A297830 -id:A297830 - cp's OEIS frontend

A297999 Solution (a(n)) of the near-complementary equation a(n) = a(1)b(n) - a(0)b(n-1) + 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, 8, 10, 12, 16, 19, 22, 23, 25, 29, 30, 34, 35, 41, 43, 44, 46, 52, 52, 54, 60, 60, 62, 64, 66, 70, 75, 77, 78, 80, 82, 84, 88, 91, 92, 94, 96, 98, 102, 105, 108, 111, 112, 114, 118, 119, 121, 123, 127, 132, 134, 137, 140, 141, 143, 147, 148, 154, 156

Offset: 0

Clark Kimberling, Feb 09 2018

The sequence (a(n)) generated by the equation a(n) = a(1)*b(n) - a(0)*b(n-1) + n, with initial values as shown, includes duplicates; e.g. a(18) = a(19) = 52.  If the duplicates are removed from (a(n)), the resulting sequence and (b(n)) are complementary. Conjectures:
(1) 0 <= a(k) - a(k-1) <= 6 for k>=1;
(2) if d is in {0,1,2,3,4,5,6}, then a(k) = a(k-1) + d for infinitely many k.
***
See A298000 and A297830 for guides to related sequences.

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

Clark Kimberling, Table of n, a(n) for n = 0..2000

Cf. A297997, A298000, A297830, A298110.

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[1]*b[n] - a[0]*b[n - 1] + n;
Table[{a[n], b[n + 1] = mex[Flatten[Map[{a[#], b[#]} &, Range[0, n]]], b[n - 0]]}, {n, 2, 3000}];
Table[a[n], {n, 0, 150}]  (* A297999 *)
Table[b[n], {n, 0, 150}]  (* A298110 *)
(* Peter J. C. Moses, Jan 16 2018 *)

A298110 Solution (b(n)) of the near-complementary equation a(n) = a(1)b(n) - a(0)b(n-1) + 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, 9, 11, 13, 14, 15, 17, 18, 20, 21, 24, 26, 27, 28, 31, 32, 33, 36, 37, 38, 39, 40, 42, 45, 47, 48, 49, 50, 51, 53, 55, 56, 57, 58, 59, 61, 63, 65, 67, 68, 69, 71, 72, 73, 74, 76, 79, 81, 83, 85, 86, 87, 89, 90, 93, 95, 97, 99, 100, 101, 103

Offset: 0

Clark Kimberling, Feb 09 2018

The sequence (a(n)) generated by the equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + n, with initial values as shown, includes duplicates; e.g. a(18) = a(19) = 51.  If the duplicates are removed from (a(n)), the resulting sequence and (b(n)) are complementary. Conjectures:
(1) 1 <= b(k) - b(k-1) <= 3 for k>=1;
(2) if d is in {1,2,3}, then b(k) = b(k-1) + d for infinitely many k.
***
See A298000 and A297830 for guides to related sequences.

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, b(2) = 5, so that a(2) = 8.
Complement: A298110 = (3,4,5,6,7,9,11,13,14,15,17, ...)

Clark Kimberling, Table of n, a(n) for n = 0..2000

Cf. A297999, A298000, A297830.

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[1]*b[n] - a[0]*b[n - 1] + n;
Table[{a[n], b[n + 1] = mex[Flatten[Map[{a[#], b[#]} &, Range[0, n]]], b[n - 0]]}, {n, 2, 3000}];
Table[a[n], {n, 0, 150}]  (* A297999 *)
Table[b[n], {n, 0, 150}]  (* A298110 *)
(* Peter J. C. Moses, Jan 16 2018 *)

A298469 a(n) = a(0)b(n) + a(1)b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2; b(1) = 4 ; b(2) = 5.

Original entry on oeis.org

1, 3, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 66, 73, 77, 82, 89, 93, 98, 105, 109, 114, 121, 125, 130, 137, 141, 146, 153, 157, 162, 169, 173, 178, 185, 189, 194, 201, 205, 210, 217, 221, 226, 233, 237, 242, 249, 253, 257

Offset: 0

Clark Kimberling, Feb 11 2018

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

			a(2) = 1*5 + 3*4 = 17.

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A298338, A298295.

mex[list_, start_] := (NestWhile[# + 1 &, start, MemberQ[list, #] &]);
aCoeffs = {1, 3}; bCoeffs = {2, 4, 5};
Table[a[n - 1] = #[[n]], {n, Length[#]}] &[aCoeffs];
Table[b[n - 1] = #[[n]], {n, Length[#]}] &[bCoeffs];
a[n_] := Hold[Sum[a[z] b[n - z], {z, 0, Length[aCoeffs] - 1}]]
Table[{a[z] = ReleaseHold[a[z]], b[z + 1] =
    mex[Join[Table[a[n], {n, 0, z}], Table[b[n], {n, 0, z}]], 1]}, {z,
    Length[aCoeffs], 1000}];
Table[a[n], {n, 0, 50}]  (* A298469 *)
Table[b[n], {n, 0, 50}]  (* complement *)
(* Peter J. C. Moses, Jan 19 2018 *)

A297835 Solution of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 2*n + 1, 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, 16, 19, 22, 25, 30, 32, 37, 39, 44, 46, 51, 53, 58, 60, 65, 67, 70, 73, 78, 82, 84, 87, 90, 95, 99, 101, 104, 107, 112, 116, 118, 121, 124, 129, 133, 135, 138, 141, 146, 150, 152, 155, 158, 163, 167, 169, 174, 176, 181, 183, 186, 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. See A297830 for a guide to related sequences.

Conjecture: a(n) - (2 +sqrt(2))*n < 7 for n >= 1.

			a(0) = 1, a(1) = 2, b(0) = 3, b(1) = 4, so that a(2) = 10.
Complement: (b(n)) = (3,4,6,7,8,9,11,12,14,15,17,18,20,...)

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

Cf. A297826, A297830.

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 + 1;
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}]  (* A297835 *)

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 *)

A298003 Solution b( ) 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.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 22, 23, 24, 25, 27, 29, 30, 31, 32, 34, 36, 37, 38, 39, 41, 43, 44, 45, 46, 48, 50, 51, 52, 53, 55, 57, 58, 60, 61, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 77, 78, 80, 81, 82, 83, 85, 86, 87, 89, 91, 92, 94

Offset: 0

Clark Kimberling, Feb 08 2018

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

Conjecture: 3/5 < a(n) - n*sqrt(2)*n < 3 for n >= 1.

Clark Kimberling, Table of n, a(n) for n = 0..10000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A297830.

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 < 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}]; (* A297830 *)
v = Table[b[n], {n, 0, k}]; (* A298003 *)
Take[u, 50]
Take[v, 50]

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]

A298005 Solution b( ) 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

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 82

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 A297837. See A297830 for a guide to related sequences.

Conjecture: 7/10 < a(n) - n*L < 3 for n >= 1, where L = -1 + sqrt(5).

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

Cf. A297830, 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] + 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}]; (* A297837 *)
v = Table[b[n], {n, 0, k}]; (* A298005 *)
Take[u, 50]
Take[v, 50]

A298006 Solution b( ) of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 2*n - 1, 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, 9, 10, 12, 13, 15, 16, 18, 19, 20, 21, 23, 25, 26, 27, 28, 30, 32, 33, 34, 35, 37, 39, 40, 41, 42, 44, 46, 47, 49, 50, 52, 53, 54, 55, 57, 58, 59, 61, 63, 64, 66, 67, 69, 70, 71, 72, 74, 75, 76, 78, 80, 81, 83, 84, 86, 87, 88, 89, 91, 92, 93

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 A297831. See A297830 for a guide to related sequences.

Conjecture: 9/10 < a(n) - n*sqrt(2) < 3 for n >= 1.

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

Cf. A297830, A297831.

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 - 1;
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}]; (* A297831 *)
v = Table[b[n], {n, 0, k}]; (* A298006 *)
Take[u, 50]
Take[v, 50]

A298007 Solution b( ) of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 2*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

3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 35, 36, 38, 39, 41, 42, 43, 44, 46, 47, 48, 50, 52, 53, 55, 56, 58, 59, 60, 61, 63, 64, 65, 67, 69, 70, 72, 73, 75, 76, 77, 78, 80, 81, 82, 84, 86, 87, 88, 89, 91, 93, 94

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 A297832. See A297830 for a guide to related sequences.

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

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

Cf. A297830, A297832.

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 - 2;
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}]; (* A297832 *)
v = Table[b[n], {n, 0, k}]; (* A298007 *)
Take[u, 50]
Take[v, 50]

cp's OEIS Frontend

A297999 Solution (a(n)) of the near-complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 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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A298110 Solution (b(n)) of the near-complementary equation a(n) = a(1)*b(n) - a(0)*b(n-1) + 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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A298469 a(n) = a(0)*b(n) + a(1)*b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2; b(1) = 4 ; b(2) = 5.

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A297835 Solution of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n + 1, 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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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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A298003 Solution b( ) 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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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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A298005 Solution b( ) 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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A298006 Solution b( ) of the complementary equation a(n) = a(1)*b(n-1) - a(0)*b(n-2) + 2*n - 1, 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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A297999 Solution (a(n)) of the near-complementary equation a(n) = a(1)b(n) - a(0)b(n-1) + 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.

A298110 Solution (b(n)) of the near-complementary equation a(n) = a(1)b(n) - a(0)b(n-1) + 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.

A298469 a(n) = a(0)b(n) + a(1)b(n-1), where a(0) = 1, a(1) = 3, b(0) = 2; b(1) = 4 ; b(2) = 5.

A297835 Solution of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 2*n + 1, 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.

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.

A298003 Solution b( ) 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.

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.

A298005 Solution b( ) 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.

A298006 Solution b( ) of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 2*n - 1, 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.

A298007 Solution b( ) of the complementary equation a(n) = a(1)b(n-1) - a(0)b(n-2) + 2*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.