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A094687
Convolution of Fibonacci and Jacobsthal numbers.
3
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
0,4
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
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
EXAMPLE
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
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 *)
PROG
(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
(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
(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
(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
CROSSREFS
Sequence in context: A289048 A297388 A115217 * A369584 A336875 A219753
KEYWORD
easy,nonn
AUTHOR
Paul Barry, May 19 2004
STATUS
approved