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A298112
Solution b( ) 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.
2
3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 72, 73, 74, 76, 77, 78, 79, 80, 82, 83, 85, 86
OFFSET
0,1
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The solution a( ) is given at A298001. See A297830 for a guide to related sequences.
Conjecture: 2/5 < a(n) - n*sqrt(2) < 3 for n >= 1.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
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 < 80000, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++]; k
u = Table[a[n], {n, 0, k}]; (* A298001 *)
v = Table[b[n], {n, 0, k}]; (* A298112 *)
Take[u, 50]
Take[v, 50]
CROSSREFS
Sequence in context: A321153 A304814 A052404 * A026417 A026421 A026481
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 09 2018
STATUS
approved