cp's OEIS Frontend

A296289 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296289 #4 Dec 14 2017 14:22:43
%S A296289 1,3,12,30,66,131,245,439,764,1302,2196,3652,6028,9888,16154,26312,
%T A296289 42770,69422,112570,182410,295440,478354,774344,1253296,2028288,
%U A296289 3282284,5311326,8594447,13906669,22502073,36409762,58912920,95323834,154237975,249563101
%N A296289 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296289 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 A296289 Clark Kimberling, <a href="/A296289/b296289.txt">Table of n, a(n) for n = 0..1000</a>
%H A296289 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 A296289 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
%e A296289 a(2) = a(0) + a(1) + 2*b(1) = 12
%e A296289 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, ...)
%t A296289 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4;
%t A296289 a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-1];
%t A296289 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296289  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296289 Table[a[n], {n, 0, k}]; (* A296289 *)
%t A296289 Table[b[n], {n, 0, 20}]    (* complement *)
%Y A296289 Cf. A001622, A296245.
%K A296289 nonn,easy
%O A296289 0,2
%A A296289 _Clark Kimberling_, Dec 14 2017