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 A080833 #29 Mar 29 2022 03:39:47
%S A080833 1,1,3,19,145,1401,16051,213403,3223809,54514225,1019601091,
%T A080833 20890209891,465156779473,11181638663209,288536019179955,
%U A080833 7953590111627371,233211718410856321,7246720953253750113,237849724555558441219,8221578401608012672435,298505383888840158941841
%N A080833 E.g.f.: exp( x/(1 - x - x^2) ).
%C A080833 From _Peter Bala_, Mar 25 2022: (Start)
%C A080833 The sequence terms are odd. 3 divides a(3*n-1), 9 divides a(9*n-1) and 27 divides a(27*n-1); 5 divides a(5*n+4), 25 divides a(25*n+9) and 125 divides a(125*n+34); 7 divides a(7*n+6), 49 divides a(49*n+34) and 343 divides a(343*n + 83); 15 divides a(15*n+14) and 17 divides a(17*n+13).
%C A080833 More generally, the congruence a(n+k) == a(n) (mod k) holds for all n and k. It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, taken modulo 7 the sequence becomes [1, 1, 3, 5, 5, 1, 0, 1, 1, 3, 3, 5, 5, 1, 0, ...], a purely periodic sequence with period 7. (End)
%H A080833 Peter Bala, <a href="/A047974/a047974_1.pdf">Integer sequences that become periodic on reduction modulo k for all k</a>
%F A080833 E.g.f.: exp( x/(1 - x - x^2) ).
%F A080833 a(n) = n!*sum{i=0..n, sum{j=0..n, C(i+j-1, j)*C(j, n-i-j)/i!}}. - _Paul Barry_, Aug 29 2005
%F A080833 E.g.f.: 1 + x*(E(0)-1)/(x+1) where E(k) = 1 + 1/(k+1)/(1-x-x^2)/(1-x/(x+1/E(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Jan 27 2013
%F A080833 Recurrence: a(n) = (2*n-1)*a(n-1) + (n-2)*(n-1)*a(n-2) - (n-2)*(n-1)*(2*n-7)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4). - _Vaclav Kotesovec_, Jun 27 2013
%F A080833 a(n) ~ ((1+sqrt(5))/2)^n*exp(2*sqrt(n)/5^(1/4)-n-1/10)*n^(n-1/4)/(sqrt(2)*5^(1/8)). - _Vaclav Kotesovec_, Jun 27 2013
%F A080833 a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * k! * Fibonacci(k) * a(n-k). - _Ilya Gutkovskiy_, Jun 07 2021
%t A080833 CoefficientList[Series[E^(x/(1-x-x^2)), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 27 2013 *)
%o A080833 (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(x/(1-x-x^2)))) \\ _Michel Marcus_, Jun 07 2021
%Y A080833 Cf. A005443.
%K A080833 easy,nonn
%O A080833 0,3
%A A080833 _Emanuele Munarini_, Mar 28 2003