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Solution of the complementary equation a(n) = 2*a(n-2) + b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.
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%I #4 Nov 19 2017 10:43:12

%S 1,3,6,11,19,30,47,70,106,153,226,321,468,659,954,1338,1929,2698,3881,

%T 5420,7787,10866,15601,21760,31231,43551,62494,87135,125022,174305,

%U 250080,348647,500198,697333,1000436,1394707,2000914,2789457,4001872,5578959,8003790

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

%C 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 The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.43..., 1.39... .

%H 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 a(0) = 1, a(1) = 3, a(2) = 2, b(0) = 2, b(1) = 4,

%e a(2) = 2*a(0) + b(1) = 6

%e Complement: (b(n)) = (2, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, ...)

%t mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;

%t a[0] = 1; a[1] = 3; b[0] = 2;

%t a[n_] := a[n] = 2 a[n - 2] + b[n - 1];

%t b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n - 1}]]];

%t Table[a[n], {n, 0, 18}] (* A295066 *)

%t Table[b[n], {n, 0, 10}]

%Y Cf. A295053, A295067.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Nov 19 2017