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A295068 Solution of the complementary equation a(n) = 2*a(n-2) - b(n-1) + n, where a(0) = 4, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295068 #4 Nov 19 2017 10:43:27
%S A295068 4,5,8,10,14,18,25,32,46,60,87,115,169,224,332,442,658,878,1310,1749,
%T A295068 2613,3491,5219,6975,10431,13942,20854,27876,41700,55744,83392,111480,
%U A295068 166776,222952,333544,445896,667080,891784,1334151,1783559,2668293,3567109
%N A295068 Solution of the complementary equation a(n) = 2*a(n-2) - b(n-1) + n, where a(0) = 4, a(1) = 5, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295068 The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295053 for a guide to related sequences.
%C A295068 The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.33..., 1.49...
%H A295068 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 A295068 a(0) = 4, a(1) = 5, b(0) = 1
%e A295068 a(2) = 2*a(0) - b(1)  + 2 = 8
%e A295068 Complement: (b(n)) = (1, 2, 3, 6, 7, 9, 11, 12, 13, 15, 16, 17, 19, ... )
%t A295068 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295068 a[0] = 4; a[1] = 5; b[0] = 1;
%t A295068 a[n_] := a[n] = 2 a[n - 2] - b[n - 1] + n;
%t A295068 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295068 Table[a[n], {n, 0, 18}]  (* A295068 *)
%t A295068 Table[b[n], {n, 0, 10}]
%Y A295068 Cf. A295053, A295069.
%K A295068 nonn,easy
%O A295068 0,1
%A A295068 _Clark Kimberling_, Nov 19 2017