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Expansion of (1 + 3x + 5x^2 + 7x^3 + ...) / (1 - 2x + 3x^2 - 4x^3 + ...).
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%I #32 Mar 12 2023 23:00:54

%S 1,5,12,20,28,36,44,52,60,68,76,84,92,100,108,116,124,132,140,148,156,

%T 164,172,180,188,196,204,212,220,228,236,244,252,260,268,276,284,292,

%U 300,308,316,324,332,340,348,356,364,372,380,388,396,404,412,420,428

%N Expansion of (1 + 3x + 5x^2 + 7x^3 + ...) / (1 - 2x + 3x^2 - 4x^3 + ...).

%C Row sums of number triangle A113128. - _Paul Barry_, Oct 14 2005

%C The Engel expansion of 1 + exp(1/8)*sqrt(2*Pi)*erf(1/(2*sqrt(2)))/5 = 1.2175306077808... - _Benedict W. J. Irwin_, Dec 16 2016

%H Leo Tavares, <a href="/A086570/a086570.jpg">Square illustration</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).

%F a(0) = 1, a(1) = 5, a(2) = 12; then a(n+1) = a(n) + 8, n > 2.

%F From _Paul Barry_, Oct 14 2005: (Start)

%F G.f.: (1+x)^3/(1-x)^2;

%F a(n) = 8n - 4 + 4*C(0, n) + C(1, n);

%F a(n) = C(n+1, n) + 3*C(n, n-1) + 3*C(n-1, n-2) + C(n-2, n-3). (End)

%F a(n) = A017113(n-1), n > 1. - _R. J. Mathar_, Sep 12 2008

%e a(6) = 44 = 8 + a(5) = 8 + 36.

%t CoefficientList[Series[(z^3 + 3*z^2 + 3*z + 1)/(z - 1)^2, {z, 0, 100}], z] (* and *) Join[{1, 5}, Table[4*(2*(n + 1) + 1), {n, 0, 100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jul 17 2011 *)

%o (PARI) a(n)=if(n>1, 8*n-4, 4*n+1) \\ _Charles R Greathouse IV_, Dec 16 2016

%Y Cf. A078370, A016754.

%K nonn,easy

%O 0,2

%A _Gary W. Adamson_, Jul 22 2003