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A296558 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - n, 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 A296558 #4 Dec 20 2017 23:33:45
%S A296558 1,3,7,13,24,41,69,114,187,306,498,809,1312,2126,3443,5574,9022,14601,
%T A296558 23628,38235,61869,100110,161985,262101,424092,686199,1110297,1796502,
%U A296558 2906805,4703313,7610124,12313443,19923573,32237022,52160601,84397630,136558238
%N A296558 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) - n, 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 A296558 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 A296558 Clark Kimberling, <a href="/A296558/b296558.txt">Table of n, a(n) for n = 0..1000</a>
%H A296558 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 A296558 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
%e A296558 a(2) = a(0) + a(1) + b(2) - 2 = 7
%e A296558 Complement: (b(n)) = (2, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, ...)
%t A296558 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A296558 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - n;
%t A296558 j = 1; While[j < 16, k = a[j] - j - 1;
%t A296558  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296558 u = Table[a[n], {n, 0, k}];  (* A296558 *)
%t A296558 Table[b[n], {n, 0, 20}] (* complement *)
%Y A296558 Cf. A001622, A296245, A296493, A296569, A296570.
%K A296558 nonn,easy
%O A296558 0,2
%A A296558 _Clark Kimberling_, Dec 20 2017