A323227 a(n) = [x^n] (1 - 2*x + x^2 - 2*x^3 + x^4)/((1 - x)^2*(1 - 2*x)).

1, 2, 4, 6, 9, 14, 23, 40, 73, 138, 267, 524, 1037, 2062, 4111, 8208, 16401, 32786, 65555, 131092, 262165, 524310, 1048599, 2097176, 4194329, 8388634, 16777243, 33554460, 67108893, 134217758, 268435487, 536870944, 1073741857, 2147483682, 4294967331, 8589934628

Offset: 0

Author

Peter Luschny, Feb 12 2019

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Crossrefs

Cf. A000051, A094373, A270841, A323211.

Programs

Magma

[n le 1 select n+1 else 2^(n-2) +(n+1) : n in [0..35]]; // G. C. Greubel, Sep 26 2024

Maple

a := proc(n) option remember; if n < 4 then return [1, 2, 4, 6][n + 1] fi;
((2 - 2*n)*a(n-2) - (5 - 3*n)*a(n-1))/(n - 2) end: seq(a(n), n=0..35);

Mathematica

A323211[n_, k_] := If[n <= 1, 1, Binomial[n - 2, k - 1] + 1];
Table[Sum[A323211[n, k], {k, 0, n}], {n, 0, 35}]

SageMath

[2^(n-2) +(n+1) -int(n==0)/4 -int(n==1)/2 for n in range(36)] # G. C. Greubel, Sep 26 2024

Formula

a(n) = Sum_{k=0..n} ( binomial(n - 2, k - 1) + 1 ), if n >= 2.

a(n) = ((2 - 2*n)*a(n-2) - (5 - 3*n)*a(n-1))/(n - 2) for n >= 4.

a(n+1) - (n + 1) = A094373(n) for n >= 0.

a(n+1) - a(n) = 2^n + 1 for n >= 2.

a(n) = A270841(n) = 2^(n-2)+n+1 for n>=2. - R. J. Mathar, Feb 14 2019

E.g.f.: (1/4)*(-(1 + 2*x) + 4*(1+x)*exp(x) + exp(2*x)). - G. C. Greubel, Sep 26 2024