cp's OEIS Frontend

A296290 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296290 #4 Dec 14 2017 14:22:52
%S A296290 1,4,11,30,65,130,243,436,759,1303,2192,3649,6021,9878,16137,26285,
%T A296290 42726,69351,112455,182224,295139,477867,773556,1252021,2026225,
%U A296290 3278946,5305925,8585708,13892529,22479194,36372743,58853022,95226917,154081160,249309369
%N A296290 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296290 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 A296290 Clark Kimberling, <a href="/A296290/b296290.txt">Table of n, a(n) for n = 0..1000</a>
%H A296290 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 A296290 a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5
%e A296290 a(2) = a(0) + a(1) + 2*b(1) = 11
%e A296290 Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, ...)
%t A296290 a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3;
%t A296290 a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-1];
%t A296290 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296290  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296290 Table[a[n], {n, 0, k}]; (* A296290 *)
%t A296290 Table[b[n], {n, 0, 20}]    (* complement *)
%Y A296290 Cf. A001622, A296245.
%K A296290 nonn,easy
%O A296290 0,2
%A A296290 _Clark Kimberling_, Dec 14 2017