cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A363003 Number of integer sequences of length n whose Gilbreath transform is (1, 1, ..., 1).

Original entry on oeis.org

1, 1, 2, 6, 26, 166, 1562, 21614, 438594, 13032614, 566069882
Offset: 0

Views

Author

Pontus von Brömssen, May 13 2023

Keywords

Comments

a(n) is even for all n >= 2, because if the sequence (x_1, ..., x_n) has Gilbreath transform (1, ..., 1), so has the sequence (2 - x_1, ..., 2 - x_n).
Negative terms are permitted.

Examples

			For n = 4, the following 13 sequences, together with the sequences obtained by replacing each term x by 2-x in each of these sequences, have Gilbreath transform (1, 1, 1, 1), so a(4) = 26.
  (1, 2, 0, -4),
  (1, 2, 0, -2),
  (1, 2, 0,  0),
  (1, 2, 0,  2),
  (1, 2, 0,  4),
  (1, 2, 2,  0),
  (1, 2, 2,  2),
  (1, 2, 2,  4),
  (1, 2, 4,  0),
  (1, 2, 4,  2),
  (1, 2, 4,  4),
  (1, 2, 4,  6),
  (1, 2, 4,  8).

Crossrefs

Cf. A080839 (increasing sequences), A363002 (nondecreasing sequences), A363004 (distinct positive integers), A363005 (distinct integers).

A080839 Number of positive increasing integer sequences of length n with Gilbreath transform (that is, the diagonal of leading successive absolute differences) given by {1,1,1,1,1,...}.

Original entry on oeis.org

1, 1, 1, 2, 6, 27, 180, 1786, 26094, 559127, 17535396, 804131875, 53833201737
Offset: 1

Views

Author

John W. Layman, Mar 28 2003

Keywords

Comments

From T. D. Noe, Feb 05 2007: (Start)
The slowest-growing sequence of length n is 1,2,4,6,...,2(n-1). The fastest-growing sequence is 1,2,4,8,...,2^(n-1).
The ratio a(n+1)a(n-1)/a(n)^2 appears to converge to a constant near 1.46, which is the approximate growth rate of A001609. Are the sequences related?
(End)
Also, a(n) is the number of (not necessarily increasing) positive integer sequences of length n-1 with Gilbreath transform (1, ..., 1). - Pontus von Brömssen, May 13 2023

Examples

			The table below shows that {1,2,4,6,10} is one of the 6 sequences of length 5 that satisfy the stated condition:
   1
   2 1
   4 2 1
   6 2 0 1
  10 4 2 2 1

Crossrefs

Cf. A001609, A036262, A363002, A363003, A363004, A363005.
Cf. also A136465, the total number of increasing sequences with the same maximum length. [From Charles R Greathouse IV, Aug 08 2010]

Extensions

More terms from T. D. Noe, Feb 05 2007
Added "positive" to definition. - N. J. A. Sloane, May 13 2023

A363004 Number of sequences of n distinct positive integers whose Gilbreath transform is (1, 1, ..., 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 7, 40, 355, 4819, 99242, 3049155, 138762035
Offset: 0

Views

Author

Pontus von Brömssen, May 13 2023

Keywords

Examples

			For n = 5, the a(5) = 7 sequences are:
  (1, 2, 4, 6,  8),
  (1, 2, 4, 6, 10),
  (1, 2, 4, 8,  6),
  (1, 2, 4, 8, 10),
  (1, 2, 4, 8, 12),
  (1, 2, 4, 8, 14),
  (1, 2, 4, 8, 16).

Crossrefs

Cf. A080839 (increasing sequences), A363002 (nondecreasing sequences), A363003, A363005 (distinct integers).

A363005 Number of sequences of n distinct integers whose Gilbreath transform is (1, 1, ..., 1).

Original entry on oeis.org

1, 1, 2, 4, 12, 56, 416, 4764, 84272, 2278740, 92890636, 5659487836
Offset: 0

Views

Author

Pontus von Brömssen, May 13 2023

Keywords

Comments

a(n) is even for all n >= 2, because if the sequence (x_1, ..., x_n) has Gilbreath transform (1, ..., 1), so has the sequence (2 - x_1, ..., 2 - x_n).
Negative terms are permitted.

Examples

			For n = 4, the following 6 sequences, together with the sequences obtained by replacing each term x by 2-x in each of these sequences, have Gilbreath transform (1, 1, 1, 1), so a(4) = 12.
  (1, 2, 0, -4),
  (1, 2, 0, -2),
  (1, 2, 0,  4),
  (1, 2, 4,  0),
  (1, 2, 4,  6),
  (1, 2, 4,  8).

Crossrefs

Cf. A080839 (increasing sequences), A363002 (nondecreasing sequences), A363003, A363004 (distinct positive integers).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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