A293349 - OEIS

A293349

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

1, 3, 8, 18, 35, 64, 112, 192, 322, 534, 878, 1436, 2340, 3804, 6174, 10010, 16219, 26266, 42524, 68831, 111398, 180274, 291719, 472042, 763812, 1235907, 1999774, 3235738, 5235571, 8471370, 13707004, 22178439, 35885511, 58064020, 93949603, 152013697

(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..35.

EXAMPLE

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

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

a(3) = a(2) + a(1) + b(1) + 3 = 18.

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; b[1] = 4;

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