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.

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A238686 Number of compositions c of n such that no three points (i,c_i), (j,c_j), (k,c_k) are collinear, where c_i denotes the i-th part.

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 19, 30, 53, 87, 148, 219, 365, 555, 884, 1379, 2098, 3152, 4865, 7051, 10884, 15681, 23637, 34062, 50336, 72425, 105738, 149781, 217625, 308859, 440889, 623823, 885116, 1241075, 1744784, 2433371, 3401728, 4719635, 6548306, 9035003, 12472106
Offset: 0

Views

Author

Joerg Arndt and Alois P. Heinz, Mar 02 2014

Keywords

Examples

			There are a(6) = 19 such compositions of 6: [6], [5,1], [4,2], [3,3], [2,4], [1,5], [4,1,1], [2,3,1], [1,4,1], [1,3,2], [3,1,2], [2,1,3], [1,1,4], [2,2,1,1], [1,2,2,1], [2,1,2,1], [1,2,1,2], [2,1,1,2], [1,1,2,2].

Links

Crossrefs

Cf. A238687 (the same for partitions).
Cf. A238423, A238432, A238569.

Programs

  • Maple
    b:= proc(n, l) local j, k, m; m:= nops(l);
      for j to m-2 do for k from j+1 to m-1 do
        if (l[m]-l[k])*(k-j)=(l[k]-l[j])*(m-k)
          then return 0 fi od od;
     `if`(n=0, 1, add(b(n-i, [l[], i]), i=1..n))
    end:
a:= n-> b(n, []):
seq(a(n), n=0..20);
  • Mathematica
    b[n_, l_] := Module[{j, k, m = Length[l]}, For[ j = 1, j <= m - 2, j++, For[k = j+1, k <= m - 1 , k++, If[(l[[m]] - l[[k]])*(k - j) == (l[[k]] - l[[j]])*(m - k), Return[0]]]]; If[n == 0, 1, Sum[b[n - i,  Append[l, i]], {i, 1, n}]]];
a[n_] := b[n, {}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

A332668 Number of strict integer partitions of n without three consecutive parts in arithmetic progression.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 11, 15, 20, 19, 26, 31, 34, 41, 50, 53, 67, 78, 84, 99, 120, 130, 154, 177, 193, 226, 262, 291, 332, 375, 419, 479, 543, 608, 676, 765, 859, 961, 1075, 1202, 1336, 1495, 1672, 1854, 2050, 2301, 2536, 2814, 3142, 3448, 3809
Offset: 0

Views

Author

Gus Wiseman, Mar 28 2020

Keywords

Comments

Also the number of strict integer partitions of n whose first differences are an anti-run, meaning there are no adjacent equal differences.

Examples

			The a(1) = 1 through a(10) = 9 partitions (A = 10):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)
                        (41)  (51)  (52)   (62)   (63)   (73)
                                    (61)   (71)   (72)   (82)
                                    (421)  (431)  (81)   (91)
                                           (521)  (621)  (532)
                                                         (541)
                                                         (631)
                                                         (721)

Links

Crossrefs

Anti-run compositions are counted by A003242.
Normal anti-runs of length n + 1 are counted by A005649.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
The non-strict version is A238424.
The version for permutations is A295370.
Anti-run compositions are ranked by A333489.
Cf. A006560, A007862, A238423, A307824, A325328, A325852, A325874, A333195.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MatchQ[Differences[#],{_,x_,x_,_}]&]],{n,0,30}]

A238432 Number of compositions of n avoiding equidistant 3-term arithmetic progressions.

Original entry on oeis.org

1, 1, 2, 3, 7, 13, 22, 41, 74, 133, 233, 400, 714, 1209, 2091, 3591, 6089, 10316, 17477, 29413, 49515, 82474, 137659, 228461, 377936, 623710, 1025445, 1680418, 2746242, 4474654, 7270430, 11774128, 19020802, 30640812, 49222427, 78857338, 126033488, 200872080
Offset: 0

Views

Author

Joerg Arndt and Alois P. Heinz, Mar 01 2014

Keywords

Examples

			The a(5) = 13 such compositions are:
01:  [ 1 1 2 1 ]
02:  [ 1 1 3 ]
03:  [ 1 2 1 1 ]
04:  [ 1 2 2 ]
05:  [ 1 3 1 ]
06:  [ 1 4 ]
07:  [ 2 1 2 ]
08:  [ 2 2 1 ]
09:  [ 2 3 ]
10:  [ 3 1 1 ]
11:  [ 3 2 ]
12:  [ 4 1 ]
13:  [ 5 ]
Note that the first and third composition contain the progression 1,1,1, but not in equidistant positions.

Links

Crossrefs

Cf. A238433 (same for partitions).
Cf. A238569 (compositions avoiding any 3-term arithmetic progression).
Cf. A238423 (compositions avoiding three consecutive parts in arithmetic progression).
Cf. A238686.

Programs

  • Maple
    b:= proc(n, l) local j;
      for j from 2 to iquo(nops(l)+1, 2) do
      if l[1]-l[j]=l[j]-l[2*j-1] then return 0 fi od;
     `if`(n=0, 1, add(b(n-i, [i, l[]]), i=1..n))
    end:
a:= n-> b(n, []):
seq(a(n), n=0..20);
  • Mathematica
    b[n_, l_] := b[n, l] = Module[{j}, For[j = 2, j <= Quotient[Length[l] + 1, 2], j++, If[l[[1]] - l[[j]] == l[[j]] - l[[2*j - 1]], Return[0]]]; If[n == 0, 1, Sum[b[n - i, Prepend[l, i]], {i, 1, n}]]];
a[n_] := b[n, {}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

A333195 Numbers with three consecutive prime indices in arithmetic progression.

Original entry on oeis.org

8, 16, 24, 27, 30, 32, 40, 48, 54, 56, 60, 64, 72, 80, 81, 88, 96, 104, 105, 108, 110, 112, 120, 125, 128, 135, 136, 144, 150, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 210, 216, 220, 224, 232, 238, 240, 243, 248, 250, 256, 264, 270, 272, 273, 280, 288
Offset: 1

Views

Author

Gus Wiseman, Mar 29 2020

Keywords

Comments

Also numbers whose first differences of prime indices do not form an anti-run, meaning there are adjacent equal differences.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of terms together with their prime indices begins:
    8: {1,1,1}          105: {2,3,4}
   16: {1,1,1,1}        108: {1,1,2,2,2}
   24: {1,1,1,2}        110: {1,3,5}
   27: {2,2,2}          112: {1,1,1,1,4}
   30: {1,2,3}          120: {1,1,1,2,3}
   32: {1,1,1,1,1}      125: {3,3,3}
   40: {1,1,1,3}        128: {1,1,1,1,1,1,1}
   48: {1,1,1,1,2}      135: {2,2,2,3}
   54: {1,2,2,2}        136: {1,1,1,7}
   56: {1,1,1,4}        144: {1,1,1,1,2,2}
   60: {1,1,2,3}        150: {1,2,3,3}
   64: {1,1,1,1,1,1}    152: {1,1,1,8}
   72: {1,1,1,2,2}      160: {1,1,1,1,1,3}
   80: {1,1,1,1,3}      162: {1,2,2,2,2}
   81: {2,2,2,2}        168: {1,1,1,2,4}
   88: {1,1,1,5}        176: {1,1,1,1,5}
   96: {1,1,1,1,1,2}    184: {1,1,1,9}
  104: {1,1,1,6}        189: {2,2,2,4}

Links

Crossrefs

Anti-run compositions are counted by A003242.
Normal anti-runs of length n + 1 are counted by A005649.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
These are the Heinz numbers of the partitions *not* counted by A238424.
Permutations avoiding triples in arithmetic progression are A295370.
Strict partitions avoiding triples in arithmetic progression are A332668.
Anti-run compositions are ranked by A333489.
Cf. A006560, A007862, A238423, A307824, A325328, A325849, A325852.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],MatchQ[Differences[primeMS[#]],{_,x_,x_,_}]&]

A333631 Number of permutations of {1..n} with three consecutive terms in arithmetic progression.

Original entry on oeis.org

0, 0, 0, 2, 6, 40, 238, 1760, 14076, 131732, 1308670, 14678452, 176166906, 2317481348, 32416648496, 490915956484, 7846449011500, 134291298372632, 2416652824505150, 46141903780094080, 922528719841017424, 19456439433050482412, 427837767407051523776, 9873256397944571377332
Offset: 0

Views

Author

Gus Wiseman, Mar 31 2020

Keywords

Comments

Also permutations whose second differences have at least one zero.

Examples

			The a(3) = 2 and a(4) = 6 permutations:
  (1,2,3)  (1,2,3,4)
  (3,2,1)  (1,4,3,2)
           (2,3,4,1)
           (3,2,1,4)
           (4,1,2,3)
           (4,3,2,1)

Links

Crossrefs

The complement is counted by A295370.
The version for prime indices is A333195.
Strict partitions with equal differences are A049980.
Partitions with equal differences are A049988.
Compositions without triples in arithmetic progression are A238423.
Partitions without triples in arithmetic progression are A238424.
Strict partitions without triples in arithmetic progression are A332668.
Cf. A000142, A007862, A175342, A325849, A325850.

Programs

  • Mathematica
    Table[Select[Permutations[Range[n]],MatchQ[Differences[#],{_,x_,x_,_}]&]//Length,{n,0,8}]

Formula

a(n) = n! - A295370(n).

Extensions

a(11)-a(21) (using A295370) from Giovanni Resta, Apr 07 2020
a(22)-a(23) (using A295370) from Alois P. Heinz, Jan 27 2024
Previous Showing 11-15 of 15 results.

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