A075664 Sum of next n cubes.

0, 1, 35, 405, 2584, 11375, 38961, 111475, 278720, 627669, 1300375, 2516921, 4604040, 8030035, 13446629, 21738375, 34080256, 52004105, 77474475, 112974589, 161603000, 227181591, 314375545, 428825915, 577295424, 767828125, 1009923551, 1314725985, 1695229480, 2166499259

Offset: 0

Author

Zak Seidov, Sep 24 2002

Keywords

• nonn
• easy

Examples

			a(1) = 1^3 = 1; a(2) = 2^3 + 3^3 = 35; a(3) = 4^3 + 5^3 + 6^3 = 64 + 125 + 216 = 405.
From _Philippe Deléham_, Mar 09 2014: (Start)
a(1) = 1*2*3/8 = 1;
a(2) = 8*5*7/8 = 35;
a(3) = 27*10*12/8 = 405;
a(4) = 64*17*19/8 = 2584;
a(5) = 125*26*28/8 = 11375; etc. (End)
		

Links

• Kelvin Voskuijl, Table of n, a(n) for n = 0..10000 (terms 1..1000 from Vincenzo Librandi)
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A000578 (cubes).

Cf. A006003, A072474 (for squares), A075665 - A075671 (4th to 10th powers), A069876 (n-th powers).

Programs

Magma

[(n^7+4*n^5+3*n^3)/8: n in [1..30]]; // Vincenzo Librandi, Mar 11 2014

Maple

A075664:=n->(n^7 + 4n^5 + 3n^3)/8; seq(A075664(n), n=1..30); # Wesley Ivan Hurt, Mar 10 2014

Mathematica

i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=3; Table[Sum[i^s, {i, i1, i2}], {n, 20}]
CoefficientList[Series[(1 + 27 x + 153 x^2 + 268 x^3 + 153 x^4 + 27 x^5 + x^6)/(1 - x)^8, {x, 0, 40}], x](* Vincenzo Librandi, Mar 11 2014 *)
With[{nn=30},Total/@TakeList[Range[(nn(nn+1))/2]^3,Range[nn]]] (* Requires Mathematica version 11 or later *) (* or *) LinearRecurrence[{8,-28,56,-70,56,-28,8,-1},{1,35,405,2584,11375,38961,111475,278720},30] (* Harvey P. Dale, Jun 05 2021 *)

PARI

a(n)=(n^7+4*n^5+3*n^3)/8 \\ Charles R Greathouse IV, Oct 07 2015

Python

def A075664(n): return n*(m:=n**2)*(m*(m+4)+3)>>3 # Chai Wah Wu, Feb 09 2025

Formula

a(n) = Sum_{i=n(n-1)/2+1..n(n-1)/2+n} i^3. [Corrected by Stefano Spezia, Jun 22 2024]

a(n) = (n^7 + 4n^5 + 3n^3)/8. - Charles R Greathouse IV, Sep 17 2009

G.f.: x*(1+27*x+153*x^2+268*x^3+153*x^4+27*x^5+x^6)/(1-x)^8. - Colin Barker, May 25 2012

a(n) = n^3*(n^2 + 1)*(n^2 + 3)/8 = A000578(n)*A002522(n)*A117950(n)/8. - Philippe Deléham, Mar 09 2014

E.g.f.: exp(x)*x*(8 + 132*x + 404*x^2 + 390*x^3 + 144*x^4 + 21*x^5 + x^6)/8. - Stefano Spezia, Jun 22 2024

Extensions

Formula from Charles R Greathouse IV, Sep 17 2009

More terms from Vincenzo Librandi, Mar 11 2014

a(0) added by Chai Wah Wu, Feb 09 2025