cp's OEIS Frontend

A296003 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) = 2, a(1) = 4, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296003 #10 Feb 21 2018 05:43:38
%S A296003 2,4,10,32,94,278,824,2440,7228,21408,63406,187800,556234,1647478,
%T A296003 4879574,14452538,42806168,126785206,375518042,1112225982,3294240212,
%U A296003 9757026674,28898794076,85593729210,253515301048,750872855508,2223968505284,6587048494582
%N A296003 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) = 2, a(1) = 4, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296003 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.  a(n)/a(n-1) -> 2.961844324... (as in A296004).  See A296000 for a guide to related sequences.
%H A296003 Clark Kimberling, <a href="/A296003/b296003.txt">Table of n, a(n) for n = 0..999</a>
%H A296003 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 A296003 a(0) = 2, a(1) = 4, b(0) = 1, b(1) = 3, so that
%e A296003 a(2) = a(0)*b(1) + a(1)*b(0) = 10
%e A296003 Complement: (b(n)) = (1, 3, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...)
%t A296003 mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
%t A296003 a[0] = 2; a[1] = 4; b[0] = 1; a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
%t A296003 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A296003 Table[a[n], {n, 0, 100}];  (* A296003 *)
%t A296003 t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
%t A296003 Take[RealDigits[Last[t], 10][[1]], 100]  (* A296004 *)
%Y A296003 Cf. A296000, A296004.
%K A296003 nonn,easy
%O A296003 0,1
%A A296003 _Clark Kimberling_, Dec 07 2017