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 1-9 of 9 results.

A274538 Number of set partitions of [n] such that each element is contained in a block whose index parity coincides with the parity of the element.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 14, 39, 95, 304, 865, 3103, 10038, 39773, 143473, 620382, 2461099, 11504723, 49658054, 249102263, 1159930119, 6205900348, 30959905841, 175763987955, 934068692102, 5602484594053, 31563436487785, 199267671153562, 1185224170637619
Offset: 0

Views

Author

Alois P. Heinz, Jun 27 2016

Keywords

Comments

All odd elements are in blocks with an odd index and all even elements are in blocks with an even index.

Examples

			a(3) = 2: 13|2, 1|2|3.
a(4) = 3: 13|24, 1|24|3, 1|2|3|4.
a(5) = 7: 135|24, 13|24|5, 15|24|3, 1|24|35, 15|2|3|4, 1|2|35|4, 1|2|3|4|5.
a(6) = 14: 135|246, 13|246|5, 13|24|5|6, 15|246|3, 15|24|3|6, 1|246|35, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46, 1|26|3|4|5, 1|2|3|46|5, 1|2|3|4|5|6.

Links

Crossrefs

Row sums of A274537.
Column k=2 of A274835.
Cf. A011655.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, 1, add(
      `if`(irem(j, 2)=t, b(n-1, max(m, j), 1-t), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..30);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j, 2] == t, b[n - 1, Max[m, j], 1 - t], 0], {j, 1, m + 1}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 23 2018, translated from Maple *)

Formula

a(n) = Sum_{k=0..n} A274537(n,k).
a(n) mod 2 = A011655(n) for n>=1.

A274836 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of three.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 9, 18, 33, 89, 215, 481, 1486, 4187, 10974, 37907, 121114, 362953, 1385575, 4924557, 16494655, 68685792, 268113185, 990074770, 4455129525, 18896355932, 76116156873, 367577989487, 1679905933299, 7313571105815, 37669220146964
Offset: 0

Views

Author

Alois P. Heinz, Jul 08 2016

Keywords

Examples

			a(5) = 3: 14|25|3, 1|25|3|4, 1|2|3|4|5.
a(6) = 4: 14|25|36, 1|25|36|4, 1|2|36|4|5, 1|2|3|4|5|6.
a(7) = 9: 147|25|36, 14|25|36|7, 17|25|36|4, 1|25|36|47, 17|2|36|4|5, 1|2|36|47|5, 17|2|3|4|5|6, 1|2|3|47|5|6, 1|2|3|4|5|6|7.

Links

Crossrefs

Column k=3 of A274835.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, 1,
     add(`if`(irem(j-t, 3)=0, b(n-1, max(m, j),
              irem(t+1, 3)), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..35);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j-t, 3] == 0, b[n-1, Max[m, j], Mod[t+1, 3]], 0], {j, 1, m+1}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

A274837 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of four.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 11, 22, 41, 72, 191, 459, 1033, 2209, 6696, 18777, 49526, 124011, 419203, 1329966, 4009931, 11544970, 43203329, 152247581, 511143253, 1644388769, 6707557342, 25952578959, 95992345048, 340793163873, 1501194339387, 6305017609678
Offset: 0

Views

Author

Alois P. Heinz, Jul 08 2016

Keywords

Examples

			a(7) = 4: 15|26|37|4, 1|26|37|4|5, 1|2|37|4|5|6, 1|2|3|4|5|6|7.
a(8) = 5: 15|26|37|48, 1|26|37|48|5, 1|2|37|48|5|6, 1|2|3|48|5|6|7, 1|2|3|4|5|6|7|8.
a(9) = 11: 159|26|37|48, 15|26|37|48|9, 19|26|37|48|5, 1|26|37|48|59, 19|2|37|48|5|6, 1|2|37|48|59|6, 19|2|3|48|5|6|7, 1|2|3|48|59|6|7, 19|2|3|4|5|6|7|8, 1|2|3|4|59|6|7|8, 1|2|3|4|5|6|7|8|9.

Links

Crossrefs

Column k=4 of A274835.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, 1,
     add(`if`(irem(j-t, 4)=0, b(n-1, max(m, j),
              irem(t+1, 4)), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..35);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j - t, 4] == 0, b[n - 1, Max[m, j], Mod[t + 1, 4]], 0], {j, 1, m + 1}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

A274838 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of five.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 13, 26, 49, 88, 151, 397, 951, 2145, 4633, 9643, 28898, 80843, 214126, 542081, 1317924, 4392295, 13871122, 41984457, 122463762, 344409561, 1273659431, 4463980333, 14994032599, 48610148069, 152506484015, 614168698264
Offset: 0

Views

Author

Alois P. Heinz, Jul 08 2016

Keywords

Examples

			a(7) = 3: 16|27|3|4|5, 1|27|3|4|5|6, 1|2|3|4|5|6|7.
a(8) = 4: 16|27|38|4|5, 1|27|38|4|5|6, 1|2|38|4|5|6|7, 1|2|3|4|5|6|7|8.
a(9) = 5: 16|27|38|49|5, 1|27|38|49|5|6, 1|2|38|49|5|6|7, 1|2|3|49|5|6|7|8, 1|2|3|4|5|6|7|8|9.

Links

Crossrefs

Column k=5 of A274835.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, 1,
     add(`if`(irem(j-t, 5)=0, b(n-1, max(m, j),
              irem(t+1, 5)), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..40);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j - t, 5] == 0, b[n - 1, Max[m, j], Mod[t + 1, 5]], 0], {j, 1, m + 1}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

A274839 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of six.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 15, 30, 57, 104, 183, 310, 811, 1939, 4377, 9497, 19987, 40883, 121620, 339817, 902822, 2301883, 5665060, 13489425, 44503335, 140080438, 425111779, 1250942834, 3575716011, 9910354002, 36376567529, 127026151621
Offset: 0

Views

Author

Alois P. Heinz, Jul 08 2016

Keywords

Examples

			a(7) = 2: 17|2|3|4|5|6, 1|2|3|4|5|6|7.
a(8) = 3: 17|28|3|4|5|6, 1|28|3|4|5|6|7, 1|2|3|4|5|6|7|8.
a(9) = 4: 17|28|39|4|5|6, 1|28|39|4|5|6|7, 1|2|39|4|5|6|7|8, 1|2|3|4|5|6|7|8|9.

Links

Crossrefs

Column k=6 of A274835.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, 1,
     add(`if`(irem(j-t, 6)=0, b(n-1, max(m, j),
              irem(t+1, 6)), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..40);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j - t, 6] == 0, b[n - 1, Max[m, j], Mod[t + 1, 6]], 0], {j, 1, m + 1}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

A274840 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of seven.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 17, 34, 65, 120, 215, 374, 629, 1641, 3919, 8849, 19241, 40707, 84203, 170229, 503902, 1407019, 3746718, 9600121, 23815820, 57408783, 134592586, 440661179, 1383544922, 4206645985, 12456581554, 36012285385
Offset: 0

Views

Author

Alois P. Heinz, Jul 08 2016

Keywords

Examples

			a(8) = 2: 18|2|3|4|5|6|7, 1|2|3|4|5|6|7|8.
a(9) = 3: 18|29|3|4|5|6|7, 1|29|3|4|5|6|7|8, 1|2|3|4|5|6|7|8|9.
a(10) = 4: 18|29|3(10)|4|5|6|7, 1|29|3(10)|4|5|6|7|8, 1|2|3(10)|4|5|6|7|8|9, 1|2|3|4|5|6|7|8|9|(10).

Links

Crossrefs

Column k=7 of A274835.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, 1,
     add(`if`(irem(j-t, 7)=0, b(n-1, max(m, j),
              irem(t+1, 7)), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..42);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j - t, 7] == 0, b[n - 1, Max[m, j], Mod[t + 1, 7]], 0], {j, 1, m + 1}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

A274841 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of eight.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 38, 73, 136, 247, 438, 757, 1268, 3303, 7883, 17801, 38745, 82179, 170907, 349341, 700517, 2066512, 5768089, 15386070, 39563059, 98692628, 239843745, 569063602, 1318211431, 4290275275, 13443268926
Offset: 0

Views

Author

Alois P. Heinz, Jul 08 2016

Keywords

Examples

			a(8) = 1: 1|2|3|4|5|6|7|8.
a(9) = 2: 19|2|3|4|5|6|7|8, 1|2|3|4|5|6|7|8|9.
a(10) = 3: 19|2(10)|3|4|5|6|7|8, 1|2(10)|3|4|5|6|7|8|9, 1|2|3|4|5|6|7|8|9|(10).

Links

Crossrefs

Column k=8 of A274835.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, 1,
     add(`if`(irem(j-t, 8)=0, b(n-1, max(m, j),
              irem(t+1, 8)), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..45);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j - t, 8] == 0, b[n - 1, Max[m, j], Mod[t + 1, 8]], 0], {j, 1, m + 1}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

A274842 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of nine.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21, 42, 81, 152, 279, 502, 885, 1524, 2547, 6629, 15815, 35713, 77769, 165155, 344379, 707693, 1434461, 2859871, 8415994, 23485835, 62727630, 161710529, 404995340, 989816263, 2367377650, 5547588797
Offset: 0

Views

Author

Alois P. Heinz, Jul 08 2016

Keywords

Examples

			a(9) = 1: 1|2|3|4|5|6|7|8|9.
a(10) = 2: 1(10)|2|3|4|5|6|7|8|9, 1|2|3|4|5|6|7|8|9|(10).
a(11) = 3: 1(10)|2(11)|3|4|5|6|7|8|9, 1|2(11)|3|4|5|6|7|8|9|(10), 1|2|3|4|5|6|7|8|9|(10)|(11).

Links

Crossrefs

Column k=9 of A274835.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, 1,
     add(`if`(irem(j-t, 9)=0, b(n-1, max(m, j),
              irem(t+1, 9)), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..45);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j - t, 9] == 0, b[n - 1, Max[m, j], Mod[t + 1, 9]], 0], {j, 1, m + 1}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)

A274843 Number of set partitions of [n] such that the difference between each element and its block index is a multiple of ten.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 23, 46, 89, 168, 311, 566, 1013, 1780, 3059, 5106, 13283, 31683, 71545, 155833, 331139, 691387, 1424525, 2902605, 5848135, 11610871, 34108236, 95170569, 254432006, 657159051, 1650540916
Offset: 0

Views

Author

Alois P. Heinz, Jul 08 2016

Keywords

Examples

			a(10) = 1: 1|2|3|4|5|6|7|8|9|(10).
a(11) = 2: 1(11)|2|3|4|5|6|7|8|9|(10), 1|2|3|4|5|6|7|8|9|(10)|(11).

Links

Crossrefs

Column k=10 of A274835.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, 1,
     add(`if`(irem(j-t, 10)=0, b(n-1, max(m, j),
              irem(t+1, 10)), 0), j=1..m+1))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..50);
  • Mathematica
    b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[If[Mod[j - t, 10] == 0, b[n - 1, Max[m, j], Mod[t + 1, 10]], 0], {j, 1, m + 1}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 15 2018, after Alois P. Heinz *)
Showing 1-9 of 9 results.

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