A297800 -id:A297800 - 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 *)

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 *)

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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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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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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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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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.

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.