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A075664
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Sum of next n cubes.
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9
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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
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OFFSET
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1,2
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).
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FORMULA
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a(1)=1; a(n)=sum(i^s, {i, n(n-1)/2+1, n(n-1)/2+1+n}).
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
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EXAMPLE
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s=3; a(1) = 1^s = 1; a(2) = 2^s + 3^s = 2^3+3^3 = 35; a(3) = 4^s + 5^s + 6^s = 64 + 125 + 216 = 405.
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. - Philippe Deléham, Mar 09 2014
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MAPLE
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A075664:=n->(n^7 + 4n^5 + 3n^3)/8; seq(A075664(n), n=1..30); # Wesley Ivan Hurt, Mar 10 2014
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MATHEMATICA
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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 *)
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PROG
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(Magma) [(n^7+4*n^5+3*n^3)/8: n in [1..30]]; // Vincenzo Librandi, Mar 11 2014
(PARI) a(n)=(n^7+4*n^5+3*n^3)/8 \\ Charles R Greathouse IV, Oct 07 2015
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CROSSREFS
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Cf. A006003 (s=1), A072474 (s=2), A075664-A075670 (s=3-10), A075671 (s=n).
Sequence in context: A225697 A133458 A238539 * A133317 A322879 A105947
Adjacent sequences: A075661 A075662 A075663 * A075665 A075666 A075667
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KEYWORD
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nonn,easy
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AUTHOR
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Zak Seidov, Sep 24 2002
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EXTENSIONS
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Formula from Charles R Greathouse IV, Sep 17 2009
More terms from Vincenzo Librandi, Mar 11 2014
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STATUS
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approved
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