A293316 - OEIS

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A293316

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

1, 3, 7, 15, 28, 50, 87, 147, 245, 404, 662, 1080, 1757, 2854, 4629, 7502, 12151, 19674, 31847, 51544, 83415, 134984, 218425, 353436, 571889, 925355, 1497275, 2422662, 3919970, 6342666, 10262671, 16605373, 26868081, 43473492, 70341612, 113815144, 184156797

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

The complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A293076 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..36.

EXAMPLE

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

a(2) = a(1) + a(0) + b(0) + 1 = 7;

a(3) = a(2) + a(1) + b(1) + 1 = 13.

Complement: (b(n)) = (2,4,5,6,8,9,10,11,12,13,14,16,...).

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 - 2] + 1;