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A097175 a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)) * 4^k. 3

%I #13 Sep 08 2022 08:45:14

%S 1,5,21,105,361,2045,6141,38865,104401,726245,1774821,13394745,

%T 30171961,244487885,512923341,4424729505,8719696801,79515368885,

%U 148234845621,1420480747785,2519992375561,25247684340125,42839870384541

%N a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)) * 4^k.

%C a(n) = (5/4)*{1, 17, 17, 289, 289, 4913, ...} - 16*{0, 1, 0, 16, 0, 256, ...} - (1/4)*{1, 1, 1, 1, 1, 1, ...}.

%H G. C. Greubel, <a href="/A097175/b097175.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,33,-33,-272,272).

%F G.f.: (1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2)).

%F a(n) = (5/8)*((1-sqrt(17))*(-sqrt(17))^n + (1+sqrt(17))*(sqrt(17))^n) - 2*(4^n - (-4)^n) - 1/4.

%F a(n) = a(n-1) + 33*a(n-2) - 33*a(n-3) - 272*a(n-4) + 272*a(n-5).

%p seq(coeff(series((1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2)), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Sep 17 2019

%t CoefficientList[Series[(1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2)), {x,0,30}], x] (* _G. C. Greubel_, Sep 17 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec((1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2))) \\ _G. C. Greubel_, Sep 17 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2)) )); // _G. C. Greubel_, Sep 17 2019

%o (Sage)

%o def A097175_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P((1+4*x-17*x^2-48*x^3)/((1-x)*(1-16*x^2)*(1-17*x^2))).list()

%o A097175_list(30) # _G. C. Greubel_, Sep 17 2019

%o (GAP) a:=[1, 5, 21, 105, 361];; for n in [6..30] do a[n]:=a[n-1] + 33*a[n-2] - 33*a[n-3] - 272*a[n-4] + 272*a[n-5]; od; a; # _G. C. Greubel_, Sep 17 2019

%Y Cf. A097176, A097177.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 30 2004

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Last modified June 16 10:42 EDT 2024. Contains 373425 sequences. (Running on oeis4.)