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 1-3 of 3 results.

A027596 Sequence satisfies T^2(a)=a, where T is defined below.

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 12, 13, 18, 21, 29, 33, 43, 49, 63, 71, 91, 103, 128, 143, 176, 198, 241, 271, 324, 363, 431, 483, 569, 636, 743, 827, 960, 1068, 1236, 1371, 1573, 1742, 1992, 2203, 2506, 2769, 3135, 3454, 3895, 4290, 4824, 5300, 5935, 6511, 7272, 7967
Offset: 1

Views

Author

N. J. A. Sloane, Dec 11 1999

Keywords

Comments

The partition transform in A007213 expands m=5k as 1/(1-x^m) = 1 + x^m + x^2m + ..., whereas the transform here expands it as 1 + x^m. Thus, if m appears as an argument to the transform, a difference will occur at n=2m due to a difference in coefficient at x^2m. The smallest such m in A007212 (and A027595) is 25, which explains why this sequences differs from A007213 from n=50 onward. - Sean A. Irvine, Nov 10 2019

References

  • S. Viswanath (student, Dept. Math, Indian Inst. Technology, Kanpur) A Note on Partition Eigensequences, preprint, 11/96.

Links

  • M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
  • M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Crossrefs

Cf. A007213, A027595.

Formula

Define T:a->b by: given a1 <= a2 <= ..., let b(n) = number of ways of partitioning n into parts from a1, a2, ... such that parts = 0 mod 5 do not occur more than once.
A027596 = T(A027595). - Sean A. Irvine, Nov 10 2019

Extensions

Revised by Sean A. Irvine, Nov 10 2019

A007213 Oscillates under partition transform.

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 12, 13, 18, 21, 29, 33, 43, 49, 63, 71, 91, 103, 128, 143, 176, 198, 241, 271, 324, 363, 431, 483, 569, 636, 743, 827, 960, 1068, 1236, 1371, 1573, 1742, 1992, 2203, 2506, 2769, 3135, 3454, 3895, 4290, 4824, 5300, 5936, 6512, 7274, 7969
Offset: 1

Views

Author

N. J. A. Sloane, Mira Bernstein

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

  • M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
  • M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
  • N. J. A. Sloane, Transforms.

Crossrefs

Cf. A007212, A027596.

Extensions

More terms from Sean A. Irvine, Nov 10 2019

A027595 Sequence satisfies T^2(a)=a, where T is defined below.

Original entry on oeis.org

1, 2, 2, 4, 4, 6, 7, 11, 12, 16, 18, 25, 28, 36, 41, 53, 59, 73, 82, 102, 115, 138, 155, 186, 209, 246, 275, 324, 363, 420, 468, 541, 605, 691, 768, 877, 976, 1103, 1222, 1380, 1530, 1716, 1895, 2122, 2343, 2609, 2872, 3192, 3514, 3890, 4269, 4716, 5172, 5697
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Georg Fischer observes that A027595 and A007212 appear to be identical - is this a theorem? - N. J. A. Sloane, Oct 17 2018
In reply to the above, no they are different, although the first difference probably does not occur until n=5935. The difference arises due to the handling of multiples of 5 in the respective transforms as explained in A027596. In particular, since A007213(50)=5936 while A027595(50)=5935, this sequence will differ from A007212 at n=5935. - Sean A. Irvine, Nov 10 2019

References

  • S. Viswanath (student, Dept. Math, Indian Inst. Technology, Kanpur) A Note on Partition Eigensequences, preprint, Nov 15 1996.

Links

  • Sean A. Irvine, Table of n, a(n) for n = 1..250
  • M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
  • M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
  • N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)

Crossrefs

Cf. A027595.

Formula

Define T:a->b by: given a1<=a2<=..., let b(n) = number of ways of partitioning n into parts from a1, a2, ... such that parts = 0 mod 5 do not occur more than once.
A027595 = T(A027596). - Sean A. Irvine, Nov 10 2019

Extensions

More terms and offset corrected by Sean A. Irvine, Nov 10 2019
Showing 1-3 of 3 results.

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