cp's OEIS Frontend

A297011 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) - b(n), where a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A297011 #9 Nov 06 2018 13:16:50
%S A297011 3,5,9,17,36,81,188,446,1068,2569,6192,14938,36052,87024,210081,
%T A297011 507166,1224392,2955928,7136225,17228354,41592908,100414144,242421169,
%U A297011 585256454,1412934048,3411124520,8235183057,19881490602,47998164228,115877819024,279753802241
%N A297011 Solution of the complementary equation a(n) = 2*a(n-1) + a(n-2) - b(n), where a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A297011 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> 1 + sqrt(2). See A296245 for a guide to related sequences.
%H A297011 Clark Kimberling, <a href="/A297011/b297011.txt">Table of n, a(n) for n = 0..1000</a>
%H A297011 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 A297011 a(0) = 3, a(1) = 5, b(0) = 1, b(1) = 2, b(2) = 4
%e A297011 a(2) = 2*a(1) + a(0) - b(2) = 9
%e A297011 Complement: (b(n)) = (1, 2, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 18, 19, ...)
%t A297011 a[0] = 3; a[1] = 5; b[0] = 1; b[1] = 2; b[2] = 4;
%t A297011 a[n_] := a[n] = 2 a[n - 1] + a[n - 2] - b[n];
%t A297011 j = 1; While[j < 9, k = a[j] - j - 1;
%t A297011 While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
%t A297011 u = Table[a[n], {n, 0, k}]; (* A297011 *)
%t A297011 Table[b[n], {n, 0, 25}] (* complement *)
%t A297011 Take[u, 30]
%Y A297011 Cf. A297012, A297013.
%K A297011 nonn,easy
%O A297011 0,1
%A A297011 _Clark Kimberling_, Jan 13 2018