A298002 - OEIS

A298002

Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 4*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

1, 2, 14, 19, 24, 29, 34, 39, 44, 49, 54, 61, 65, 70, 75, 82, 86, 91, 96, 103, 107, 112, 117, 124, 128, 133, 138, 145, 149, 154, 159, 166, 170, 175, 180, 187, 191, 196, 201, 208, 212, 217, 222, 229, 233, 238, 243, 248, 253, 260, 264, 269, 276, 280, 285, 290

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OFFSET

0,2

COMMENTS

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. See A298000 for a guide to related sequences.

Conjecture: a(n) - n*L < 4 for n >= 1, where L = 3 + sqrt(5).

LINKS

Clark Kimberling, Table of n, a(n) for n = 0..10000

EXAMPLE

a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, so that a(2) = 14.

Complement: (b(n)) = (3,4,5,6,7,8,9,10,11,13,15,17,18,20...)

MATHEMATICA

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;

a[n_] := a[n] = a[1]*b[n] - a[0]*b[n - 1] + 4 n;

j = 1; While[j < 100, k = a[j] - j - 1;

While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k

Table[a[n], {n, 0, k}] (* A298002 *)