A295063 - OEIS

%I #4 Nov 19 2017 10:42:45

%S 1,3,10,21,51,97,219,405,896,1643,3609,6599,14465,26427,57893,105743,

%T 231609,423011,926478,1692089,3705959,6768405,14823887,27073673,

%U 59295603,108294749,237182471

%N Solution of the complementary equation a(n) = 4*a(n-2) + b(n-1) + b(n-2), 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 2.19... and 1.82... .

%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, b(0) = 2, b(1) = 4

%e a(2) = 4*a(0) + b(1) + b(0) = 10

%e Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, ...)

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

%t a[0] = 1; a[1] = 3; b[0] = 2; b[1]=4;

%t a[n_] := a[n] = 4 a[n - 2] + b[n - 1] + b[n - 2];

%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}]  (* A295063 *)

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

%Y Cf. A295053.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Nov 19 2017