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A298001
Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 3*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.
3
1, 2, 12, 16, 20, 24, 28, 32, 36, 42, 45, 49, 55, 58, 62, 68, 71, 75, 81, 84, 88, 94, 97, 101, 107, 110, 114, 120, 123, 127, 131, 135, 141, 144, 150, 153, 157, 163, 166, 170, 174, 178, 184, 187, 193, 196, 200, 206, 209, 213, 217, 221, 227, 230, 236, 239, 243
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 = (5 + sqrt(13))/2.
LINKS
EXAMPLE
a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, so that a(2) = 12.
Complement: (b(n)) = (3,4,5,6,8,9,10,11,14,15,17,18,19,21...)
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] + 3 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}] (* A298001 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 08 2018
STATUS
approved