A026743 a(n) = Sum_{j=0..n} T(n,j), T given by A026736.

1, 2, 4, 8, 17, 34, 73, 146, 314, 628, 1350, 2700, 5798, 11596, 24872, 49744, 106573, 213146, 456169, 912338, 1950697, 3901394, 8334539, 16669078, 35582783, 71165566, 151809737, 303619474, 647279131, 1294558262, 2758310121

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A026736.

Programs

Magma

R:=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

Mathematica

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 *)

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

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

Formula

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

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

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