This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A094687 #35 Aug 28 2025 13:24:39
%S A094687 0,0,1,2,6,13,30,64,137,286,594,1221,2498,5084,10313,20858,42094,
%T A094687 84797,170582,342760,688105,1380390,2767546,5546037,11109786,22248228,
%U A094687 44542825,89160674,178442742,357081901,714481614,1429477456,2859786953
%N A094687 Convolution of Fibonacci and Jacobsthal numbers.
%C A094687 Also convolution of A008346(n-1) and A000079(n).
%C A094687 Also difference of Fibonacci and Jacobsthal numbers shifted left: a(n) = A000045(n+1) - A001045(n+1). - _David Callan_, Jul 22 2008
%H A094687 G. C. Greubel, <a href="/A094687/b094687.txt">Table of n, a(n) for n = 0..1000</a>
%H A094687 Tamás Szakács, <a href="https://hdl.handle.net/2437/381856">Linear recursive sequences and factorials</a>, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 28.
%H A094687 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-3,-2).
%F A094687 G.f.: x^2/((1-x-x^2)*(1-x-2*x^2)).
%F A094687 a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - 2*a(n-4).
%F A094687 a(n) = Sum_{k=0..n} A000045(k)*A001045(n-k).
%F A094687 a(n+1) = a(n) + 2*a(n-1) + A000045(n). - _Philippe Deléham_, Mar 06 2013
%F A094687 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
%F A094687 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
%e A094687 a(2) = 0 + 2*0 + 1 = 1
%e A094687 a(3) = 1 + 2*0 + 1 = 2
%e A094687 a(4) = 2 + 2*1 + 2 = 6
%e A094687 a(5) = 6 + 2*2 + 3 = 13
%e A094687 a(6) = 13 + 2*6 + 5 = 30
%e A094687 a(7) = 30 + 2*13 + 8 = 64
%e A094687 a(8) = 64 + 2*30 + 13 = 137
%e A094687 a(9) = 137 + 2*64 + 21 = 286
%e A094687 ... - _Philippe Deléham_, Mar 06 2013
%p A094687 with(combstruct):
%p A094687 TSU := [T, { T = Sequence(S, card > 1), S = Sequence(U, card > 0), U = Sequence(Z, card > 1)}, unlabeled]:
%p A094687 seq(count(TSU, size = j+2), j=0..32); # _Peter Luschny_, Jan 04 2020
%t A094687 LinearRecurrence[{2,2,-3,-2}, {0,0,1,2}, 40] (* _G. C. Greubel_, Mar 06 2019 *)
%o A094687 (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 A094687 (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 A094687 (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 A094687 (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
%Y A094687 Cf. A000045, A000079, A001045, A008346.
%K A094687 easy,nonn,changed
%O A094687 0,4
%A A094687 _Paul Barry_, May 19 2004