cp's OEIS Frontend

A296001 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296001 #16 Jun 25 2018 17:37:29
%S A296001 1,2,10,43,185,796,3425,14737,63411,272845,1174000,5051498,21735632,
%T A296001 93524277,402417118,1731524071,7450417675,32057725596,137938276110,
%U A296001 593522081260,2553812262104,10988566855385,47281706383454,203444160458068,875381402033582
%N A296001 Solution of the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296001 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 4.302809183918588... (as in A296002). See A296000 for a guide to related sequences.
%H A296001 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling/kimberling26.html">Complementary equations</a>, J. Int. Seq. 19 (2007), 1-13.
%e A296001 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that
%e A296001 a(2) = a(0)*b(1) + a(1)*b(0) = 10
%e A296001 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, ...)
%t A296001 mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
%t A296001 a[0] = 1; a[1] = 2; b[0] = 3; a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
%t A296001 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A296001 Table[a[n], {n, 0, 100}];  (* A296001 *)
%t A296001 t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
%t A296001 Take[RealDigits[Last[t], 10][[1]], 100]  (* A296002 *)
%Y A296001 Cf. A296000, A296002.
%K A296001 nonn,easy
%O A296001 0,2
%A A296001 _Clark Kimberling_, Dec 04 2017
%E A296001 Conjectured g.f. removed by _Alois P. Heinz_, Jun 25 2018