A134522 a(n) = 2^n + ceiling(n/2).

1, 3, 5, 10, 18, 35, 67, 132, 260, 517, 1029, 2054, 4102, 8199, 16391, 32776, 65544, 131081, 262153, 524298, 1048586, 2097163, 4194315, 8388620, 16777228, 33554445, 67108877, 134217742, 268435470, 536870927, 1073741839, 2147483664, 4294967312, 8589934609

Offset: 0

Author

Gary W. Adamson, Oct 29 2007

Keywords

• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,2).

Crossrefs

Row sums of A134521.

Cf. A000079, A110654.

Programs

Mathematica

CoefficientList[Series[(1 - 3*x^2 + x^3)/((1 - x)^2*(1 + x)*(1 - 2*x)),{x,0,33}],x] (* James C. McMahon, Apr 07 2025 *)

PARI

a(n)=2^n + (n+1)\2; \\ Andrew Howroyd, Aug 10 2018

PARI

Vec((1 - 3*x^2 + x^3)/((1 - x)^2*(1 + x)*(1 - 2*x)) + O(x^40)) \\ Andrew Howroyd, Aug 10 2018

Formula

From Andrew Howroyd, Aug 10 2018: (Start)

a(n) = A000079(n) + A110654(n).

a(n) = 3*a(n-1) - a(n-2) - 3*a(n-3) + 2*a(n-4).

G.f.: (1 - 3*x^2 + x^3)/((1 - x)^2*(1 + x)*(1 - 2*x)). (End)

a(n) = Sum_{k=1..n+1} C(n+1,k)^(k mod 2). - Wesley Ivan Hurt, Nov 20 2021

E.g.f.: (x*cosh(x) + 2*cosh(2*x) + (1 + x + 4*cosh(x))*sinh(x))/2. - Stefano Spezia, Jun 14 2025

Extensions

Name changed, a(8) inserted and a(14)-a(33) from Andrew Howroyd, Aug 10 2018