

A295061


Solution of the complementary equation a(n) = 4*a(n2) + b(n1), where a(0) = 1, a(1) = 3, b(0) = 2, and (a(n)) and (b(n)) are increasing complementary sequences.


2



1, 3, 8, 17, 38, 75, 161, 310, 655, 1252, 2633, 5022, 10547, 20104, 42206, 80435, 168844, 321761, 675398, 1287067, 2701616, 5148293, 10806490, 20593199, 43225988, 82372825, 172903982
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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 1.09... and 2.09... .


LINKS

Table of n, a(n) for n=0..26.
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 113.


EXAMPLE

a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4
a(2) = 4*a(0) + b(1) = 8
Complement: (b(n)) = (2, 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] = 3; b[0] = 2;
a[n_] := a[n] = 4 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}] (* A295061 *)
Table[b[n], {n, 0, 10}]


CROSSREFS

Cf. A295053.
Sequence in context: A097391 A202554 A034481 * A247374 A336512 A046994
Adjacent sequences: A295058 A295059 A295060 * A295062 A295063 A295064


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, Nov 18 2017


STATUS

approved



