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A296843 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, b(3) = 6, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296843 #9 Nov 06 2018 13:16:44
%S A296843 1,2,9,18,35,63,109,184,306,504,825,1345,2187,3551,5758,9330,15110,
%T A296843 24463,39597,64085,103708,167820,271556,439405,710991,1150427,1861450,
%U A296843 3011910,4873394,7885340,12758771,20644149,33402959,54047148,87450148,141497338
%N A296843 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, b(3) = 6, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296843 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 A296843 Clark Kimberling, <a href="/A296843/b296843.txt">Table of n, a(n) for n = 0..1000</a>
%H A296843 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 A296843 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, b(3) = 6
%e A296843 a(2) = a(0) + a(1) + b(3) = 9
%e A296843 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, ...)
%t A296843 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5; b[3] = 6;
%t A296843 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n + 1];
%t A296843 j = 1; While[j < 16, k = a[j] - j - 1;
%t A296843 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296843 u = Table[a[n], {n, 0, k}];  (* A296843 *)
%t A296843 Table[b[n], {n, 0, 20}] (* complement *)
%Y A296843 Cf. A001622, A296245, A296844, A296845.
%K A296843 nonn,easy
%O A296843 0,2
%A A296843 _Clark Kimberling_, Jan 12 2018