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%I A296776 #8 Feb 11 2018 04:27:13
%S A296776 1,3,13,28,56,102,179,305,511,846,1391,2274,3705,6022,9773,15844,
%T A296776 25669,41568,67295,108924,176283,285274,461627,746974,1208678,1955732,
%U A296776 3164493,5120311,8284893,13405296,21690284,35095678,56786063,91881845,148668015,240549970
%N A296776 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 2*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 A296776 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 A296776 Clark Kimberling, <a href="/A296776/b296776.txt">Table of n, a(n) for n = 0..1000</a>
%H A296776 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 A296776 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
%e A296776 a(2) = a(0) + a(1) + b(2) + 4 = 13
%e A296776 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, ...)
%t A296776 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A296776 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 2 n;
%t A296776 j = 1; While[j < 16, k = a[j] - j - 1;
%t A296776 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296776 u = Table[a[n], {n, 0, k}];  (* A296776 *)
%t A296776 Table[b[n], {n, 0, 20}] (* complement *)
%Y A296776 Cf. A001622, A296245, A298171, A298172.
%K A296776 nonn,easy
%O A296776 0,2
%A A296776 _Clark Kimberling_, Jan 06 2018

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