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%I #18 Sep 23 2020 10:51:02
%S 1,3,10,37,135,493,1800,6572,23996,87614,319895,1167997,4264577,
%T 15570774,56851829,207576737,757901769,2767242128,10103722287,
%U 36890593353,134694505577,491795012865,1795636233585,6556206140806,23937943641806,87401941533192
%N 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) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
%C The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 3.651188... (as in A295999). Guide for the complementary equation a(n) = a(0)*b(n-1) + a(1)*b(n-2) + ... + a(n-1)*b(0):
%C A296000: a(0) = 1, a(1) = 3, b(0) = 2, limiting ratio of a(n)/a(n-1): A295999
%C A296001: a(0) = 1, a(1) = 2, b(0) = 3, limiting ratio of a(n)/a(n-1): A296002
%C A296003: a(0) = 2, a(1) = 4, b(0) = 1, limiting ratio of a(n)/a(n-1): A296004
%C A296005: a(0) = 2, a(1) = 3, b(0) = 1, limiting ratio of a(n)/a(n-1): A296006
%H 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 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that
%e a(2) = a(0)*b(1) + a(1)*b(0) = 10
%e Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, ...)
%t $RecursionLimit = Infinity;
%t mex[list_] := NestWhile[# + 1 &, 1, MemberQ[list, #] &];
%t a[0] = 1; a[1] = 3; b[0] = 2; a[n_] := a[n] = Sum[a[k]*b[n - k - 1], {k, 0, n - 1}];
%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t Table[a[n], {n, 0, 100}] (* A296000 *)
%t t = N[Table[a[n]/a[n - 1], {n, 1, 500, 100}], 200]
%t Take[RealDigits[Last[t], 10][[1]], 100] (* A295999 *)
%Y Cf. A295999, A296001.
%K nonn,easy
%O 0,2
%A _Clark Kimberling_, Dec 04 2017
%E Incorrect conjectured g.f. removed by _Georg Fischer_, Sep 23 2020
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