|
%I #13 Sep 27 2023 05:14:58
%S 1,1,1,1,1,2,3,4,5,8,13,20,29,44,70,112,175,272,430,690,1107,1766,
%T 2822,4542,7347,11886,19222,31150,50647,82518,134542,219542,358808,
%U 587430,962898,1579686,2593967,4264292,7017800,11559548,19055420,31437318,51908076,85775954,141841207
%N G.f. A(x) satisfies: A(x) = 1 + x + x^2 + x^3 + x^4*A(x)^2.
%C Shifts 4 places left when convolved with itself.
%H Seiichi Manyama, <a href="/A307971/b307971.txt">Table of n, a(n) for n = 0..4000</a>
%F G.f.: 1/(1 - x/(1 - x^4/(1 - x^4/(1 - x/(1 - x^4/(1 - x^4/(1 - x/(1 - x^4/(1 - x^4/(1 - ...)))))))))), a continued fraction.
%F Recurrence: a(n+4) = Sum_{k=0..n} a(k)*a(n-k).
%F G.f.: (1 - sqrt(1 - 4*x^4 - 4*x^5 - 4*x^6 - 4*x^7))/(2*x^4). - _Vaclav Kotesovec_, Sep 27 2023
%e G.f.: A(x) = 1 + x + x^2 + x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 8*x^9 + 13*x^10 + ...
%p a:= proc(n) option remember; `if`(n<4, 1,
%p add(a(j)*a(n-4-j), j=0..n-4))
%p end:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, May 08 2019
%t terms = 44; A[_] = 0; Do[A[x_] = 1 + x + x^2 + x^3 + x^4 A[x]^2 + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
%t a[n_] := a[n] = Sum[a[k] a[n - k - 4], {k, 0, n - 4}]; a[0] = a[1] = a[2] = a[3] = 1; Table[a[n], {n, 0, 44}]
%t CoefficientList[Series[(1 - Sqrt[1 - 4*x^4 - 4*x^5 - 4*x^6 - 4*x^7])/(2*x^4), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Sep 27 2023 *)
%Y Cf. A000108, A007477, A307970, A307972.
%K nonn
%O 0,6
%A _Ilya Gutkovskiy_, May 08 2019
|
