A298002 -id:A298002 - cp's OEIS frontend

A298000 Solution of the complementary equation a(n) = a(1)b(n) - a(0)b(n-1) + 2*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, 10, 13, 16, 19, 22, 27, 29, 34, 36, 41, 43, 48, 50, 55, 57, 60, 63, 68, 72, 74, 77, 80, 85, 89, 91, 94, 97, 102, 106, 108, 111, 114, 119, 123, 125, 128, 131, 136, 140, 142, 147, 149, 154, 156, 159, 162, 167, 169, 172, 177, 181, 183, 188, 190, 195, 197

Offset: 0

Clark Kimberling, Feb 04 2018

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.
Conjectures:  a(n) - (2 +sqrt(2))*n < 4 for n >= 1.  Guide to related sequences having initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, where (b(n)) is the increasing sequence of positive integers not in (a(n)):
***
a(n) = a(1)*b(n) - a(0)*b(n-1) + n     (a(n)) = A297999; (b(n)) = A298110
a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*n   (a(n)) = A298000; (b(n)) = A298111
a(n) = a(1)*b(n) - a(0)*b(n-1) + 3*n   (a(n)) = A298001; (b(n)) = A298112
a(n) = a(1)*b(n) - a(0)*b(n-1) + 4*n   (a(n)) = A298002; (b(n)) = A298113

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 10.
Complement: (b(n)) = (3,4,5,6,8,9,11,12,14,15,17,18,20,...)

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

Cf. A297826, A297836, A297837.

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] + 2 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}]  (* A298000 *)

A298113 Solution b( ) 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.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 40, 41, 42, 43, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81

Offset: 0

Clark Kimberling, Feb 09 2018

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. The solution a( ) is given at A298002. See A297830 for a guide to related sequences.

Conjecture: 1/2 < a(n) - n*sqrt(2) < 3 for n >= 1.

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

Cf. A297830, A298002.

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 < 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}]; (* A298002 *)
v = Table[b[n], {n, 0, k}]; (* A298113 *)
Take[u, 50]
Take[v, 50]

cp's OEIS Frontend

A298000 Solution of the complementary equation a(n) = a(1)b(n) - a(0)b(n-1) + 2*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.

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A298113 Solution b( ) 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.

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cp's OEIS Frontend

A298000 Solution of the complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 2*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.

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A298113 Solution b( ) 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.

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A298000 Solution of the complementary equation a(n) = a(1)b(n) - a(0)b(n-1) + 2*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.

A298113 Solution b( ) 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.