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A115008 a(n) = a(n-1) + a(n-3) + a(n-4).

%I A115008 #8 Aug 26 2025 08:52:53
%S A115008 1,0,2,4,5,7,13,22,34,54,89,145,233,376,610,988,1597,2583,4181,6766,
%T A115008 10946,17710,28657,46369,75025,121392,196418,317812,514229,832039,
%U A115008 1346269,2178310,3524578,5702886,9227465,14930353,24157817,39088168
%N A115008 a(n) = a(n-1) + a(n-3) + a(n-4).
%C A115008 a(n+2) - a(n+1) - a(n) gives match to A000034, apart from signs.
%H A115008 G. C. Greubel, <a href="/A115008/b115008.txt">Table of n, a(n) for n = 0..1000</a>
%H A115008 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,1).
%F A115008 a(2*n) = A000045(2*n+1) = A001519(n).
%F A115008 G.f.: (1-x+2*x^2+x^3)/((1+x^2)*(1-x-x^2)).
%F A115008 a(2*n+1) = (-1)^(n+1) + A001906(n+1) (compare with a similar property for A116697) - _Creighton Dement_, Mar 31 2006
%F A115008 From _G. C. Greubel_, Aug 24 2025: (Start)
%F A115008 a(n) = A000045(n+1) - i^(n-1)*(n mod 2).
%F A115008 E.g.f.: exp(x/2)*(cosh(p*x) + (1/(2*p))*sinh(p*x)) - sin(x), where 2*p = sqrt(5). (End)
%t A115008 Table[Fibonacci[n+1] -I^(n-1)*Mod[n,2], {n,0,50}] (* _G. C. Greubel_, Aug 24 2025 *)
%o A115008 (Magma)
%o A115008 A115008:= func< n | Fibonacci(n+1) - (n mod 2) + 2*0^((n+1) mod 4) >;
%o A115008 [A115008(n): n in [0..50]]; // _G. C. Greubel_, Aug 24 2025
%o A115008 (SageMath)
%o A115008 def A115008(n): return fibonacci(n+1) -i**(n-1)*(n%2)
%o A115008 print([A115008(n) for n in range(51)]) # _G. C. Greubel_, Aug 24 2025
%Y A115008 Cf. A000034, A000045, A001519, A001906, A006498, A116697, A116698, A116699.
%K A115008 easy,nonn,changed
%O A115008 0,3
%A A115008 _Creighton Dement_, Feb 23 2006