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

1,1

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).

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)

Mathematica

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

Prog

(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
(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

Crossrefs

Cf. A005803, A135835.

Sequence in context: A055550 A075204 A106150 * A004305 A275290 A368478

Adjacent sequences: A135833 A135834 A135835 * A135837 A135838 A135839

Keyword

nonn,easy

Author

John W. Layman, Nov 30 2007

Extensions

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

Status

approved