cp's OEIS Frontend

A295958 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295958 #8 Feb 22 2018 22:42:29
%S A295958 2,3,11,21,40,70,120,201,334,549,898,1463,2378,3859,6256,10135,16412,
%T A295958 26570,43006,69601,112633,182261,294922,477212,772164,1249407,2021603,
%U A295958 3271043,5292680,8563758,13856474,22420269,36276781,58697089,94973910,153671041
%N A295958 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n) + 1, where a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295958 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).
%C A295958 See A295862 for a guide to related sequences.
%H A295958 Clark Kimberling, <a href="/A295958/b295958.txt">Table of n, a(n) for n = 0..2000</a>
%H A295958 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 A295958 a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5
%e A295958 b(3) = 6 (least "new number")
%e A295958 a(2) = a(1) + a(0) + b(2) + 1 = 11
%e A295958 Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, ...)
%t A295958 a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4; b[2] = 5;
%t A295958 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] + 1;
%t A295958 j = 1; While[j < 6, k = a[j] - j - 1;
%t A295958 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A295958 Table[a[n], {n, 0, k}];  (* A295958 *)
%Y A295958 Cf. A001622, A000045, A295862.
%K A295958 nonn,easy
%O A295958 0,1
%A A295958 _Clark Kimberling_, Dec 08 2017