A135836 Column three of the triangular matrix in A135835.

3, 22, 82, 254, 677, 1692, 3972, 9052, 19975, 43394, 92534, 195546, 408489, 848584, 1749544, 3594104, 7345547, 14976366, 30424986, 61706038, 124829101, 252226676, 508704716, 1025115156, 2062984719, 4149086938, 8336437438, 16742227730, 33599246513, 67406551968

Offset: 1

Author

John W. Layman, Nov 30 2007

Keywords

• nonn
• easy

Comments

Column two of the associated matrix is A005803.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-10,15,-6).

Crossrefs

Cf. A005803, A135835.

Programs

Magma

[(1/12)*(330 +78*n +3*2^(n+8) -(1-(-1)^n)*106*3^((n+3)/2) -(1+(-1)^n)*61*3^(2 +n/2)): n in [1..40]]; // G. C. Greubel, Feb 07 2022

Mathematica

LinearRecurrence[{4,-2,-10,15,-6}, {3,22,82,254,677}, 40] (* G. C. Greubel, Feb 07 2022 *)

SageMath

def a(n):
    if (n%2==0): return (1/2)*(55 + 13*n + 2^(n+7) -61*3^(n/2+1))
    else: return (1/2)*(55 + 13*n + 2^(n+7) - 106*3^((n+1)/2))
[a(n) for n in (1..40)] # G. C. Greubel, Feb 07 2022

Formula

From G. C. Greubel, Feb 07 2022: (Start)

a(n) = (1/4)*(110 + 26*n + 2^(n+8) - (1 - (-1)^n)*106*3^((n+1)/2) - (1 + (-1)^n)*61*3^(1+n/2)).

a(2*n) = (1/2)*(55 + 26*n + 2^(2*n+7) - 61*3^(n+1)).

a(2*n+1) = (1/2)*(68 + 26*n + 4^(n+4) - 106*3^(n+1)).

G.f.: x*(3 + 10*x)/((1-x)^2*(1 - 2*x - 3*x^2 + 6*x^3)).

E.g.f.: (1/2)*( (55 + 13*x)*exp(x) + 128*exp(2*x) - 183*cosh(sqrt(3)*x) - 106*sqrt(3)*sinh(sqrt(3)*x) ). (End)

Extensions

Terms a(14) onward added by G. C. Greubel, Feb 07 2022