%I #26 Sep 08 2022 08:45:13

%S 0,0,1,2,6,13,30,64,137,286,594,1221,2498,5084,10313,20858,42094,

%T 84797,170582,342760,688105,1380390,2767546,5546037,11109786,22248228,

%U 44542825,89160674,178442742,357081901,714481614,1429477456,2859786953

%N Convolution of Fibonacci and Jacobsthal numbers.

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

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

%H G. C. Greubel, <a href="/A094687/b094687.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-3,-2).

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

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

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

%F a(n+1) = a(n) + 2*a(n-1) + A000045(n). - _Philippe Deléham_, Mar 06 2013

%F 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 a(2) = 0 + 2*0 + 1 = 1

%e a(3) = 1 + 2*0 + 1 = 2

%e a(4) = 2 + 2*1 + 2 = 6

%e a(5) = 6 + 2*2 + 3 = 13

%e a(6) = 13 + 2*6 + 5 = 30

%e a(7) = 30 + 2*13 + 8 = 64

%e a(8) = 64 + 2*30 + 13 = 137

%e a(9) = 137 + 2*64 + 21 = 286

%e ... - _Philippe Deléham_, Mar 06 2013

%p with(combstruct):

%p TSU := [T, { T = Sequence(S, card > 1), S = Sequence(U, card > 0), U = Sequence(Z, card > 1)}, unlabeled]:

%p seq(count(TSU, size = j+2), j=0..32); # _Peter Luschny_, Jan 04 2020

%t LinearRecurrence[{2,2,-3,-2}, {0,0,1,2}, 40] (* _G. C. Greubel_, Mar 06 2019 *)

%o (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

%o (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

%o (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

%o (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

%K easy,nonn

%O 0,4

%A _Paul Barry_, May 19 2004