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A295067

Solution of the complementary equation a(n) = 2*a(n-2) + b(n-2), where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 4, 11, 14, 29, 36, 67, 82, 146, 177, 307, 370, 631, 758, 1281, 1536, 2583, 3094, 5189, 6212, 10403, 12450, 20833, 24928, 41696, 49887, 83424, 99807, 166882, 199649, 333801, 399336, 667641, 798712, 1335323, 1597466, 2670689, 3194976, 5341423, 6389998

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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 1.19..., 1.67...

LINKS

Table of n, a(n) for n=0..40.

Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

EXAMPLE

a(0) = 1, a(1) = 3, a(2) = 3, b(0) = 2, b(1) = 5

a(2) = 2*a(0) + b(0) = 4

Complement: (b(n)) = (2, 5, 6, 7, 8, 9, 10, 12, 13, 15, ... )

MATHEMATICA

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

a[0] = 1; a[1] = 3; b[0] = 2; b[1]=5;

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