cp's OEIS Frontend

A296288 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296288 #4 Dec 14 2017 14:22:35
%S A296288 1,2,11,28,63,126,237,426,743,1277,2150,3581,5911,9700,15849,25819,
%T A296288 41972,68131,110481,179030,289971,469505,760026,1230129,1990803,
%U A296288 3221657,5213240,8435734,13649870,22086561,35737451,57825097,93563700,151390018,244955010
%N A296288 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-1), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296288 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 A296288 Clark Kimberling, <a href="/A296288/b296288.txt">Table of n, a(n) for n = 0..1000</a>
%H A296288 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 A296288 a(0) = 1, a(1) = 2, b(0) = 1, b(1) = 3, b(2) = 4
%e A296288 a(2) = a(0) + a(1) + 2*b(0) = 11
%e A296288 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, ...)
%t A296288 a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
%t A296288 a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-1];
%t A296288 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296288  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296288 Table[a[n], {n, 0, k}]; (* A296288 *)
%t A296288 Table[b[n], {n, 0, 20}]    (* complement *)
%Y A296288 Cf. A001622, A296245.
%K A296288 nonn,easy
%O A296288 0,2
%A A296288 _Clark Kimberling_, Dec 14 2017