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 1-6 of 6 results.

A182722 a(n) = A005291(n+1)-A182712(n).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 14, 26, 36, 60, 83, 128, 175, 261, 351, 504, 674, 943, 1247, 1711, 2243
Offset: 0

Views

Author

Omar E. Pol, Nov 28 2010, Nov 29 2010

Keywords

Comments

The difference between two apparently unrelated sequences which happen to have the same initial terms. - N. J. A. Sloane, Dec 01 2010

Crossrefs

Cf. A005291, A087787, A100818, A135010, A138121, A182703, A182712.

Formula

a(n) = A005291(n+1)-A182712(n)

A182725 Bisection of A005291.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 28, 51, 92, 163, 285, 490, 833, 1396, 2313, 3789
Offset: 1

Views

Author

Omar E. Pol, Dec 02 2010

Keywords

Comments

This sequence contains the string "1, 2, 4, 8, 15, 28, 51, 92" the same as A005682 and A029907. The next terms is similar. I think that A005291 is a sequence formed from two similar mechanisms but not from only one.

Crossrefs

Cf. A005291, A182712, A182722, A182726.

A182726 Bisection of A005291.

Original entry on oeis.org

0, 0, 1, 3, 7, 15, 30, 58, 108, 196, 348, 605, 1034, 1740, 2887, 4730
Offset: 1

Views

Author

Omar E. Pol, Dec 02 2010

Keywords

Crossrefs

Cf. A005291, A182712, A182722, A182725.

A195308 a(n) = A005291(n) + A005291(n+1).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 58, 81, 109, 150, 200, 271, 359, 481, 633, 838, 1095, 1438, 1867, 2430, 3136, 4053, 5200, 6676, 8519, 10871, 13802, 17514, 22129, 27940, 35141, 44155, 55299, 69179, 86286, 107495, 133562, 165744, 205188, 253691, 312975, 385619
Offset: 1

Views

Author

Omar E. Pol, Feb 03 2012

Keywords

Comments

This sequence arises from A005291 in the same way as A000041 arises from A182712.
Observation: a(3)..a(13) coincide with a sequence related to Stirling's numbers from the Jordan's book.

References

  • Charles Jordan, Calculus of finite differences, Chelsea Publishing Co., 1965, chapter IV, pp. 153-155.

Crossrefs

Cf. A000041, A005291, A087787, A100818, A182712, A194439, A195017.

Formula

a(n) = A000041(n-2), 2 <= n <= 11. - Omar E. Pol, Feb 24 2013

Extensions

More terms from Amiram Eldar, May 17 2025

A182712 Number of 2's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409
Offset: 0

Views

Author

Omar E. Pol, Nov 28 2010

Keywords

Comments

Essentially the same as A087787 but here a(n) is the number of 2's in the last section of n, not n-2. See also A100818.
Note that a(1)..a(11) coincide with a(2)..a(12) of A005291.
Also number of 2's in all partitions of n that do not contain 1's as a part, if n >= 1. Also 0 together with the first differences of A024786. - Omar E. Pol, Nov 13 2011
Also number of 2's in the n-th section of the set of partitions of any integer >= n. For the definition of "section" see A135010. - Omar E. Pol, Dec 01 2013

Examples

			a(6) = 4 counts the 2's in 6 = 4+2 = 2+2+2. The 2's in 6 = 3+2+1 = 2+2+1+1 = 2+1+1+1+1 do not count. - _Omar E. Pol_, Nov 13 2011
From _Omar E. Pol_, Oct 27 2012: (Start)
----------------------------------
Last section               Number
of the set of                of
partitions of 6             2's
----------------------------------
6 .......................... 0
3 + 3 ...................... 0
4 + 2 ...................... 1
2 + 2 + 2 .................. 3
.   1 ...................... 0
.       1 .................. 0
.       1 .................. 0
.           1 .............. 0
.           1 .............. 0
.               1 .......... 0
.                   1 ...... 0
---------------------------------
.   8 - 4 =                  4
.
In the last section of the set of partitions of 6 the difference between the sum of the second column and the sum of the third column is 8 - 4 = 4, the same as the number of 2's, so a(6) = 4 (see also A024786).
(End)

Links

Crossrefs

Column 2 of A194812.
Cf. A005291, A087787, A100818, A135010, A138121, A182703, A182713, A182714.

Programs

  • Mathematica
    Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 2], {n, 0, 49}] (* Robert Price, May 15 2020 *)
  • Sage
    A182712 = lambda n: sum(list(p).count(2) for p in Partitions(n) if 1 not in p) # Omar E. Pol, Nov 13 2011

Formula

It appears that A000041(n) = a(n+1) + a(n+2), n >= 0. - Omar E. Pol, Feb 04 2012
G.f.: (x^2/(1 + x))*Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Jun 02 2018

A259965 Triangle read by rows: representation degeneracies of bosonic strings.

Original entry on oeis.org

1, -1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 2, 0, 1, 2, 1, 3, 2, 2, 0, 1, 0, 4, 3, 4, 2, 2, 0, 1, 3, 3, 7, 4, 5, 2, 2, 0, 1, 1, 8, 6, 10, 5, 5, 2, 2, 0, 1, 5, 7, 14, 10, 11, 6, 5, 2, 2, 0, 1, 2, 15, 15, 19, 13, 12, 6, 5, 2, 2, 0, 1, 10, 15, 26, 23, 24, 14, 13, 6, 5, 2, 2, 0, 1, 5, 28, 31, 38, 29, 27, 15, 13, 6, 5, 2, 2, 0, 1, 16, 30, 51, 47, 48, 34, 28, 16, 13, 6, 5, 2, 2, 0, 1
Offset: 0

Views

Author

N. J. A. Sloane, Jul 11 2015

Keywords

Examples

			Triangle begins:
1,
-1,1,
0,0,1,
0,1,0,1,
1,0,2,0,1,
0,2,1,2,0,1,
2,1,3,2,2,0,1,
0,4,3,4,2,2,0,1,
3,3,7,4,5,2,2,0,1,
1,8,6,10,5,5,2,2,0,1,
5,7,14,10,11,6,5,2,2,0,1,
2,15,15,19,13,12,6,5,2,2,0,1,
10,15,26,23,24,14,13,6,5,2,2,0,1,
5,28,31,38,29,27,15,13,6,5,2,2,0,1,
16,30,51,47,48,34,28,16,13,6,5,2,2,0,1,
...

References

  • T. L. Curtright and C. B. Thorn, Symmetry patterns in the mass spectra of dual string models, Nuclear Phys. B 274 (1986), 520-588. See Table 6b.

Crossrefs

The leading columns are A005290, A005291, A005292, A005293, A005294.
Showing 1-6 of 6 results.

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

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