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%I #45 Aug 24 2022 08:50:52
%S 36,100,224,432,748,1196,1800,2584,3572,4788,6256,8000,10044,12412,
%T 15128,18216,21700,25604,29952,34768,40076,45900,52264,59192,66708,
%U 74836,83600,93024,103132,113948,125496,137800,150884,164772
%N a(n) = n^3 + (n+1)^3 + (n+2)^3 + (n+3)^3.
%C Sums of four consecutive cubes. - _Wesley Ivan Hurt_, Dec 16 2015
%H Vincenzo Librandi, <a href="/A027603/b027603.txt">Table of n, a(n) for n = 0..1000</a>
%H Patrick De Geest, <a href="http://www.worldofnumbers.com/sumcube.htm">Palindromic Sums of Cubes of Consecutive Integers</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4) for n>=4.
%F a(n) = 4*n^3 + 18*n^2 + 42*n + 36 = 4*A173965(n+2).
%F From _Bruno Berselli_, Jan 24 2011: (Start)
%F G.f.: 4*(9 - 11*x + 10*x^2 - 2*x^3)/(1-x)^4.
%F a(n) = A027689(n+1) * A016825(n+1). (End)
%F E.g.f.: 2*(18 + 32*x + 15*x^2 + 2*x^3)*exp(x). - _G. C. Greubel_, Aug 24 2022
%p A027603:=n->n^3 + (n+1)^3 + (n+2)^3 + (n+3)^3: seq(A027603(n), n=0..50); # _Wesley Ivan Hurt_, Dec 16 2015
%t Table[n^3 +(n+1)^3 +(n+2)^3 +(n+3)^3, {n, 0, 33}] (* or *)
%t Table[Plus@@(Range[n, n + 3]^3), {n, 0, 33}] (* _Alonso del Arte_, Jan 24 2011 *)
%o (Sage) [n^3+(n+1)^3+(n+2)^3+(n+3)^3 for n in range(0,40)] # _Zerinvary Lajos_, Jul 03 2008
%o (Magma) [4*n^3 + 18*n^2 + 42*n + 36: n in [0..40]]; // _Vincenzo Librandi_, Jun 04 2011
%o (Python)
%o A027603_list, m = [], [24, 12, 28, 36]
%o for _ in range(10**2):
%o A027603_list.append(m[-1])
%o for i in range(3):
%o m[i+1] += m[i] # _Chai Wah Wu_, Dec 15 2015
%o (PARI) Vec(-4*(-9+11*x-10*x^2+2*x^3)/(1-x)^4 + O(x^100)) \\ _Altug Alkan_, Dec 16 2015
%Y Cf. A000578, A005898, A027602, A027604.
%Y Cf. A016825, A027689, A173965.
%K nonn,easy
%O 0,1
%A _Patrick De Geest_
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