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A296556 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 A296556 #4 Dec 20 2017 14:41:15
%S A296556 1,3,11,23,45,81,141,239,400,661,1085,1772,2885,4687,7604,12325,19965,
%T A296556 32328,52333,84704,137082,221833,358964,580848,939865,1520768,2460690,
%U A296556 3981517,6442268,10423848,16866181,27290096,44156346,71446513,115602932,187049520
%N A296556 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 A296556 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 A296556 Clark Kimberling, <a href="/A296556/b296556.txt">Table of n, a(n) for n = 0..1000</a>
%H A296556 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 A296556 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
%e A296556 a(2) = a(0) + a(1) + b(2) + 2 = 11
%e A296556 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, ...)
%t A296556 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A296556 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + n;
%t A296556 j = 1; While[j < 16, k = a[j] - j - 1;
%t A296556  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296556 u = Table[a[n], {n, 0, k}];  (* A296556 *)
%t A296556 Table[b[n], {n, 0, 20}] (* complement *)
%Y A296556 Cf. A001622, A296245, A296493, A296565, A296566.
%K A296556 nonn,easy
%O A296556 0,2
%A A296556 _Clark Kimberling_, Dec 20 2017