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A295963 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295963 #8 Feb 22 2018 22:43:39
%S A295963 1,2,7,14,28,50,87,147,245,404,663,1082,1761,2860,4639,7518,12177,
%T A295963 19716,31915,51654,83593,135272,218891,354191,573111,927332,1500474,
%U A295963 2427838,3928345,6356217,10284597,16640850,26925484,43566372,70491895,114058307,184550243
%N A295963 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - 1, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295963 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 A295963 See A295862 for a guide to related sequences.
%H A295963 Clark Kimberling, <a href="/A295963/b295963.txt">Table of n, a(n) for n = 0..2000</a>
%H A295963 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 A295963 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
%e A295963 b(3) = 6 (least "new number")
%e A295963 a(2) = a(1) + a(0) + b(2) - 1 = 7
%e A295963 Complement: (b(n)) = (3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, ...)
%t A295963 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A295963 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
%t A295963 j = 1; While[j < 6, k = a[j] - j - 1;
%t A295963 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A295963 Table[a[n], {n, 0, k}];  (* A295963 *)
%Y A295963 Cf. A001622, A000045, A295862.
%K A295963 nonn,easy
%O A295963 0,2
%A A295963 _Clark Kimberling_, Dec 08 2017