cp's OEIS Frontend

A296279 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1)*b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296279 #4 Dec 13 2017 18:39:51
%S A296279 1,3,44,167,421,924,1849,3493,6332,11145,19193,32522,54445,90327,
%T A296279 148852,244075,398741,649656,1056377,1715273,2782276,4509693,7305769,
%U A296279 11831062,19154381,31005099,50181404,81210863,131419237,212659860,344111833,556807597,900958700
%N A296279 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-2)*b(n-1)*b(n), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296279 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 A296279 Clark Kimberling, <a href="/A296279/b296279.txt">Table of n, a(n) for n = 0..1000</a>
%H A296279 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 A296279 a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, b(2) = 5
%e A296279 a(2) = a(0) + a(1) + b(0)*b(1)*b(2) = 44
%e A296279 Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...)
%t A296279 a[0] = 1; a[1] = 3; b[0] = 2; b[1] = 4; b[2] = 5;
%t A296279 a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n-2] b[n - 1] b[n];
%t A296279 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296279  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296279 Table[a[n], {n, 0, k}]; (* A296279 *)
%t A296279 Table[b[n], {n, 0, 20}]    (* complement *)
%Y A296279 Cf. A001622, A296245.
%K A296279 nonn,easy
%O A296279 0,2
%A A296279 _Clark Kimberling_, Dec 13 2017