

A295139


Solution of the complementary equation a(n) = 3*a(n2) + b(n2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.


2



1, 2, 6, 10, 23, 37, 77, 120, 242, 372, 739, 1130, 2232, 3406, 6713, 10236, 20158, 30728, 60495, 92206, 181509, 276643, 544553, 829956, 1633687, 2489897, 4901091, 7469722, 14703305, 22409199, 44109949, 67227632, 132329883
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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 1.52..., 1.96...


LINKS

Table of n, a(n) for n=0..32.
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 113.


EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4
a(2) =3*a(0) + b(0) = 6
Complement: (b(n)) = (3, 4, 5, 7, 8, 9, 11, 12, 13, 14, 15, ...)


MATHEMATICA

mex := First[Complement[Range[1, Max[#1] + 1], #1]] &;
a[0] = 1; a[1] = 2; b[0] = 3; b[1]=4;
a[n_] := a[n] = 3 a[n  2] + b[n  2];
b[n_] := b[n] = mex[Flatten[Table[Join[{a[n]}, {a[i], b[i]}], {i, 0, n  1}]]];
Table[a[n], {n, 0, 18}] (* A295139 *)
Table[b[n], {n, 0, 10}]


CROSSREFS

Cf. A295053.
Sequence in context: A200572 A342136 A049750 * A134016 A072297 A183036
Adjacent sequences: A295136 A295137 A295138 * A295140 A295141 A295142


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Nov 19 2017


STATUS

approved



