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.

A183374 T(n,k)=Number of nXk binary arrays with no element equal to the mod 3 sum of its diagonal and antidiagonal neighbors.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 4, 4, 1, 1, 9, 6, 9, 1, 1, 16, 16, 16, 16, 1, 1, 25, 30, 49, 30, 25, 1, 1, 49, 64, 64, 64, 64, 49, 1, 1, 81, 165, 144, 195, 144, 165, 81, 1, 1, 144, 361, 729, 625, 625, 729, 361, 144, 1, 1, 256, 875, 2304, 2580, 3969, 2580, 2304, 875, 256, 1, 1, 441, 2116
Offset: 1

Views

Author

R. H. Hardin Jan 04 2011

Keywords

Comments

Table starts
.1...1....1.....1......1.......1.........1..........1...........1.............1
.1...4....4.....9.....16......25........49.........81.........144...........256
.1...4....6....16.....30......64.......165........361.........875..........2116
.1...9...16....49.....64.....144.......729.......2304........6724.........22500
.1..16...30....64....195.....625......2580.......8836.......34903........148996
.1..25...64...144....625....3969.....22801.....104329......580644.......3763600
.1..49..165...729...2580...22801....285516....1999396....15732530.....141039376
.1..81..361..2304...8836..104329...1999396...22743361...233203441....2719518201
.1.144..875..6724..34903..580644..15732530..233203441..3414837564...58892126329
.1.256.2116.22500.148996.3763600.141039376.2719518201.58892126329.1467194393284

Examples

			Some solutions for 6X5
..0..1..0..0..1....1..1..0..1..0....0..0..1..1..0....0..0..1..1..0
..0..1..0..0..1....0..0..0..1..0....1..1..0..1..0....1..1..0..1..0
..0..0..0..0..1....0..0..1..0..1....0..0..1..0..0....0..0..1..1..0
..1..0..0..0..0....0..1..1..1..0....0..0..1..0..0....0..0..0..0..0
..1..0..0..1..0....0..1..1..1..0....1..1..0..1..0....0..1..0..0..0
..1..0..0..1..0....0..0..1..0..0....0..0..1..1..0....0..1..0..1..1

Links

Crossrefs

Column 2 is A133037(n+6)

A329227 Products of consecutive terms of the Padovan sequence A000931.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 2, 4, 6, 12, 20, 35, 63, 108, 192, 336, 588, 1036, 1813, 3185, 5590, 9804, 17214, 30200, 53000, 93015, 163215, 286440, 502656, 882096, 1547992, 2716504, 4767161, 8365777, 14680890, 25763220, 45211238, 79340228, 139232412, 244335771
Offset: 0

Views

Author

David Nacin, Nov 08 2019

Keywords

Examples

			For n=5, a(5) = A000931(5)*A000931(6) = 1*1.

Links

Crossrefs

Cf. A000931, A133037.

Programs

  • Mathematica
    Times@@@Partition[LinearRecurrence[{0,1,1},{1,0,0},50],2,1] (* Harvey P. Dale, Jul 05 2021 *)
  • Python
    p = lambda x:[1,0,0][x] if x<3 else p(x-2)+p(x-3)
a = lambda x:p(x)*p(x+1)

Formula

a(n) = A000931(n)*A000931(n+1).
a(n+2) = Sum_{i=0..n} A000931(i)*A000931(i+2).
a(n) - a(n-2) - a(n-3) - a(n-4) = A133037(n-2) + A133037(n-3) for n>3.
G.f.: x^5 / ((1 - 2*x + x^2 - x^3)*(1 + x - x^3)) (conjectured). - Colin Barker, Nov 08 2019

A329178 Sum of the products of pairs of Padovan numbers which are two apart, starting from A000931(5).

Original entry on oeis.org

1, 3, 5, 11, 19, 34, 62, 107, 191, 335, 587, 1035, 1812, 3184, 5589, 9803, 17213, 30199, 52999, 93014, 163214, 286439, 502655, 882095, 1547991, 2716503, 4767160, 8365776, 14680889, 25763219, 45211237, 79340227, 139232411, 244335770, 428779502, 752455475
Offset: 0

Views

Author

David Nacin, Nov 08 2019

Keywords

Examples

			For n=3, a(3) = 1*1 + 1*2 + 1*2 + 2*3 = 11.

Crossrefs

Cf. A000931, A133037, A329227.

Programs

  • Python
    p = lambda x:[1, 1, 1][x] if x<3 else p(x-2)+p(x-3)
a = lambda x:sum(p(i)*p(i+2) for i in range(x+1))

Formula

a(n) = Sum_{i=5..n+5} A000931(i)*A000931(i+2).
a(n) = A329227(n+7) - 1.
Conjectures from Colin Barker, Nov 09 2019: (Start)
G.f.: (1 + x - x^2 + x^3 - x^4) / ((1 - x)*(1 - 2*x + x^2 - x^3)*(1 + x - x^3)).
a(n) = 2*a(n-1) - 2*a(n-4) + 2*a(n-5) - 2*a(n-6) + a(n-7) for n>6.
(End)
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.