

A295138


Solution of the complementary equation a(n) = 3*a(n2) + b(n1), where a(0) = 1, a(1) = 2, b(0) = 3, and (a(n)) and (b(n)) are increasing complementary sequences.


2



1, 2, 7, 11, 27, 41, 90, 133, 282, 412, 860, 1251, 2596, 3770, 7806, 11329, 23438, 34008, 70336, 102047, 211032, 306166, 633122, 918526, 1899395, 2755608, 5698216, 8266856, 17094681, 24800602, 51284078, 74401842
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OFFSET

0,2


COMMENTS

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.
The sequence a(n+1)/a(n) appears to have two convergent subsequences, with limits 1.45..., 2.06...


LINKS

Table of n, a(n) for n=0..31.
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 113.


EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3
a(2) =3*a(0) + b(1) = 7
Complement: (b(n)) = (3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, ... )


MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3;
a[n_] := a[n] = 3 a[n  2] + b[n  1];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n  1}]]];
Table[a[n], {n, 0, 18}] (* A295138 *)
Table[b[n], {n, 0, 10}]


CROSSREFS

Cf. A295053.
Sequence in context: A201630 A023862 A024479 * A023864 A024857 A024481
Adjacent sequences: A295135 A295136 A295137 * A295139 A295140 A295141


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Nov 19 2017


STATUS

approved



