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

Author

Paul Barry, May 16 2003

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..500
• Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Crossrefs

Cf. A001045, A078008, A084153.

Except for initial terms, same as A015249 and A084175.

Programs

Magma

[(4^n-2+(-2)^n)/18: n in [0..35]]; // Vincenzo Librandi, Jul 05 2011

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 *)

Python

def A084152(n): return ((m:=1<>1|1)//3) # Chai Wah Wu, Apr 25 2025

SageMath

[(4^n-2+(-2)^n)/18 for n in range(41)] # G. C. Greubel, Oct 11 2022

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