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.

A183009 a(n) = 24*n*p(n) = 24*n*A000041(n).

Original entry on oeis.org

24, 96, 216, 480, 840, 1584, 2520, 4224, 6480, 10080, 14784, 22176, 31512, 45360, 63360, 88704, 121176, 166320, 223440, 300960, 399168, 529056, 692760, 907200, 1174800, 1520064, 1950480, 2498496, 3177240, 4034880, 5090448, 6412032
Offset: 1

Views

Author

Omar E. Pol, Jan 22 2011

Keywords

Comments

a(n) is also the sum of the partition number of n and the "trace" Tr(n) of A183011. a(n) = p(n) + Tr(n).
a(n) is also the number of "sectors" or "half-periods" in all partitions of n in some versions of the shell model of partitions of A135010.

Examples

			The number of partitions of 6 is p(6) = A000041(6) = 11, so a(6) = 24*6*11 = 1584.
Also the trace Tr(6) = A183011(6) = 1573, so a(6) = p(6) + Tr(6) = 11 + 1573 = 1584.

Crossrefs

Partial sums of A183006. Column 24 of A182728.
Cf. A000041, A000796, A008606, A018253, A066186, A135010, A182742, A182743, A183008, A183010, A183011.

Programs

  • Mathematica
    Table[24n*PartitionsP[n],{n,40}] (* Harvey P. Dale, Mar 07 2019 *)

Formula

a(n) = A008606(n)*A000041(n) = 24*A066186(n) = n*A183008(n).
a(n) = p(n) + Tr(n) = A000041(n) + A183011(n).
a(n) = 12*M_2(n) = 24*spt(n) + 12*N_2(n) = 12*A220909(n) = 24*A092269(n) + 12*A220908(n). - Omar E. Pol, Feb 17 2013

A182727 Sum of largest parts of the shell model of partitions with n regions.

Original entry on oeis.org

1, 3, 6, 8, 12, 15, 20, 22, 26, 29, 35, 38, 43, 47, 54, 56, 60, 63, 69, 74, 78, 86, 89, 94, 98, 105, 108, 114, 119, 128, 130, 134, 137, 143, 148, 152, 160, 164, 171, 177, 182, 192, 195, 200, 204, 211, 214, 220, 225, 234, 239, 243, 251, 258, 264, 275, 277, 281
Offset: 1

Views

Author

Omar E. Pol, Jan 25 2011

Keywords

Comments

Question: Is there some connection with fractals?

Examples

			For n = 6 the largest parts of the first six regions of the shell model of partitions are 1, 2, 3, 2, 4, 3, so a(6) = 1+2+3+2+4+3 = 15.
Written as a triangle begins:
1;
3;
6;
8,   12;
15,  20;
22,  26, 29, 35;
38,  43, 47, 54;
56,  60, 63, 69, 74, 78, 86;
89,  94, 98,105,108,114,119,128;
130,134,137,143,148,152,160,164,171,177,182,192;
195,200,204,211,214,220,225,234,239,243,251,258,264,275;

Links

Crossrefs

Partial sums of A141285. Row j has length A187219(j). Right border gives A006128.
Cf. A135010, A141285, A182728, A182729, A186114, A206437, A182181, A182244, A210990.

Formula

a(A000041(n)) = A182181(A000041(n)) = A006128(n). - Omar E. Pol, May 24 2012

Extensions

New name from Omar E. Pol, Apr 26 2012

A182729 Square array T(n,k) = (n*k-1)*A000041(n) read by antidiagonals upwards.

Original entry on oeis.org

0, 2, 1, 6, 6, 2, 15, 15, 10, 3, 28, 35, 24, 14, 4, 55, 63, 55, 33, 18, 5, 90, 121, 98, 75, 42, 22, 6, 154, 195, 187, 133, 95, 51, 26, 7, 240, 330, 300, 253, 168, 115, 60, 30, 8, 378, 510, 506, 405, 319, 203, 135, 69, 34, 9
Offset: 1

Views

Author

Omar E. Pol, Jan 22 2011

Keywords

Examples

			Square array T(n,k) begins:
   0,   1,   2,   3,   4,   5, ...
   2,   6,  10,  14,  18,  22, ...
   6,  15,  24,  33,  42,  51, ...
  15,  35,  55,  75,  95, 115, ...
  28,  63,  98, 133, 168, 203, ...
  55, 121, 187, 253, 319, 385, ...

Crossrefs

Cf. A000041, A135010, A182724, A182728, A183009, A183010, A183011.

Programs

  • Maple
    T:= (n,k)-> (n*k-1)*combinat[numbpart](n):
seq (seq (T(d-k, k), k=1..d-1), d=2..11);
  • Mathematica
    Table[With[{n = m - k + 1}, (n k - 1) PartitionsP[n]], {m, 10}, {k, m}] // Flatten (* Michael De Vlieger, Nov 02 2017 *)

Formula

T(n,1) = A182724(n).
T(n,24) = A183011(n).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.