A293357 - OEIS

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A293357

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

1, 3, 9, 20, 39, 71, 124, 211, 354, 586, 963, 1574, 2564, 4167, 6762, 10962, 17759, 28758, 46557, 75357, 121958, 197361, 319367, 516778, 836197, 1353029, 2189282, 3542369, 5731711, 9274142, 15005917, 24280125, 39286110, 63566305, 102852487, 166418866

(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) + 3 = 9;

a(3) = a(2) + a(1) + b(1) + 4 = 20.

Complement: (b(n)) = (2, 4, 5, 6, 7, 8, 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 + 1;