cp's OEIS Frontend

A296284 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) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A296284 #4 Dec 13 2017 18:40:31
%S A296284 1,2,9,23,52,105,199,360,639,1098,1857,3098,5123,8416,13763,22434,
%T A296284 36485,59242,96087,155728,252255,408487,661292,1070377,1732317,
%U A296284 2803394,4536465,7340669,11878002,19219599,31098591,50319244,81418955,131739387,213159600
%N A296284 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) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A296284 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 A296284 Clark Kimberling, <a href="/A296284/b296284.txt">Table of n, a(n) for n = 0..1000</a>
%H A296284 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 A296284 a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5
%e A296284 a(2) = a(0) + a(1) + 2*b(0) = 9
%e A296284 Complement: (b(n)) = (3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, ...)
%t A296284 a[0] = 1; a[1] = 2; b[0] = 3;
%t A296284 a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n-2];
%t A296284 j = 1; While[j < 10, k = a[j] - j - 1;
%t A296284  While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A296284 Table[a[n], {n, 0, k}]; (* A296284 *)
%t A296284 Table[b[n], {n, 0, 20}]    (* complement *)
%Y A296284 Cf. A001622, A296245.
%K A296284 nonn,easy
%O A296284 0,2
%A A296284 _Clark Kimberling_, Dec 13 2017