*(e+2)/2)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 20 2020 *)

a(n)=sumdiv(n,d,numdiv(d))^2 \\ Charles R Greathouse IV, Jan 22 2013

a(n)=sumdiv(n, d, numdiv(d)^3); \\ Michel Marcus, Nov 21 2013

a(n) = (Sum_{d dividing n} tau(d))^2 = (A007425(n))^2.
Multiplicative with a(p^e) = ((e+1)*(e+2)/2)^2. - Amiram Eldar, Sep 20 2020
Dirichlet g.f.: zeta(s)^5 * Product_{p prime} (1 + 4/p^s + 1/p^(2*s)). - Amiram Eldar, Sep 14 2023

More terms from Carl Najafi, Oct 19 2011

Entry revised by N. J. A. Sloane, May 22 2012

A101101 a(1)=1, a(2)=5, and a(n)=6 for n >= 3.

Original entry on oeis.org

1, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 15 2004

Previous name was: The first summation of row 3 of Euler's triangle - a row that will recursively accumulate to the power of 3.

Decimal expansion of 47/30. - Elmo R. Oliveira, Aug 09 2024

D. J. Pengelley, The bridge between the continuous and the discrete via original sources in Study the Masters: The Abel-Fauvel Conference [pdf], Kristiansand, 2002, (ed. Otto Bekken et al.), National Center for Mathematics Education, University of Gothenburg, Sweden, in press.
C. Rossiter, Depictions, Explorations and Formulas of the Euler/Pascal Cube. [broken link: domain now owned by a domain grabber]
Eric Weisstein, Link to section of MathWorld: Worpitzky's Identity of 1883.
Eric Weisstein, Link to section of MathWorld: Eulerian Number.
Eric Weisstein, Link to section of MathWorld: Nexus number.
Eric Weisstein, Link to section of MathWorld: Finite Differences.
Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
Index entries for linear recurrences with constant coefficients, signature (1).

Within the "cube" of related sequences with construction based upon MaginNKZ formula, with n downward, k rightward and z backward:
Before: this_sequence, A008458, A003215, A000578, A000537, A024166 or A024166, A101094, A101097, A101102.
Above: this_sequence, below: A101104, A101100.
Within the "cube" of related sequences with construction based upon SeriesAtLevelR formula, with n downward, x rightward and r backward:
Above: this_sequence, below: A101103, A101096.

MagicNKZ=Sum[(-1)^j*Binomial[n+1-z, j]*(k-j+1)^n, {j, 0, k+1}];Table[MagicNKZ, {n, 3, 3}, {z, 1, 1}, {k, 0, 34}] (* OR *)
SeriesAtLevelR = Sum[Eulerian[n, i - 1]*Binomial[n + x - i + r, n + r], {i, 1, n}]; Table[SeriesAtLevelR, {n, 3, 3}, {r, -3, -3}, {x, 4, 35}]
Join[{1, 5},LinearRecurrence[{1},{6},78]] (* Ray Chandler, Sep 23 2015 *)

G.f.: x*(1+4*x+x^2)/(1-x). - L. Edson Jeffery, Jan 29 2012

I wish the sequence was as interesting as the list of references! - N. J. A. Sloane

New name from Joerg Arndt, Nov 30 2014

A108674 a(n) = (n+1)^2 * (n+2)^2 * (2*n+3) / 12.

Original entry on oeis.org

1, 15, 84, 300, 825, 1911, 3920, 7344, 12825, 21175, 33396, 50700, 74529, 106575, 148800, 203456, 273105, 360639, 469300, 602700, 764841, 960135, 1193424, 1470000, 1795625, 2176551, 2619540, 3131884, 3721425, 4396575, 5166336, 6040320, 7028769, 8142575

Offset: 0

Emeric Deutsch, Jun 17 2005

Kekulé numbers for certain benzenoids.
This is the case P(3,n) of the family of sequences defined in A132458. - Ottavio D'Antona (dantona(AT)dico.unimi.it), Oct 31 2007
Using the triangular numbers 0, 1, 3, ..., create a sequence of advancing sums of k-tuples with k=n*(n+1)/2 of the odd numbers: 0, 1, 15, 84, 300, 825, 1911, 3920, ... . This begins 0, then 1, then 3+5+7=15, then 9+11+13+15+17+19=84, then 21+23+...+39=300 and so on. - J. M. Bergot, Dec 08 2014
Partial sums of A008354. - J. M. Bergot, Dec 19 2014
Coefficients in the terminating series identity 1 - 15*n/(n + 4) + 84*n*(n - 1)/((n + 4)*(n + 5)) - 300*n*(n - 1)*(n - 2)/((n + 4)*(n + 5)*(n + 6)) + ... = 0 for n = 2,3,4,.... Cf. A000330. - Peter Bala, Feb 12 2019

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (p. 231, # 33).

Muniru A Asiru, 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. A000537, A126274.

Cf. A000217, A000330, A008354 (first diffs.), A132458.

List([0..40],k->(k+1)^2*(k+2)^2*(2*k+3)/12); # Muniru A Asiru, Feb 18 2019

A108674 :=n->(n+1)^2*(n+2)^2*(2*n+3)/12;
seq(A108674(n),n=0..35);

Table[(n^2+4*n^3+5*n^4+2*n^5)/12, {n, 40}] (* Enrique Pérez Herrero, Feb 27 2013 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{1,15,84,300,825,1911},40] (* Harvey P. Dale, Apr 04 2024 *)

a(n)=(n+1)^2*(n+2)^2*(2*n+3)/12 \\ Charles R Greathouse IV, Feb 27 2013

G.f.: (1+z)*(1+8*z+z^2)/(1-z)^6.
a(n) = Sum_{j=1..n+1} j^2 Sum_{i=1..n+1} i. - Alexander Adamchuk, Jun 25 2006
a(n) = A000330(n+1) * A000217(n+1). - Daniel Suteu, Nov 26 2020
E.g.f.: exp(x)*(12 + 168*x + 330*x^2 + 184*x^3 + 35*x^4 + 2*x^5)/12. - Stefano Spezia, Mar 02 2022
From Amiram Eldar, May 29 2022: (Start)
Sum_{n>=0} 1/a(n) 192*log(2) - 132.
Sum_{n>=0} (-1)^n/a(n) = 2*Pi^2 - 48*Pi + 132. (End)

A153976 a(n) = n^3 + (n+2)^3.

Original entry on oeis.org

8, 28, 72, 152, 280, 468, 728, 1072, 1512, 2060, 2728, 3528, 4472, 5572, 6840, 8288, 9928, 11772, 13832, 16120, 18648, 21428, 24472, 27792, 31400, 35308, 39528, 44072, 48952, 54180, 59768, 65728, 72072, 78812, 85960, 93528, 101528, 109972, 118872, 128240

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..765
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000537, A000578, A003215, A005898, A006007, A027602.

[n^3+(n+2)^3: n in [0..60]]; // Vincenzo Librandi, Apr 26 2011

f[n_]:=n^3;lst={};Do[AppendTo[lst,(f[n]+f[n+2])],{n,0,6!}];lst
Array[#^3+(#+2)^3&,40,0] (* or *) LinearRecurrence[{4,-6,4,-1},{8,28,72,152},40] (* Harvey P. Dale, Aug 02 2011 *)

def a(n): return n**3 + (n+2)**3
print([a(n) for n in range(40)]) # Michael S. Branicky, Aug 28 2021

For n>3, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Aug 02 2011
G.f.: 4*( 2-x+2*x^2 ) / (x-1)^4 . - R. J. Mathar, Apr 11 2016
a(n) = 4*A229183(n+1). - R. J. Mathar, Apr 11 2016

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 26 2011

A153978 a(n) = n(n-1)(n+1)(3n-2)/12.

Original entry on oeis.org

0, 2, 14, 50, 130, 280, 532, 924, 1500, 2310, 3410, 4862, 6734, 9100, 12040, 15640, 19992, 25194, 31350, 38570, 46970, 56672, 67804, 80500, 94900, 111150, 129402, 149814, 172550, 197780, 225680, 256432, 290224, 327250, 367710, 411810, 459762

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Partial sums of A011379.

Antidiagonal sums of the convolution array A213819. - Clark Kimberling, Jul 04 2012

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

Cf. A003215, A000537, A000578, A005898, A027602, A006007, A153976, A153977, A011379, A052149, A213819.

With[{r=Range[0,50]},Accumulate[r^2+r^3]] (* Harvey P. Dale, Jan 16 2011 *)
Rest[CoefficientList[Series[-2 x^2 * (2 x + 1)/(x - 1)^5, {x, 0, 40}], x]] (* Vincenzo Librandi, Jun 30 2014 *)
LinearRecurrence[{5,-10,10,-5,1}, {0,2,14,50,130}, 25] (* G. C. Greubel, Sep 01 2016 *)

concat(0, Vec(-2*x^2*(2*x+1)/(x-1)^5 + O(x^100))) \\ Colin Barker, Jun 28 2014

a(n) = n*(n-1)*(n+1)*(3*n-2)/12 \\ Charles R Greathouse IV, Sep 01 2016

a(n) = 2 * A001296(n-1) = (n-1)*n*(n+1)*(3*n-2)/12 (n>0). - Bruno Berselli, Apr 21 2010
a(n) = Sum_{i=1..n-1} binomial(i+1,i)*i^2. - Enrique Pérez Herrero, Jun 28 2014
G.f.: 2*x^2*(2*x+1) / (1 - x)^5. - Colin Barker, Jun 28 2014
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4. - Vincenzo Librandi, Jun 30 2014
a(n) = Sum_{k=1..n-1}k*((n-1)*n/2 + k) for n > 1. - J. M. Bergot, Feb 16 2018
From Amiram Eldar, Aug 23 2022: (Start)
Sum_{n>=2} 1/a(n) = 141/5 - 9*sqrt(3)*Pi/5 - 81*log(3)/5.
Sum_{n>=2} (-1)^n/a(n) = 18*sqrt(3)*Pi/5 + 48*log(2)/5 - 129/5. (End)

Edited by Bruno Berselli, Jun 15 2010

Simpler definition as suggested by Wesley Ivan Hurt, Jun 29 2014

A254371 Sum of cubes of the first n even numbers (A016743).

Original entry on oeis.org

0, 8, 72, 288, 800, 1800, 3528, 6272, 10368, 16200, 24200, 34848, 48672, 66248, 88200, 115200, 147968, 187272, 233928, 288800, 352800, 426888, 512072, 609408, 720000, 845000, 985608, 1143072, 1318688, 1513800, 1729800, 1968128, 2230272, 2517768, 2832200, 3175200

Offset: 0

Luciano Ancora, Mar 16 2015

Property: for n >= 2, each (a(n), a(n)+1, a(n)+2) is a triple of consecutive terms that are the sum of two nonzero squares; precisely: a(n) = (n*(n + 1))^2 + (n*(n + 1))^2, a(n)+1 = (n^2+2n)^2 + (n^2-1)^2 and a(n)+2 = (n^2+n+1)^2 + (n^2+n-1)^2 (see Diophante link). - Bernard Schott, Oct 05 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Luciano Ancora, The Square Pyramidal Number and other figurate numbers, ch. 3.
Diophante, Une miniature avec trois entiers consécutifs (in French).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000537 (sum of first n cubes); A002593 (sum of first n odd cubes).
Cf. A060300 (2*a(n)).
First bisection of A105636; second bisection of A212892.
Cf. A000217, A000415, A002378, A035287, A163102.

List([0..35],n->2*(n*(n+1))^2); # Muniru A Asiru, Oct 24 2018

[2*n^2*(n+1)^2: n in [0..40]]; // Bruno Berselli, Mar 23 2015

A254371:=n->2*n^2*(n + 1)^2: seq(A254371(n), n=0..50); # Wesley Ivan Hurt, Apr 28 2017

Table[2 n^2 (n+1)^2, {n, 0, 40}] (* or *) LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 72, 288, 800}, 40]
Accumulate[Range[0,80,2]^3] (* Harvey P. Dale, Jun 26 2017 *)

a(n)=sum(i=0, n, 8*i^3); \\ Michael B. Porter, Mar 16 2015

G.f.: 8*x*(1 + 4*x + x^2)/(1 - x)^5.
a(n) = 2*n^2*(n + 1)^2.
a(n) = 2*A035287(n+1) = 2*A002378(n)^2 = 8*A000217(n)^2. - Bruce J. Nicholson, Apr 23 2017
a(n) = 8*A000537(n). - Michel Marcus, Apr 23 2017
From Amiram Eldar, Aug 25 2022: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/6 - 3/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3/2 - 2*log(2). (End)
From Elmo R. Oliveira, Aug 14 2025: (Start)
E.g.f.: 2*x*(2 + x)*(2 + 6*x + x^2)*exp(x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 4*A163102(n) = A060300(n)/2. (End)

A254869 Seventh partial sums of cubes (A000578).

Original entry on oeis.org

1, 15, 111, 561, 2211, 7293, 21021, 54483, 129558, 286858, 598026, 1184118, 2242266, 4083366, 7184166, 12257850, 20348031, 32951985, 52179985, 80958735, 123288165, 184562235, 271965915, 394962165, 565884540, 800652996, 1119632580, 1548656956

Offset: 1

Luciano Ancora, Feb 17 2015

			2nd differences:   0,  6,  12,  18,   24,   30, ... (A008588)
1st differences:   1,  7,  19,  37,   61,   91, ... (A003215)
-------------------------------------------------------------------
The cubes:         1,  8,  27,  64,  125,  216, ... (A000578)
-------------------------------------------------------------------
1st partial sums:  1,  9,  36, 100,  225,  441, ... (A000537)
2nd partial sums:  1, 10,  46, 146,  371,  812, ... (A024166)
3rd partial sums:  1, 11,  57, 203,  574, 1386, ... (A101094)
4th partial sums:  1, 12,  69, 272,  846, 2232, ... (A101097)
5th partial sums:  1, 13,  82, 354, 1200, 3432, ... (A101102)
6th partial sums:  1, 14,  96, 450, 1650, 5082, ... (A254469)
7th partial sums:  1, 15, 111, 561, 2211, 7293, ... (this sequence)

Luciano Ancora, Table of n, a(n) for n = 1..1000
Luciano Ancora, Partial sums of m-th powers with Faulhaber polynomials.
Luciano Ancora, Pascal's triangle and recurrence relations for partial sums of m-th powers.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 15.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A000537, A000578, A003215, A024166, A101094, A101097, A101102, A254469, A254870, A254871, A254872.

[n*(1+n)*(2+n)*(3+n)*(4+n)*(5+n)*(6+n)*(7+n)*(7+7*n+n^2)/604800: n in [1..30]]; // Vincenzo Librandi, Feb 19 2015

Table[n (1 + n) (2 + n) (3 + n) (4 + n) (5 + n) (6 + n) (7 + n) (7 + 7 n + n^2)/604800, {n, 26}] (* or *)
CoefficientList[Series[(- 1 - 4 x - x^2)/(- 1 + x)^11, {x, 0, 25}], x]
Nest[Accumulate,Range[30]^3,7] (* or *) LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,15,111,561,2211,7293,21021,54483,129558,286858,598026},30] (* Harvey P. Dale, Apr 24 2017 *)

vector(50, n, n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 7*n + n^2)/604800) \\ Derek Orr, Feb 19 2015

G.f.: x*(1 + 4*x + x^2)/(1 - x)^11.
a(n) = n*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n)*(7 + n)*(7 + 7*n + n^2)/604800.
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) + n^3.
Sum_{n>=1} 1/a(n) = 1920*sqrt(3/7)*Pi*tan(sqrt(21)*Pi/2) - 251488/49. - Amiram Eldar, Jan 26 2022

A346642 a(n) = Sum_{j=1..n} Sum_{i=1..j} j^3*i^3.

Original entry on oeis.org

0, 1, 73, 1045, 7445, 35570, 130826, 399738, 1063290, 2539515, 5564515, 11362351, 21875503, 40068860, 70321460, 118921460, 194681076, 309689493, 480223005, 727832905, 1080632905, 1574809126, 2256376958, 3183210350, 4427370350, 6077760975, 8243141751

Offset: 0

Roudy El Haddad, Jan 24 2022

a(n) is the sum of all products of two cubes of positive integers up to n, i.e., the sum of all products of two elements from the set of cubes {1^3, ..., n^3}.

			For n=3,
a(3) = (1*1)^3+(2*1)^3+(2*2)^3+(3*1)^3+(3*2)^3+(3*3)^3 = 1045,
a(3) = 1^3*(1^3)+2^3*(1^3+2^3)+3^3*(1^3+2^3+3^3) = 1045.

Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000537 (sum of first n cubes), A347107 (for distinct cubes).

Cf. A001296 (for power 1), A060493 (for squares).

CoefficientList[Series[-(8 x^5 + 179 x^4 + 584 x^3 + 424 x^2 + 64 x + 1) x/(x - 1)^9, {x, 0, 26}], x] (* Michael De Vlieger, Feb 04 2022 *)

{a(n) = n*(n+1)*(n+2)*(21n^5+69n^4+45n^3-21n^2-6n+4)/672};

a(n) = sum(i=1, n, sum(j=1, i, i^3*j^3)); \\ Michel Marcus, Jan 27 2022

def A346642(n): return n*(n**2*(n*(n*(n*(n*(21*n + 132) + 294) + 252) + 21) - 56) + 8)//672 # Chai Wah Wu, Feb 17 2022

a(n) = n*(n+1)*(n+2)*(21*n^5+69*n^4+45*n^3-21*n^2-6*n+4)/672 (from the recurrent form of Faulhaber's formula).

G.f.: -(8*x^5+179*x^4+584*x^3+424*x^2+64*x+1)*x/(x-1)^9. - Alois P. Heinz, Jan 27 2022

A347107 a(n) = Sum_{1 <= i < j <= n} j^3*i^3.

Original entry on oeis.org

0, 0, 8, 251, 2555, 15055, 63655, 214918, 616326, 1561110, 3586110, 7612385, 15139553, 28506101, 51229165, 88438540, 147420940, 238291788, 374813076, 575377095, 864177095, 1272587195, 1840775123, 2619572626, 3672629650, 5078879650, 6935344650, 9360309933

Offset: 0

Roudy El Haddad, Jan 27 2022

a(n) is the sum of all products of two distinct cubes of positive integers up to n, i.e., the sum of all products of two distinct elements from the set of cubes {1^3, ..., n^3}.

			For n=3, a(3) = (2*1)^3+(3*1)^3+(3*2)^3 = 251.

Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Cf. A000537, A000578.
Cf. A346642 (for nondistinct cubes).
Cf. A000217 (for power 0), A000914 (for power 1), A000596 (for squares).

CoefficientList[Series[-(x^5 + 64 x^4 + 424 x^3 + 584 x^2 + 179 x + 8) x^2/(x - 1)^9, {x, 0, 27}], x] (* Michael De Vlieger, Feb 04 2022 *)
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,0,8,251,2555,15055,63655,214918,616326},30] (* Harvey P. Dale, Jul 07 2025 *)

a(n) = sum(i=2, n, sum(j=1, i-1, i^3*j^3));

{a(n) = n*(n+1)*(n-1)*(21*n^5+36*n^4-21*n^3-48*n^2+8)/672};

def A347107(n): return n*(n**2*(n*(n*(n*(n*(21*n + 36) - 42) - 84) + 21) + 56) - 8)//672 # Chai Wah Wu, Feb 17 2022

a(n) = Sum_{j=2..n} Sum_{i=1..j-1} j^3*i^3.
a(n) = n*(n+1)*(n-1)*(21*n^5+36*n^4-21*n^3-48*n^2+8)/672 (from the generalized form of Faulhaber's formula).
From Alois P. Heinz, Jan 27 2022: (Start)
a(n) = Sum_{i=1..n} A000578(i)*A000537(i-1) = Sum_{i=1..n} i^3*(i*(i-1)/2)^2.
G.f.: -(x^5+64*x^4+424*x^3+584*x^2+179*x+8)*x^2/(x-1)^9. (End)

A078618 a(n) = floor(average of first n cubes).

Original entry on oeis.org

1, 4, 12, 25, 45, 73, 112, 162, 225, 302, 396, 507, 637, 787, 960, 1156, 1377, 1624, 1900, 2205, 2541, 2909, 3312, 3750, 4225, 4738, 5292, 5887, 6525, 7207, 7936, 8712, 9537, 10412, 11340, 12321, 13357, 14449, 15600, 16810, 18081, 19414, 20812, 22275

Offset: 1

Joseph L. Pe, Dec 10 2002

			a(3) = floor((1 + 8 + 27)/3) = 12.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1,1,-3,3,-1).

Cf. A000537.

ZL:=n->sum(i^3, i=1..n): a:=n->floor(numer(ZL(n))/n): seq(a(n), n=1..44); # Zerinvary Lajos, Mar 28 2007

s = 0; a = {}; For[i = 1, i <= 100, i++, s = s + i^3; a = Append[a, Floor[(1/i) s]]]; a

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

a(n) = floor((1/n)*(Sum_{i=1..n} i^3)) = floor(n*(n+1)^2/4).

G.f.: x*(1+x+x^4+3*x^2) / ( (1+x)*(x^2+1)*(x-1)^4 ). - R. J. Mathar, Feb 20 2011

cp's OEIS Frontend

A097988 a(n) = Sum_{d dividing n} tau(d)^3 = (Sum_{d dividing n} tau(d))^2.

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A101101 a(1)=1, a(2)=5, and a(n)=6 for n >= 3.

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A108674 a(n) = (n+1)^2 * (n+2)^2 * (2*n+3) / 12.

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A153976 a(n) = n^3 + (n+2)^3.

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A153978 a(n) = n*(n-1)*(n+1)*(3*n-2)/12.

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A254371 Sum of cubes of the first n even numbers (A016743).

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A153978 a(n) = n(n-1)(n+1)(3n-2)/12.