A297836 -id:A297836 - cp's OEIS frontend

A297830 Solution of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

1, 2, 9, 12, 15, 18, 21, 26, 28, 33, 35, 40, 42, 47, 49, 54, 56, 59, 62, 67, 71, 73, 76, 79, 84, 88, 90, 93, 96, 101, 105, 107, 110, 113, 118, 122, 124, 127, 130, 135, 139, 141, 146, 148, 153, 155, 158, 161, 166, 168, 171, 176, 180, 182, 187, 189, 194, 196

Offset: 0

Clark Kimberling, Feb 04 2018

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. Conjecture: a(n) - (2 +sqrt(2))*n < 3 for n >= 1.
Guide to related sequences having initial values a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, where (b(n)) is the increasing sequence of positive integers not in (a(n)):
***
a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n           (a(n)) = A297826; (b(n)) = A297997
a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n         (a(n)) = A297830; (b(n)) = A298003
a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 3*n         (a(n)) = A297836; (b(n)) = A298004
a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*n         (a(n)) = A297837; (b(n)) = A298005
a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1     (a(n)) = A297831; (b(n)) = A298006
a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 2     (a(n)) = A297832; (b(n)) = A298007
a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 3     (a(n)) = A297833; (b(n)) = A298108
a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 4     (a(n)) = A297834; (b(n)) = A298109
a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n + 1     (a(n)) = A297835;
a(n) = a(1)*b(n-1) - a(0)*b(n-2)+floor(5*n/2)  (a(n)) = A297998;
***
For sequences (a(n)) and (b(n)) associated with equations of the form a(n) = a(1)*b(n) - a(0)*b(n-1), see the guide at A297800.

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 9.
Complement: (b(n)) = (3,4,5,6,8,10,11,13,14,16,17,19,...)

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;
a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 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}]  (* A297830 *)

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.

Original entry on oeis.org

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 *)

A297837 Solution of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 4*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

Original entry on oeis.org

1, 2, 13, 18, 23, 28, 33, 38, 43, 48, 53, 60, 64, 69, 74, 81, 85, 90, 95, 102, 106, 111, 116, 123, 127, 132, 137, 144, 148, 153, 158, 165, 169, 174, 179, 186, 190, 195, 200, 207, 211, 216, 221, 228, 232, 237, 242, 247, 252, 259, 263, 268, 275, 279, 284, 289

Offset: 0

Clark Kimberling, Feb 04 2018

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. For a guide to related sequences, see A297830.

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

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 13.
Complement: (b(n)) = (3,4,5,6,7,8,9,10,11,12,14,15,16,17,19,20,...)

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

Cf. A297826, A297830, A297836.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 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}]  (* A297836 *)

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.

Original entry on oeis.org

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

Clark Kimberling, Feb 08 2018

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.

			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...)

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

Cf. A297800, 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] + 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 *)

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.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Feb 08 2018

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).

			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...)

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

Cf. A297800, 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] + 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 *)

A297998 Solution of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + floor(5*n/2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

Original entry on oeis.org

1, 2, 10, 13, 17, 20, 24, 27, 33, 35, 41, 43, 47, 52, 55, 60, 63, 66, 72, 74, 80, 82, 86, 89, 93, 98, 103, 105, 109, 112, 116, 121, 126, 128, 132, 137, 140, 143, 147, 152, 155, 160, 163, 166, 170, 175, 178, 183, 186, 191, 194, 197, 201, 204, 210, 214, 217

Offset: 0

Clark Kimberling, Feb 04 2018

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values.

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 10.
Complement: (b(n)) = (3,4,5,6,7,8,9,11,12,14,15,16,18,19,21,...)

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

Cf. A297826, A297830, A297836.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + Floor[5/2];
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}]  (* A297998 *)

A298004 Solution b( ) of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 3*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, 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, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 39, 40, 42, 43, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 71, 72, 73, 75, 76, 77, 78, 79, 81, 82, 84, 85, 86

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 A297836. See A297830 for a guide to related sequences.

Conjecture: 7/10 < a(n) - n*L < 3 for n >= 1, where L = (-1 + sqrt(13))/2.

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

Cf. A297830, A297836.

a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4;
a[n_] := a[n] = a[1]*b[n - 1] - a[0]*b[n - 2] + 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}]; (* A297836 *)
v = Table[b[n], {n, 0, k}]; (* A298004 *)
Take[u, 50]
Take[v, 50]

cp's OEIS Frontend

A297830 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, 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.

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A297837 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 4*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

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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.

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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.

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A297998 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + floor(5*n/2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

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A298004 Solution b( ) of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 3*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

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A297830 Solution of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 2*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

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.

A297837 Solution of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 4*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

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.

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.

A297998 Solution of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + floor(5*n/2), where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.

A298004 Solution b( ) of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 3*n, where a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, and (b(n)) is the increasing sequence of positive integers not in (a(n)). See Comments.