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 1-10 of 10 results.

A161943 Number of different equations that can be made by summing numbers from 1 to n and using every number not more than once.

Original entry on oeis.org

0, 0, 1, 3, 7, 17, 43, 108, 273, 708, 1867, 4955, 13256, 35790, 97340, 266240, 732014, 2022558, 5612579, 15634288, 43702232, 122550885, 344661924, 971908613, 2747404212, 7784038617, 22100387619, 62869809733, 179173559128, 511497066733, 1462522478549, 4188024794407
Offset: 1

Views

Author

Tanya Khovanova, Jun 22 2009

Keywords

Comments

The summands of each side are in increasing order and the minimum of all summands is on the left side.

Examples

			a(3) = 1, as the only equation we can make by summing numbers from the set {1, 2, 3} is 1+2=3. a(4) = 3, as we can make three equations: 1+2=3, 1+3=4, 1+4=2+3.

Links

Crossrefs

Column k=2 of A196231.

Programs

  • Maple
    b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
      if n>m then 0
    elif n=m then 1
    else b(n, i-1) +b(abs(n-i), i-1) +b(n+i, i-1)
      fi
    end:
a:= proc(n) option remember;
      `if`(n>2, b(n, n-1)+ a(n-1), 0)
    end:
seq(a(n), n=1..40); # Alois P. Heinz, Aug 31 2009, revised Sep 16 2011
  • Mathematica
    Table[(Length[ Select[Range[0, 3^n - 1], Apply[Plus, Pick[Range[n], PadLeft[IntegerDigits[ #, 3], n], 1]] == Apply[Plus, Pick[Range[n], PadLeft[IntegerDigits[ #, 3], n], 2]] &]] - 1)/ 2, {n, 14}]

Formula

a(n) ~ 3^(n+1) / (4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Sep 11 2014

Extensions

More terms from Alois P. Heinz, Aug 31 2009

A164934 Number of different ways to select 3 disjoint subsets from {1..n} with equal element sum.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 8, 22, 63, 157, 502, 1562, 4688, 15533, 50953, 165054, 562376, 1911007, 6467143, 22447463, 78021923, 271410289, 957082911, 3384587525, 11998851674, 42876440587, 153684701645, 552421854011, 1995875594696, 7231871165277, 26274832876337
Offset: 1

Views

Author

Alois P. Heinz, Aug 31 2009

Keywords

Comments

a(5) = 1, because {1,4}, {2,3}, {5} are disjoint subsets of {1..5} with element sum 5.
a(6) = 3: {1,4}, {2,3}, {5} have element sum 5, {1,5}, {2,4}, {6} have element sum 6, and {1,6}, {2,5}, {3,4} have element sum 7.

Links

Crossrefs

Column k=3 of A196231.
Cf. A161943, A164949, A232534.

Programs

  • Maple
    b:= proc(n, k, i) option remember; local m;
      m:= i*(i+1)/2;
      if k>n then b(k, n, i)
    elif k>=0 and n+k>m or k<0 and n-2*k>m then 0
    elif [n, k, i] = [0, 0, 0] then 1
    else b(n, k, i-1)+b(n+i, k+i, i-1)+b(n-i, k, i-1)+b(n, k-i, i-1)
      fi
    end:
a:= proc(n) option remember;
      `if`(n>2, b(n, n, n-1)/2+ a(n-1), 0)
    end:
seq(a(n), n=1..20);
  • Mathematica
    b[n_, k_, i_] := b[n, k, i] = Module[{m = i*(i+1)/2}, Which[k>n , b[k, n, i], k >= 0 && n+k>m || k<0 && n-2*k > m, 0, {n, k, i} == {0, 0, 0}, 1, True, b[n, k, i-1] + b[n+i, k+i, i-1] + b[n-i, k, i-1] + b[n, k-i, i-1]]]; a[n_] := a[n] = If[n>2, b[n, n, n-1]/2 + a[n-1], 0]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

Formula

Conjecture: a(n) ~ 4^n / (Pi * sqrt(3) * n^3). - Vaclav Kotesovec, Oct 16 2014

A164949 Number of different ways to select 4 disjoint subsets from {1..n} with equal element sum.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 9, 23, 67, 203, 693, 2584, 9929, 37480, 137067, 522854, 2052657, 8199728, 33456333, 137831268, 574295984, 2392149818, 9950364020, 41860671346, 177512155194, 757447761138, 3254519322231, 14049972380612, 60960849334377, 265354255338637
Offset: 1

Views

Author

Alois P. Heinz, Sep 01 2009

Keywords

Examples

			a(7) = 1, because {1,6}, {2,5}, {3,4}, {7} are disjoint subsets of {1..7} with element sum 7.
a(8) = 3: {1,6}, {2,5}, {3,4}, {7} have element sum 7, {1,7}, {2,6}, {3,5}, {8} have element sum 8, and {1,8}, {2,7}, {3,6}, {4,5} have element sum 9.

Crossrefs

Column k=4 of A196231.
Cf. A161943, A164934.

Programs

  • Maple
    b:= proc() option remember; local i, j; `if`(args[1]=0 and args[2]=0 and args[3]=0 and args[4]=0, 1, `if`(add(args[j], j=1..4)> args[5] *(args[5]-1)/2, 0, b(args[j]$j=1..4, args[5]-1)) +add(`if`(args[j] -args[5]<0, 0, b(sort([seq(args[i] -`if`(i=j, args[5], 0), i=1..4)])[], args[5]-1)), j=1..4)) end: a:= n-> add(b(k$4, n), k=7..floor(n*(n+1)/8)) /24: seq(a(n), n=1..20);
  • Mathematica
    b[l_, n_, k_] := b[l, n, k] = Module[{i, j}, If[l == Array[0&, k], 1, If[ Total[l] > n(n-1)/2, 0, b[l, n-1, k]] + Sum[If[l[[j]]-n < 0, 0, b[Sort[ Table[l[[i]] - If[i==j, n, 0], {i, 1, k}]], n-1, k]], {j, 1, k}]]];
T[n_, k_] := Sum[b[Array[t&, k], n, k], {t, 2k-1, Floor[n(n+1)/(2k)]}]/k!;
a[n_] := T[n, 4];
Array[a, 20] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz's Maple code in A196231 *)

A196232 Number of different ways to select 5 disjoint subsets from {1..n} with equal element sum.

Original entry on oeis.org

1, 3, 10, 26, 83, 322, 1182, 3971, 15662, 69371, 328016, 1460297, 6080910, 26901643, 123926071, 598722099, 2838432721, 13220493552, 63710261040, 312134646974, 1554373859464, 7673048166979, 37597940705361, 186986406578372
Offset: 9

Views

Author

Alois P. Heinz, Sep 29 2011

Keywords

Examples

			a(10) = 3: {1,8}, {2,7}, {3,6}, {4,5}, {9} have element sum 9; {1,9}, {2,8}, {3,7}, {4,6}, {10} have element sum 10; {1,10}, {2,9}, {3,8}, {4,7}, {5,6} have element sum 11.

Crossrefs

Column k=5 of A196231. Cf. A000225, A161943, A164934, A164949, A196233, A196234, A196235, A196236, A196237.

Programs

  • Mathematica
    b[l_, n_, k_] := b[l, n, k] = Module[{i, j}, If[l == Array[0 &, k], 1, If[Total[l] > n*(n - 1)/2, 0, b[l, n - 1, k]] + Sum[If[l[[j]] - n < 0, 0, b[Sort[Table[l[[i]] - If[i == j, n, 0], {i, 1, k}]], n-1, k]], {j, 1, k}] ]];
T[n_, k_] := Sum[b[Array[t &, k], n, k], {t, 2*k - 1, Floor[n*(n + 1)/(2*k) ]}]/k!;
a[n_] := T[n, 5];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 9, 25}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)

Extensions

a(26)-a(28) from Alois P. Heinz, Sep 25 2014
a(29)-a(32) from Bert Dobbelaere, Sep 05 2019

A196233 Number of different ways to select 6 disjoint subsets from {1..n} with equal element sum.

Original entry on oeis.org

1, 3, 11, 30, 113, 330, 1284, 5342, 23976, 141836, 604359, 2977297, 15970382, 80990028, 384959038, 1943894348, 10652582085, 53759893907, 292581087499, 1608101020113, 8896321349456, 51394417812545
Offset: 11

Views

Author

Alois P. Heinz, Sep 29 2011

Keywords

Examples

			a(12) = 3: {1,10}, {2,9}, {3,8}, {4,7}, {5,6}, {11} have element sum 11; {1,11}, {2,10}, {3,9}, {4,8}, {5,7}, {12} have element sum 12; {1,12}, {2,11}, {3,10}, {4,9}, {5,8}, {6,7} have element sum 13.

Crossrefs

Column k=6 of A196231. Cf. A000225, A161943, A164934, A164949, A196232, A196234, A196235, A196236, A196237.

Programs

  • Mathematica
    b[l_, n_, k_] := b[l, n, k] = Module[{i, j}, If[l == Array[0&, k], 1, If[Total[l] > n*(n - 1)/2, 0, b[l, n - 1, k]] + Sum[If[l[[j]] - n < 0, 0, b[Sort[Table[l[[i]] - If[i == j, n, 0], {i, 1, k}]], n-1, k]], {j, 1, k}] ]];
T[n_, k_] := Sum[b[Array[t&, k], n, k], {t, 2*k - 1, Floor[n*(n + 1)/(2*k) ]}]/k!;
a[n_] := T[n, 6];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 11, 25}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)

Extensions

a(26) from Alois P. Heinz, Sep 25 2014
a(27)-a(32) from Bert Dobbelaere, Sep 05 2019

A196234 Number of different ways to select 7 disjoint subsets from {1..n} with equal element sum.

Original entry on oeis.org

1, 3, 12, 33, 114, 403, 1618, 8946, 45917, 189428, 979841, 5497818, 31708309, 178006222, 1091681487, 6207647636, 32636979255, 184162388392, 1069147827024, 6446977283374
Offset: 13

Views

Author

Alois P. Heinz, Sep 29 2011

Keywords

Examples

			a(14) = 3:
{1,12}, {2,11}, {3,10}, {4,9}, {5,8}, {6,7}, {13} have element sum 13; {1,13}, {2,12}, {3,11}, {4,10}, {5,9}, {6,8}, {14} have element sum 14; {1,14}, {2,13}, {3,12}, {4,11}, {5,10}, {6,9}, {7,8} have element sum 15.

Crossrefs

Column k=7 of A196231. Cf. A000225, A161943, A164934, A164949, A196232, A196233, A196235, A196236, A196237.

Programs

  • Mathematica
    b[l_, n_, k_] := b[l, n, k] = Module[{i, j}, If[l == Array[0 &, k], 1, If[Total[l] > n*(n - 1)/2, 0, b[l, n - 1, k]] + Sum[If[l[[j]] - n < 0, 0, b[Sort[Table[l[[i]] - If[i == j, n, 0], {i, 1, k}]], n-1, k]], {j, 1, k} ]]];
T[n_, k_] := Sum[b[Array[t&, k], n, k], {t, 2*k - 1, Floor[n*(n+1)/(2*k) ]}]/k!;
a[n_] := T[n, 7];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 13, 25}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)

Extensions

a(26)-a(28) from Alois P. Heinz, Sep 26 2014
a(29)-a(32) from Bert Dobbelaere, Sep 02 2019

A196235 Number of different ways to select 8 disjoint subsets from {1..n} with equal element sum.

Original entry on oeis.org

1, 3, 13, 37, 134, 466, 1916, 9409, 46006, 255714, 1525052, 9524779, 58944302, 355219704, 2315784192, 14568780212, 97993669291, 619342933593
Offset: 15

Views

Author

Alois P. Heinz, Sep 29 2011

Keywords

Examples

			a(16) = 3: {1,14}, {2,13}, {3,12}, {4,11}, {5,10}, {6,9}, {7,8}, {15} have element sum 15; {1,15}, {2,14}, {3,13}, {4,12}, {5,11}, {6,10}, {7,9}, {16} have element sum 16; {1,16}, {2,15}, {3,14}, {4,13}, {5,12}, {6,11}, {7,10}, {8,9} have element sum 17.

Crossrefs

Column k=8 of A196231. Cf. A000225, A161943, A164934, A164949, A196232, A196233, A196234, A196236, A196237.

Programs

  • Mathematica
    b[l_, n_, k_] := b[l, n, k] = Module[{i, j}, If[l == Array[0 &, k], 1, If[Total[l] > n*(n - 1)/2, 0, b[l, n - 1, k]] + Sum[If[l[[j]] - n < 0, 0, b[Sort[Table[l[[i]] - If[i == j, n, 0], {i, 1, k}]], n - 1, k]], {j, 1, k}]]];
T[n_, k_] := Sum[b[Array[t &, k], n, k], {t, 2*k - 1, Floor[n*(n + 1)/(2*k) ]}]/k!;
a[n_] := T[n, 8];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 15, 25}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)

Extensions

a(27)-a(28) from Alois P. Heinz, Nov 05 2014
a(29)-a(32) from Bert Dobbelaere, Sep 01 2019

A196236 Number of different ways to select 9 disjoint subsets from {1..n} with equal element sum.

Original entry on oeis.org

1, 3, 14, 40, 156, 554, 2369, 11841, 60654, 498320, 2987689, 15177178, 96041346, 656938806, 4640699138, 31263742313, 221075005249
Offset: 17

Views

Author

Alois P. Heinz, Sep 29 2011

Keywords

Examples

			a(18) = 3: {1,16}, {2,15}, {3,14}, {4,13}, {5,12}, {6,11}, {7,10}, {8,9}, {17} have element sum 17; {1,17}, {2,16}, {3,15}, {4,14}, {5,13}, {6,12}, {7,11}, {8,10}, {18} have element sum 18; {1,18}, {2,17}, {3,16}, {4,15}, {5,14}, {6,13}, {7,12}, {8,11}, {9,10} have element sum 19.

Crossrefs

Column k=9 of A196231. Cf. A000225, A161943, A164934, A164949, A196232, A196233, A196234, A196235, A196237.

Programs

  • Mathematica
    b[l_, n_, k_] := b[l, n, k] = Module[{i, j}, If[l == Array[0 &, k], 1, If[Total[l] > n*(n - 1)/2, 0, b[l, n - 1, k]] + Sum[If[l[[j]] - n < 0, 0, b[Sort[Table[l[[i]] - If[i == j, n, 0], {i, 1, k}]], n - 1, k]], {j, 1, k}]]];
T[n_, k_] := Sum[b[Array[t &, k], n, k], {t, 2*k - 1, Floor[n*(n+1)/(2*k) ]}]/k!;
a[n_] := T[n, 9];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 17, 25}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)

Extensions

a(29) from Alois P. Heinz, Nov 05 2014
a(30)-a(33) from Bert Dobbelaere, Sep 02 2019

A196237 Number of different ways to select 10 disjoint subsets from {1..n} with equal element sum.

Original entry on oeis.org

1, 3, 15, 44, 179, 741, 2989, 13932, 79433, 456134, 3096812, 21083037, 151022325, 1119202826, 8627014654
Offset: 19

Views

Author

Alois P. Heinz, Sep 29 2011

Keywords

Examples

			a(20) = 3: {1,18}, {2,17}, {3,16}, {4,15}, {5,14}, {6,13}, {7,12}, {8,11}, {9,10}, {19} have element sum 19; {1,19}, {2,18}, {3,17}, {4,16}, {5,15}, {6,14}, {7,13}, {8,12}, {9,11}, {20} have element sum 20; {1,20}, {2,19}, {3,18}, {4,17}, {5,16}, {6,15}, {7,14}, {8,13}, {9,12}, {10,11} have element sum 21.

Crossrefs

Column k=10 of A196231. Cf. A000225, A161943, A164934, A164949, A196232, A196233, A196234, A196235, A196236.

Programs

  • Mathematica
    b[l_, n_, k_] := b[l, n, k] = Module[{i, j}, If[l == Array[0 &, k], 1, If[Total[l] > n*(n - 1)/2, 0, b[l, n - 1, k]] + Sum[If[l[[j]] - n < 0, 0, b[Sort[Table[l[[i]] - If[i == j, n, 0], {i, 1, k}]], n - 1, k]], {j, 1, k}]]];
T[n_, k_] := Sum[b[Array[t &, k], n, k], {t, 2*k - 1, Floor[n*(n + 1)/(2*k) ]}]/k!;
a[n_] := T[n, 10];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 19, 30}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)

Extensions

a(31)-a(33) from Bert Dobbelaere, Sep 02 2019

A196534 Number of different ways to select disjoint nonempty subsets from {1..n} with equal element sum.

Original entry on oeis.org

1, 3, 8, 18, 39, 83, 179, 388, 857, 1914, 4494, 10844, 26923, 70645, 192297, 538646, 1579602, 4793718, 15010425, 48941642, 164010913, 566065123, 2025354291, 7450901462, 27986863322, 107940691328
Offset: 1

Views

Author

Alois P. Heinz, Oct 03 2011

Keywords

Comments

A000225(n) <= a(n) <= A058692(n+1).

Examples

			a(3) = 8: {{1}}, {{2}}, {{3}}, {{1,2}}, {{1,3}}, {{2,3}}, {{1,2,3}}, {{1,2},{3}}. Element sums are 1, 2, 3, 3, 4, 5, 6, and 3, respectively.

Crossrefs

Row sums of A196231. Cf. A000225, A161943, A164934, A164949, A196232, A196233, A196234, A196235, A196236, A196237, A058692.

Programs

  • Maple
    b:= proc(l, n, k) option remember; local i, j; `if`(l=[0$k], 1, `if`(add(j, j=l)>n*(n-1)/2, 0, b(l, n-1, k))+ add(`if`(l[j]-n<0, 0, b(sort([seq(l[i] -`if`(i=j, n, 0), i=1..k)]), n-1, k)), j=1..k)) end: a:= n-> add(add(b([t$k], n, k), t=2*k-1..floor(n*(n+1)/(2*k)))/k!, k=1..n): seq(a(n), n=1..15);
  • Mathematica
    b[l_, n_, k_] := b[l, n, k] = If[l == Array[0&, k], 1, If[Total[l] > n*(n-1)/2, 0, b[l, n-1, k]] + Sum[If[l[[j]]-n < 0, 0, b[Sort[Table[ l[[i]] - If[i == j, n, 0], {i, 1, k}]], n-1, k]], {j, 1, k}]];
a[n_] := Sum[Sum[b[Array[t&, k], n, k], {t, 2*k-1, Floor[n*(n+1)/(2*k)]} ]/k!, {k, 1, Ceiling[n/2]}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 25}] (* Jean-François Alcover, Jun 01 2022, after Alois P. Heinz *)

Extensions

a(26) from Alois P. Heinz, Oct 20 2014
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