cp's OEIS Frontend

A295965 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 A295965 #6 Feb 22 2018 22:44:16
%S A295965 2,3,9,17,32,56,97,163,271,446,730,1190,1935,3142,5095,8256,13371,
%T A295965 21648,35041,56712,91777,148514,240317,388858,629203,1018090,1647323,
%U A295965 2665445,4312801,6978280,11291116,18269432,29560585,47830055,77390679,125220774,202611494
%N A295965 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 A295965 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 A295965 See A295862 for a guide to related sequences.
%H A295965 Clark Kimberling, <a href="/A295965/b295965.txt">Table of n, a(n) for n = 0..2000</a>
%H A295965 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 A295965 a(0) = 2, a(1) = 3, b(0) = 1, b(1) = 4, b(2) = 5
%e A295965 b(3) = 6 (least "new number")
%e A295965 a(2) = a(1) + a(0) + b(2) - 1 = 9
%e A295965 Complement: (b(n)) = (1, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, ...)
%t A295965 a[0] = 2; a[1] = 3; b[0] = 1; b[1] = 4; b[2] = 5;
%t A295965 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n] - 1;
%t A295965 j = 1; While[j < 6, k = a[j] - j - 1;
%t A295965 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A295965 Table[a[n], {n, 0, k}];  (* A295965 *)
%Y A295965 Cf. A001622, A000045, A295862.
%K A295965 nonn,easy
%O A295965 0,1
%A A295965 _Clark Kimberling_, Dec 08 2017