A075664 - OEIS

%I #64 Feb 10 2025 10:14:44

%S 0,1,35,405,2584,11375,38961,111475,278720,627669,1300375,2516921,

%T 4604040,8030035,13446629,21738375,34080256,52004105,77474475,

%U 112974589,161603000,227181591,314375545,428825915,577295424,767828125,1009923551,1314725985,1695229480,2166499259

%N Sum of next n cubes.

%H Kelvin Voskuijl, <a href="/A075664/b075664.txt">Table of n, a(n) for n = 0..10000</a> (terms 1..1000 from Vincenzo Librandi)

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).

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

%F a(n) = (n^7 + 4n^5 + 3n^3)/8. - _Charles R Greathouse IV_, Sep 17 2009

%F 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

%F a(n) = n^3*(n^2 + 1)*(n^2 + 3)/8 = A000578(n)*A002522(n)*A117950(n)/8. - _Philippe Deléham_, Mar 09 2014

%F 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

%e a(1) = 1^3 = 1; a(2) = 2^3 + 3^3 = 35; a(3) = 4^3 + 5^3 + 6^3 = 64 + 125 + 216 = 405.

%e From _Philippe Deléham_, Mar 09 2014: (Start)

%e a(1) = 1*2*3/8 = 1;

%e a(2) = 8*5*7/8 = 35;

%e a(3) = 27*10*12/8 = 405;

%e a(4) = 64*17*19/8 = 2584;

%e a(5) = 125*26*28/8 = 11375; etc. (End)

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

%t i1 := n(n-1)/2+1; i2 := n(n-1)/2+n; s=3; Table[Sum[i^s, {i, i1, i2}], {n, 20}]

%t 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 *)

%t 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 *)

%o (Magma) [(n^7+4*n^5+3*n^3)/8: n in [1..30]]; // _Vincenzo Librandi_, Mar 11 2014

%o (PARI) a(n)=(n^7+4*n^5+3*n^3)/8 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Python)

%o def A075664(n): return n*(m:=n**2)*(m*(m+4)+3)>>3 # _Chai Wah Wu_, Feb 09 2025

%Y Cf. A000578 (cubes).

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

%K nonn,easy

%O 0,3

%A _Zak Seidov_, Sep 24 2002

%E Formula from _Charles R Greathouse IV_, Sep 17 2009

%E More terms from _Vincenzo Librandi_, Mar 11 2014

%E a(0) added by _Chai Wah Wu_, Feb 09 2025