A094687 Convolution of Fibonacci and Jacobsthal numbers.

0, 0, 1, 2, 6, 13, 30, 64, 137, 286, 594, 1221, 2498, 5084, 10313, 20858, 42094, 84797, 170582, 342760, 688105, 1380390, 2767546, 5546037, 11109786, 22248228, 44542825, 89160674, 178442742, 357081901, 714481614, 1429477456, 2859786953

Offset: 0

Author

Paul Barry, May 19 2004

Keywords

• easy
• nonn
• changed

Comments

Also convolution of A008346(n-1) and A000079(n).

Also difference of Fibonacci and Jacobsthal numbers shifted left: a(n) = A000045(n+1) - A001045(n+1). - David Callan, Jul 22 2008

Examples

			a(2) =   0 + 2*0  +  1 =   1
a(3) =   1 + 2*0  +  1 =   2
a(4) =   2 + 2*1  +  2 =   6
a(5) =   6 + 2*2  +  3 =  13
a(6) =  13 + 2*6  +  5 =  30
a(7) =  30 + 2*13 +  8 =  64
a(8) =  64 + 2*30 + 13 = 137
a(9) = 137 + 2*64 + 21 = 286
... - _Philippe Deléham_, Mar 06 2013
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 28.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-3,-2).

Crossrefs

Cf. A000045, A000079, A001045, A008346.

Programs

GAP

a:=[0,0,1,2];; for n in [5..40] do a[n]:=2*a[n-1]+2*a[n-2] - 3*a[n-3]-2*a[n-4]; od; a; # G. C. Greubel, Mar 06 2019

Magma

I:=[0,0,1,2]; [n le 4 select I[n] else 2*Self(n-1) + 2*Self(n-2) -3*Self(n-3) -2*Self(n-4): n in [1..40]]; // G. C. Greubel, Mar 06 2019

Maple

with(combstruct):
TSU := [T, { T = Sequence(S, card > 1), S = Sequence(U, card > 0), U = Sequence(Z, card > 1)}, unlabeled]:
seq(count(TSU, size = j+2), j=0..32); # Peter Luschny, Jan 04 2020

Mathematica

LinearRecurrence[{2,2,-3,-2}, {0,0,1,2}, 40] (* G. C. Greubel, Mar 06 2019 *)

PARI

my(x='x+O('x^40)); concat([0,0], Vec(x^2/((1-x-x^2)*(1-x-2*x^2)))) \\ G. C. Greubel, Mar 06 2019

Sage

(x^2/((1-x-x^2)*(1-x-2*x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Mar 06 2019

Formula

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

a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - 2*a(n-4).

a(n) = Sum_{k=0..n} A000045(k)*A001045(n-k).

a(n+1) = a(n) + 2*a(n-1) + A000045(n). - Philippe Deléham, Mar 06 2013

a(n) = J(n+1) - F(n+1) = Sum_{k=0..n} F(k)*J(n-k), where J=A001045, F=A000045. - Yuchun Ji, Mar 05 2019

E.g.f.: exp(x/2)*(5*(3*cosh(3*x/2) - 3*cosh(sqrt(5)*x/2) + sinh(3*x/2)) - 3*sqrt(5)*sinh(sqrt(5)*x/2))/15. - Stefano Spezia, Aug 28 2025