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A296557 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - n, 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 A296557 #4 Dec 20 2017 14:41:22
%S A296557 1,2,6,11,21,36,61,102,168,275,448,728,1181,1914,3100,5019,8125,13150,
%T A296557 21281,34437,55724,90167,145897,236070,381973,618049,1000028,1618083,
%U A296557 2618117,4236206,6854330,11090543,17944880,29035430,46980317,76015754,122996078
%N A296557 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - n, 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 A296557 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). See A296245 for a guide to related sequences.
%H A296557 Clark Kimberling, <a href="/A296557/b296557.txt">Table of n, a(n) for n = 0..1000</a>
%H A296557 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 A296557 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
%e A296557 a(2) = a(0) + a(1) + b(2) - 2 = 6
%e A296557 Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, ...)
%t A296557 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296557 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - n;
%t A296557 j = 1; While[j < 16, k = a[j] - j - 1;
%t A296557  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296557 u = Table[a[n], {n, 0, k}];  (* A296557 *)
%t A296557 Table[b[n], {n, 0, 20}] (* complement *)
%Y A296557 Cf. A001622, A296245, A296493, A296567, A296568.
%K A296557 nonn,easy
%O A296557 0,2
%A A296557 _Clark Kimberling_, Dec 20 2017