A294417 - OEIS

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A294417

Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1) + b(n-2) - n, where a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4.

1, 3, 8, 17, 32, 57, 99, 168, 280, 462, 757, 1235, 2009, 3262, 5291, 8575, 13889, 22488, 36402, 58916, 95345, 154289, 249663, 403982, 653676, 1057690, 1711399, 2769123, 4480558, 7249719, 11730316, 18980075, 30710432, 49690549, 80401024, 130091617

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OFFSET

0,2

COMMENTS

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A294414 for a guide to related sequences.

Conjecture: a(n)/a(n-1) -> (1 + sqrt(5))/2, the golden ratio.

LINKS

Table of n, a(n) for n=0..35.

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

EXAMPLE

a(0) = 1, a(1) = 3, b(0) = 2, b(1) = 4, so that

a(2) = a(1) + a(0) + b(1) + b(0) - 2 = 8

Complement: (b(n)) = (2, 4, 5, 6, 7, 9, 10, 11, 13, 14,...)

MATHEMATICA

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

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

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