A298000 -id:A298000 - cp's OEIS frontend

A298295 Solution a( ) of the complementary equation a(n) = a(0)b(n) + a(1)b(n-1), 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, 13, 16, 19, 22, 25, 28, 31, 34, 38, 43, 47, 52, 56, 61, 65, 70, 74, 79, 83, 88, 92, 97, 101, 106, 109, 113, 118, 121, 124, 128, 133, 136, 140, 145, 148, 151, 155, 160, 163, 167, 172, 175, 178, 182, 187, 190, 194, 199, 202, 205, 209, 214, 217, 221, 226

Offset: 0

Clark Kimberling, Feb 09 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, b(2) = 5, so that a(2) = 13.
Complement: (3,4,5,6,7,8,9,10,11,12,14,15,17,...)

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A298296, A297830, A298000.

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[0]*b[n] + a[1]*b[n - 1]
Table[{a[n],
   b[n + 1] = mex[Flatten[Map[{a[#], b[#]} &, Range[0, n]]], b[n - 0]]}, {n, 2, 1010}];
Table[a[n], {n, 0, 150}]  (* A298295 *)
Table[b[n], {n, 0, 150}]  (* A298296 *)
(* Peter J. C. Moses, Jan 16 2018 *)

A298296 Solution b( ) of the complementary equation a(n) = a(0)b(n) + a(1)b(n-1), 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, 14, 15, 17, 18, 20, 21, 23, 24, 26, 27, 29, 30, 32, 33, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87

Offset: 0

Clark Kimberling, Feb 09 2018

The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values; b(n)-b(n-1) is in {1,2} for all n >= 1.

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

Clark Kimberling, Table of n, a(n) for n = 0..1000
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.

Cf. A298295, A297830, A298000.

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[0]*b[n] + a[1]*b[n - 1]
Table[{a[n], b[n + 1] = mex[Flatten[Map[{a[#], b[#]} &, Range[0, n]]], b[n - 0]]}, {n, 2, 1010}];
Table[a[n], {n, 0, 150}]  (* A298295 *)
Table[b[n], {n, 0, 150}]  (* A298296 *)
(* Peter J. C. Moses, Jan 16 2018 *)

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.

Original entry on oeis.org

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 *)

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 *)

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 *)

A298111 Solution b( ) 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

3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 25, 26, 28, 30, 31, 32, 33, 35, 37, 38, 39, 40, 42, 44, 45, 46, 47, 49, 51, 52, 53, 54, 56, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 75, 76, 78, 79, 81, 82, 83, 84, 86, 87, 88, 90, 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 A298000. See A297830 for a guide to related sequences.

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

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

Cf. A297830, A298000.

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 < 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}]; (* A298000 *)
v = Table[b[n], {n, 0, k}]; (* A298111 *)
Take[u, 50]
Take[v, 50]

cp's OEIS Frontend

A298295 Solution a( ) of the complementary equation a(n) = a(0)*b(n) + a(1)*b(n-1), 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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A298296 Solution b( ) of the complementary equation a(n) = a(0)*b(n) + a(1)*b(n-1), 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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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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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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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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A298111 Solution b( ) 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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A298295 Solution a( ) of the complementary equation a(n) = a(0)b(n) + a(1)b(n-1), 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.

A298296 Solution b( ) of the complementary equation a(n) = a(0)b(n) + a(1)b(n-1), 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.

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.

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.

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.

A298111 Solution b( ) 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.