A295364 - OEIS

A295364

Solution of the complementary equation a(n) = a(n-1) + a(n-2) + b(n-1)*b(n-2), where a(0) = 1, a(1) = 3, a[2] = 5, b(0) = 2, b(1) = 4, and (a(n)) and (b(n)) are increasing complementary sequences.

1, 3, 5, 32, 79, 167, 318, 575, 1003, 1710, 2869, 4761, 7840, 12841, 20953, 34100, 55395, 89875, 145690, 236027, 382223, 618802, 1001625, 1621077, 2623404, 4245237, 6869453, 11115560, 17985943, 29102526, 47089591

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

OFFSET

0,2

COMMENTS

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A295357 for a guide to related sequences.

LINKS

Table of n, a(n) for n=0..30.

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

FORMULA

a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622).

EXAMPLE

a(0) = 1, a(1) = 3, a(2) = 5, b(0) = 2, b(1) = 4, so that

b(2) = 6 (least "new number")

a(3) = a(2) + a(1) + b(2)*b(1) = 32

Complement: (b(n)) = (2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...)

MATHEMATICA

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