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A295966 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 5, b(0) = 2, b(1) = 3, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295966 #6 Feb 22 2018 22:44:37
%S A295966 1,5,9,19,34,60,103,173,287,472,772,1258,2045,3319,5381,8719,14120,
%T A295966 22860,37002,59885,96911,156821,253758,410606,664392,1075027,1739449,
%U A295966 2814507,4553988,7368529,11922552,19291117,31213706,50504861,81718606,132223507,213942154
%N A295966 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 5, b(0) = 2, b(1) = 3, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295966 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 A295966 See A295862 for a guide to related sequences.
%H A295966 Clark Kimberling, <a href="/A295966/b295966.txt">Table of n, a(n) for n = 0..2000</a>
%H A295966 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 A295966 a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5
%e A295966 b(3) = 6 (least "new number")
%e A295966 a(2) = a(1) + a(0) + b(2) - 1 = 9
%e A295966 Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...)
%t A295966 a[0] = 1; a[1] = 5; b[0] = 2; b[1] = 3; b[2] = 4;
%t A295966 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
%t A295966 j = 1; While[j < 6, k = a[j] - j - 1;
%t A295966 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A295966 Table[a[n], {n, 0, k}];  (* A295966 *)
%Y A295966 Cf. A001622, A000045, A295862.
%K A295966 nonn,easy
%O A295966 0,2
%A A295966 _Clark Kimberling_, Dec 08 2017