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A295069 Solution of the complementary equation a(n) = 2*a(n-2) - b(n-2) + n, where a(0) = 3, a(1) = 4, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.

%I A295069 #5 Nov 19 2017 10:43:33
%S A295069 3,4,7,9,13,17,24,31,45,59,86,114,168,223,331,441,657,877,1309,1748,
%T A295069 2612,3490,5218,6974,10430,13941,20853,27875,41699,55743,83391,111479,
%U A295069 166775,222951,333543,445895,667079,891783,1334150,1783558,2668292,3567108
%N A295069 Solution of the complementary equation a(n) = 2*a(n-2) - b(n-2) + n, where a(0) = 3, a(1) = 4, b(0) = 1, and (a(n)) and (b(n)) are increasing complementary sequences.
%C A295069 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 A295069 The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.33..., 1.49...
%H A295069 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 A295069 a(0) = 3, a(1) = 4, b(0) = 1
%e A295069 a(2) = 2*a(0) - b(0)  + 2 = 7
%e A295069 Complement: (b(n)) = (1, 2, 5, 6, 8, 10, 11, 12, 14, 15, 16, 18, ... )
%t A295069 mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
%t A295069 a[0] = 3; a[1] = 4; b[0] = 1;
%t A295069 a[n_] := a[n] = 2 a[n - 2] + b[n - 1] + n;
%t A295069 b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];
%t A295069 Table[a[n], {n, 0, 18}]  (* A295069 *)
%t A295069 Table[b[n], {n, 0, 10}]
%Y A295069 Cf. A295053, A295068.
%K A295069 nonn,easy
%O A295069 0,1
%A A295069 _Clark Kimberling_, Nov 19 2017