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.

Previous Showing 21-24 of 24 results.

A206556 Number of 6's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 5, 4, 8, 9, 14, 16, 26, 28, 42, 50, 69, 82, 114, 133, 179, 215, 279, 335, 434, 516, 657, 789, 987, 1182, 1473, 1754, 2164, 2583, 3154, 3755, 4567, 5414, 6542, 7753, 9307, 11000, 13158, 15501, 18456, 21712, 25731, 30196, 35677
Offset: 1

Views

Author

Omar E. Pol, Feb 09 2012

Keywords

Comments

Zero together with the first differences of A024790. Also number of occurrences of 6 in all partitions of n that do not contain 1 as a part. For the definition of "last section of n" see A135010. It appears that the sums of six successive terms give the partition numbers A000041.

Crossrefs

Column 6 of A182703 and of A194812.
Cf. A000041, A135010, A138121, A182712-A182714, A206556-A206560.

Programs

  • Sage
    A206556 = lambda n: sum(list(p).count(6) for p in Partitions(n) if 1 not in p)

Formula

It appears that A000041(n) = Sum_{j=1..6} a(n+j), n >= 0.

A206557 Number of 7's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 23, 28, 39, 48, 64, 79, 104, 128, 165, 204, 258, 317, 399, 487, 606, 739, 912, 1105, 1356, 1637, 1994, 2400, 2906, 3485, 4199, 5016, 6015, 7164, 8553, 10151, 12076, 14286, 16930, 19974, 23588, 27749
Offset: 1

Views

Author

Omar E. Pol, Feb 09 2012

Keywords

Comments

Zero together with the first differences of A024791. Also number of occurrences of 7 in all partitions of n that do not contain 1 as a part. For the definition of "last section of n" see A135010. It appears that the sums of seven successive terms give the partition numbers A000041.

Crossrefs

Column 7 of A182703 and of A194812.
Cf. A000041, A135010, A138121, A182712-A182714, A206555, A206556, A206558-A206560.

Programs

  • Sage
    A206557 = lambda n: sum(list(p).count(7) for p in Partitions(n) if 1 not in p)

Formula

It appears that A000041(n) = Sum_{j=1..7} a(n+j), n >= 0.

A207377 Triangle read by rows in which row n lists the parts of the last section of the set of partitions of n in nondecreasing order.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6, 8
Offset: 1

Views

Author

Omar E. Pol, Feb 23 2012

Keywords

Comments

Starting from the first row; it appears that the total numbers of occurrences of k in k successive rows give the sequence A000041. For more information see A182703.

Examples

			Written as a triangle:
1;
1,2;
1,1,3;
1,1,1,2,2,4;
1,1,1,1,1,2,3,5;
1,1,1,1,1,1,1,2,2,2,2,3,3,4,6;
1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,5,7;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,5,6,8;

Links

Crossrefs

Triangle similar to A135010. Mirror of A207378. Row n has length A138137(n). Row sums give A138879. Right border is A000027.
Cf. A000041, A006128, A066186, A138121, A182703, A182712-A182714, A194812.

A194711 Triangle read by rows: T(n,k) = number of partitions or zones in the last section of the set of partitions of n that contains k as a part.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 1, 1, 0, 1, 7, 2, 1, 1, 0, 1, 11, 2, 2, 1, 1, 0, 1, 15, 4, 2, 1, 1, 1, 0, 1
Offset: 1

Views

Author

Omar E. Pol, Feb 10 2012

Keywords

Comments

It appears that row n lists A000041(n)-1 together with the row n-2 of the triangle A116598, if n >= 2.

Examples

			Triangle begins:
1,
1, 1,
2, 0, 1,
3, 1, 0, 1,
5, 1, 1, 0, 1,
7, 2, 1, 1, 0, 1,
11, 2, 2, 1, 1, 0, 1,
15, 4, 2, 1, 1, 1, 0, 1,

Crossrefs

Column 1 is A000041. Columns >= 2 are A002865.
Cf. A027293, A116598, A135010, A138121, A182703, A194812.
Previous Showing 21-24 of 24 results.

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