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a(n) = Sum_{j=0..n} T(n,j), T given by A026736.
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%I #22 Mar 08 2023 05:33:38

%S 1,2,4,8,17,34,73,146,314,628,1350,2700,5798,11596,24872,49744,106573,

%T 213146,456169,912338,1950697,3901394,8334539,16669078,35582783,

%U 71165566,151809737,303619474,647279131,1294558262,2758310121

%N a(n) = Sum_{j=0..n} T(n,j), T given by A026736.

%H G. C. Greubel, <a href="/A026743/b026743.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: ((1-3*x^2)*sqrt((1+2*x)/(1-2*x)) + (1+2*x)*(1+x^2))/(2*(1 -4*x^2 - x^4)). - _David Callan_, Jan 17 2016

%F Conjecture D-finite with recurrence n*a(n) -2*a(n-1) +(-11*n+20)*a(n-2) +14*a(n-3) +(39*n-152)*a(n-4) -22*a(n-5) +(-41*n+268)*a(n-6) -6*a(n-7) +12*(-n+6)*a(n-8)=0. - _R. J. Mathar_, Jan 13 2023

%F a(n) ~ ((1 + (-1)^n)*phi^(3/2) + 2*(1 - (-1)^n)) * phi^((3*n + 1)/2) / (2*sqrt(5)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Mar 08 2023

%t CoefficientList[Normal[Series[((1-3x^2)Sqrt[(1+2x)/(1-2x)] +(1 + 2x)(1+ x^2))/(2(1-4x^2-x^4)), {x,0,40}]], x] (* _David Callan_, Jan 17 2016 *)

%o (PARI) my(x='x+O('x^40)); Vec(((1-3*x^2)*sqrt((1+2*x)/(1-2*x)) +(1+2*x)*(1+x^2))/(2*(1-4*x^2-x^4))) \\ _G. C. Greubel_, Jul 16 2019

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( ((1 -3*x^2)*Sqrt((1+2*x)/(1-2*x)) +(1+2*x)*(1+x^2))/(2*(1-4*x^2-x^4)) )); // _G. C. Greubel_, Jul 16 2019

%o (Sage) (((1-3*x^2)*sqrt((1+2*x)/(1-2*x)) + (1+2*x)*(1+x^2))/(2*(1-4*x^2 - x^4))).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 16 2019

%Y Cf. A026736.

%K nonn

%O 0,2

%A _Clark Kimberling_