cp's OEIS Frontend

A296555 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 A296555 #5 Dec 19 2017 22:08:43
%S A296555 1,2,10,21,42,76,133,227,380,629,1033,1688,2749,4467,7248,11749,19033,
%T A296555 30821,49895,80759,130699,211505,342253,553809,896115,1449979,2346151,
%U A296555 3796189,6142401,9938653,16081119,26019839,42101027,68120937,110222037,178343049
%N A296555 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 A296555 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 A296555 Clark Kimberling, <a href="/A296555/b296555.txt">Table of n, a(n) for n = 0..1000</a>
%H A296555 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 A296555 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
%e A296555 a(2) = a(0) + a(1) + b(2) + 2 = 10
%e A296555 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)
%t A296555 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
%t A296555 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + n;
%t A296555 j = 1; While[j < 16, k = a[j] - j - 1;
%t A296555  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296555 u = Table[a[n], {n, 0, k}];  (* A296555 *)
%t A296555 Table[b[n], {n, 0, 20}] (* complement *)
%Y A296555 Cf. A001622, A296245, A296493, A296496.
%K A296555 nonn,easy
%O A296555 0,2
%A A296555 _Clark Kimberling_, Dec 19 2017