A378676 a(n) = J(n) * J(n+2) where J(n) = Jacobsthal(n) = A001045(n).

0, 3, 5, 33, 105, 473, 1785, 7353, 28985, 116793, 465465, 1865273, 7454265, 29830713, 119295545, 477236793, 1908837945, 7635570233, 30541844025, 122168249913, 488671252025, 1954688503353, 7818747022905, 31275002072633, 125099980328505, 500399977238073, 2001599797104185, 8006399412112953, 32025597201059385

Offset

0,2

Links

Table of n, a(n) for n=0..28.

Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Formula

a(n) = (2^n - (-1)^n) * (2^(n+2) - (-1)^n) / 9 = (4 * 4^n - 5 * (-2)^n + 1) / 9.

G.f.: x * (3 - 4*x) / ((1-x) * (1+2*x) * (1-4*x)).

a(n) = 3 * a(n-1) + 6 * a(n-2) - 8 * a(n-3) for n > 2 with initial values a(0) = 0, a(1) = 3, and a(2) = 5.

Sum_{k=1..n-1} 2^(k-1) / a(k) = 1 - 2^(n-1) / A084175(n) for n > 0.

Sum_{k>0} 2^(k-1) / a(k) = 1.

E.g.f.: exp(x)*(1 - cosh(3*x) + 9*sinh(3*x))/9. - Stefano Spezia, Dec 06 2024

Mathematica

a[n_] := (4^(n+1) - 5*(-2)^n + 1)/9; Array[a, 30, 0] (* Amiram Eldar, Dec 06 2024 *)

Prog

(PARI) a(n)=(4^(n+1)-5*(-2)^n+1)/9

Crossrefs

Cf. A001045, A084175.

Sequence in context: A103010 A225866 A332704 * A263295 A222484 A354849

Adjacent sequences: A378670 A378672 A378673 * A378677 A378678 A378679

Keyword

nonn,easy,new

Author

Werner Schulte, Dec 03 2024

Status

approved