A084152 Exponential self-convolution of Jacobsthal numbers (divided by 2).

0, 0, 1, 3, 15, 55, 231, 903, 3655, 14535, 58311, 232903, 932295, 3727815, 14913991, 59650503, 238612935, 954429895, 3817763271, 15270965703, 61084037575, 244335800775, 977343902151, 3909374210503, 15637499638215, 62549992960455

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

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

Formula

a(n) = (4^n - 2 + (-2)^n)/18.

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

a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3).

E.g.f.: (exp(2*x) - exp(-x))^2/18 = (exp(4*x) - 2*exp(x) + exp(-x))/18.

Binomial transform of 0, 0, 1, 0, 9, 0, 81, ... .

a(n) = A001045(n)*A078008(n)/2.

a(n) = floor(2^n/3)ceiling(2^n/3)/2. - Paul Barry, Apr 28 2004

Mathematica

Join[{a=0, b=0}, Table[c=2*b+8*a+1; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 05 2011*)
LinearRecurrence[{3, 6, -8}, {0, 0, 1}, 30] (* Harvey P. Dale, Nov 11 2011 *)

Prog

(Magma) [(4^n-2+(-2)^n)/18: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011
(SageMath) [(4^n-2+(-2)^n)/18 for n in range(41)] # G. C. Greubel, Oct 11 2022

Crossrefs

Cf. A001045, A078008, A084153.

Except for initial terms, same as A015249 and A084175.

Sequence in context: A007973 A261737 A015249 * A084175 A081951 A367520

Adjacent sequences: A084149 A084150 A084151 * A084153 A084154 A084155

Keyword

easy,nonn

Author

Paul Barry, May 16 2003

Status

approved