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%I #36 Sep 08 2022 08:45:47
%S 0,45,35,285,30,725,255,1365,110,2205,675,3245,240,4485,1295,5925,420,
%T 7565,2115,9405,650,11445,3135,13685,930,16125,4355,18765,1260,21605,
%U 5775,24645,1640,27885,7395,31325,2070,34965,9215,38805,2550,42845,11235,47085
%N Quintisection A061037(5*n+2).
%C A trisection of A061037 is in A142590. These (2k+1)-sections A061037(2+n*(2k+1)) are multiples of 2k+1.
%H G. C. Greubel, <a href="/A165248/b165248.txt">Table of n, a(n) for n = 0..5000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,3,0,0,0,-3,0,0,0,1).
%F Conjecture: a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12), n>11. - _R. J. Mathar_, Mar 02 2010
%F The conjecture is equivalent to a(4n) = 5n*(5n+1), a(4n+1) = 5*(20n+9)*(4n+1), a(4n+2) = 5*(10n+7)*(2n+1) and a(4n+3) = 5*(20n+19)*(4n+3). - _R. J. Mathar_, Feb 13 2011
%F The conjectures can be proved by taking the closed form of A061037, and writing up the quadrisections case by case. - _Bruno Berselli_, Feb 20 2011
%F From _Ilya Gutkovskiy_, Apr 19 2016: (Start)
%F G.f.: 5*x*(9 + 7*x + 57*x^2 + 6*x^3 + 118*x^4 + 30*x^5 + 102*x^6 + 4*x^7 + 33*x^8 + 3*x^9 + x^10)/((1 - x)^3*(1 + x)^3*(1 + x^2)^3).
%F a(n) = -5*n (5*n + 4)*(27*(-1)^n + 6*cos((Pi*n)/2) - 37)/64. (End)
%t CoefficientList[Series[5*x*(9 + 7*x + 57*x^2 + 6*x^3 + 118*x^4 + 30*x^5 + 102*x^6 +4*x^7 + 33*x^8 + 3*x^9 +x^10)/((1 - x)^3*(1 + x)^3*(1 + x^2)^3), {x, 0, 50}], x] (* _G. C. Greubel_, Sep 19 2018 *)
%t LinearRecurrence[{0,0,0,3,0,0,0,-3,0,0,0,1},{0,45,35,285,30,725,255,1365,110,2205,675,3245},50] (* _Harvey P. Dale_, Sep 18 2021 *)
%o (PARI) a(n) = numerator(1/4 - 1/(5*n+2)^2); \\ _Altug Alkan_, Apr 19 2016
%o (PARI) x='x+O('x^50); concat([0], Vec(5*x*(9 + 7*x + 57*x^2 + 6*x^3 + 118*x^4 + 30*x^5 + 102*x^6 + 4*x^7 + 33*x^8 + 3*x^9 + x^10)/((1 - x)^3*(1 + x)^3*(1 + x^2)^3))) \\ _G. C. Greubel_, Sep 19 2018
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(5*x*(9 + 7*x + 57*x^2 + 6*x^3 + 118*x^4 + 30*x^5 + 102*x^6 + 4*x^7 + 33*x^8 + 3*x^9 + x^10)/((1 - x)^3*(1 + x)^3*(1 + x^2)^3))); // _G. C. Greubel_, Sep 19 2018
%K nonn,easy,less
%O 0,2
%A _Paul Curtz_, Sep 10 2009
%E Extended by _R. J. Mathar_, Mar 02 2010
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