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A295953 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295953 #11 Feb 22 2018 22:40:35
%S A295953 1,3,10,20,38,67,115,194,322,530,867,1413,2297,3728,6044,9792,15858,
%T A295953 25673,41555,67253,108834,176114,284976,461119,746125,1207275,1953432,
%U A295953 3160740,5114206,8274981,13389223,21664241,35053502,56717783,91771326,148489151
%N A295953 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295953 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).
%C A295953 See A295862 for a guide to related sequences.
%H A295953 Clark Kimberling, <a href="/A295953/b295953.txt">Table of n, a(n) for n = 0..2000</a>
%H A295953 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 A295953 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
%e A295953 b(3) = 6 (least "new number")
%e A295953 a(2) = a(1) + a(0) + b(2) + 1 = 10
%e A295953 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, ...)
%t A295953 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A295953 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 1;
%t A295953 j = 1; While[j < 6, k = a[j] - j - 1;
%t A295953 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A295953 Table[a[n], {n, 0, k}];  (* A295953 *)
%Y A295953 Cf. A001622, A000045, A295862.
%K A295953 nonn,easy
%O A295953 0,2
%A A295953 _Clark Kimberling_, Dec 08 2017