cp's OEIS Frontend

A296285 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296285 #4 Dec 13 2017 18:40:39
%S A296285 1,2,11,25,56,111,209,376,657,1123,1900,3166,5234,8595,14053,22903,
%T A296285 37244,60470,98074,158943,257457,416883,674868,1092349,1767865,
%U A296285 2860914,4629533,7491257,12121658,19613843,31736491,51351388,83088999,134441575,217531832
%N A296285 Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n-2), where a(0) = 1, a(1) = 2, b(0) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296285 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 A296285 Clark Kimberling, <a href="/A296285/b296285.txt">Table of n, a(n) for n = 0..1000</a>
%H A296285 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 A296285 a(0) = 1, a(1) = 2, b(0) = 4, b(1) = 3, b(2) = 5
%e A296285 a(2) = a(0) + a(1) + 2*b(0) = 11
%e A296285 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, ...)
%t A296285 a[0] = 1; a[1] = 2; b[0] = 4;
%t A296285 a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-2];
%t A296285 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296285  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296285 Table[a[n], {n, 0, k}]; (* A296285 *)
%t A296285 Table[b[n], {n, 0, 20}]    (* complement *)
%Y A296285 Cf. A001622, A296245.
%K A296285 nonn,easy
%O A296285 0,2
%A A296285 _Clark Kimberling_, Dec 13 2017