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 1-5 of 5 results.

A303586 Duplicate of A211694.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 11, 23, 47, 103, 226, 518, 1200, 2867, 6946, 17234, 43393, 111419, 290242, 768901, 2065172, 5630083, 15549403, 43527487, 123343911, 353864422, 1026935904, 3014535166, 8945274505, 26829206798, 81293234754, 248805520401, 768882019073, 2398686176048, 7552071250781
Offset: 0

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Author

Keywords

Links

A211700 T(n,k)=Number of nonnegative integer arrays of length n+2k-2 with new values introduced in order 0 upwards and every value appearing only in runs of at least k.

Original entry on oeis.org

1, 1, 2, 1, 2, 5, 1, 2, 3, 15, 1, 2, 3, 6, 52, 1, 2, 3, 4, 11, 203, 1, 2, 3, 4, 7, 23, 877, 1, 2, 3, 4, 5, 12, 47, 4140, 1, 2, 3, 4, 5, 8, 19, 103, 21147, 1, 2, 3, 4, 5, 6, 13, 33, 226, 115975, 1, 2, 3, 4, 5, 6, 9, 20, 59, 518, 678570, 1, 2, 3, 4, 5, 6, 7, 14, 29, 102, 1200, 4213597, 1, 2, 3, 4
Offset: 1

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Author

R. H. Hardin, Apr 19 2012

Keywords

Examples

			Table starts
..........1.....1....1...1...1...1..1..1..1..1..1..1..1..1
..........2.....2....2...2...2...2..2..2..2..2..2..2..2..2
..........5.....3....3...3...3...3..3..3..3..3..3..3..3..3
.........15.....6....4...4...4...4..4..4..4..4..4..4..4..4
.........52....11....7...5...5...5..5..5..5..5..5..5..5..5
........203....23...12...8...6...6..6..6..6..6..6..6..6..6
........877....47...19..13...9...7..7..7..7..7..7..7..7..7
.......4140...103...33..20..14..10..8..8..8..8..8..8..8..8
......21147...226...59..29..21..15.11..9..9..9..9..9..9..9
.....115975...518..102..45..30..22.16.12.10.10.10.10.10.10
.....678570..1200..182..73..41..31.23.17.13.11.11.11.11.11
....4213597..2867..334.118..59..42.32.24.18.14.12.12.12.12
...27644437..6946..608.185..89..55.43.33.25.19.15.13.13.13
..190899322.17234.1121.294.136..75.56.44.34.26.20.16.14.14
.1382958545.43393.2109.480.205.107.71.57.45.35.27.21.17.15
All solutions for n=5 k=4
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..0....0....0....0....0
..1....0....0....0....0
..1....0....0....0....1
..1....0....1....0....1
..1....0....1....1....1
..1....0....1....1....1
..1....0....1....1....1
..1....0....1....1....1

Links

Crossrefs

Cf. A000110 (column 1), A211694 (column 2), A211695 (column 3).

A363181 Number of permutations p of [n] such that for each i in [n] we have: (i>1) and |p(i)-p(i-1)| = 1 or (i

Original entry on oeis.org

1, 0, 2, 2, 8, 14, 54, 128, 498, 1426, 5736, 18814, 78886, 287296, 1258018, 4986402, 22789000, 96966318, 461790998, 2088374592, 10343408786, 49343711666, 253644381032, 1268995609502, 6756470362374, 35285321738624, 194220286045506, 1054759508543554
Offset: 0

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Author

Alois P. Heinz, May 19 2023

Keywords

Comments

Number of permutations p of [n] such that each element in p has at least one neighbor whose value is smaller or larger by one.
Number of permutations of [n] having n occurrences of the 1-box pattern.

Examples

			a(0) = 1: (), the empty permutation.
a(1) = 0.
a(2) = 2: 12, 21.
a(3) = 2: 123, 321.
a(4) = 8: 1234, 1243, 2134, 2143, 3412, 3421, 4312, 4321.
a(5) = 14: 12345, 12354, 12543, 21345, 21543, 32145, 32154, 34512, 34521, 45123, 45321, 54123, 54312, 54321.
a(6) = 54: 123456, 123465, 123654, 124356, 124365, 125634, 125643, 126534, 126543, 213456, 213465, 214356, 214365, 215634, 215643, 216534, 216543, 321456, 321654, 341256, 341265, 342156, 342165, 345612, 345621, 346512, 346521, 431256, 431265, 432156, 432165, 435612, 435621, 436512, 436521, 456123, 456321, 561234, 561243, 562134, 562143, 563412, 563421, 564312, 564321, 651234, 651243, 652134, 652143, 653412, 653421, 654123, 654312, 654321.

Links

Crossrefs

Main diagonal of A346462.
Cf. A002464, A003274, A011655, A095816, A127697, A179957, A229730, A271212, A211694, A333833, A363236.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<4, [1, 0, 2$2][n+1],
      3/2*a(n-1)+(n-3/2)*a(n-2)-(n-5/2)*a(n-3)+(n-4)*a(n-4))
    end:
seq(a(n), n=0..30);

Formula

a(n) = A346462(n,n).
a(n)/2 mod 2 = A011655(n-1) for n>=1.
a(n) ~ sqrt(Pi) * n^((n+1)/2) / (2 * exp(n/2 - sqrt(n)/2 + 7/16)) * (1 - 119/(192*sqrt(n))). - Vaclav Kotesovec, May 26 2023

A303587 Number of partitions of n that contain exactly one isolated singleton.

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 5, 12, 24, 56, 123, 292, 682, 1667, 4079, 10288, 26159, 68026, 178823, 478659, 1296271, 3564911, 9919320, 27978084, 79816424, 230520511, 673071482, 1987599262, 5930739339, 17883932293, 54464027956, 167512285647, 520076498672, 1629804156975
Offset: 1

Views

Author

N. J. A. Sloane, May 19 2018

Keywords

Links

Crossrefs

Cf. A211694, A303588.

Programs

  • Maple
    f:=proc(n,r) local j;
add(combinat:-bell(j-1)*binomial(n-j-1,j-r-1),j=1..floor((n+r)/2));
end;
[seq(f(n,1),n=1..40)];

A303588 Number of partitions of n that contain exactly two isolated singletons.

Original entry on oeis.org

1, 0, 2, 2, 7, 12, 32, 67, 169, 390, 985, 2412, 6209, 15871, 41867, 110797, 299836, 817612, 2268640, 6354409, 18058764, 51838340, 150704087, 442550971, 1314527780, 3943140077, 11953192954, 36580095663, 113048257691, 352564213025, 1109727658303
Offset: 2

Views

Author

N. J. A. Sloane, May 19 2018

Keywords

Links

Crossrefs

Cf. A211694, A303587.

Programs

  • Maple
    f:=proc(n,r) local j;
add(combinat:-bell(j-1)*binomial(n-j-1,j-r-1),j=1..floor((n+r)/2));
end;
[seq(f(n,2),n=2..40)];
Showing 1-5 of 5 results.

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

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