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 1-4 of 4 results.

A325589 Number of compositions of n whose circular differences are all 1 or -1.

Original entry on oeis.org

0, 0, 2, 0, 2, 2, 2, 4, 4, 2, 8, 6, 8, 10, 12, 16, 18, 20, 28, 34, 42, 48, 62, 78, 92, 112, 146, 174, 216, 264, 326, 412, 500, 614, 770, 944, 1166, 1444, 1784, 2214, 2730, 3366, 4182, 5164, 6386, 7898, 9770, 12098, 14950, 18488, 22894, 28312, 35020, 43330, 53606
Offset: 1

Views

Author

Gus Wiseman, May 11 2019

Keywords

Comments

A composition of n is a finite sequence of positive integers summing to n.
The circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).

Examples

			The a(3) = 2 through a(11) = 8 compositions (empty columns not shown):
  (12)  (23)  (1212)  (34)  (1232)  (45)      (2323)  (56)
  (21)  (32)  (2121)  (43)  (2123)  (54)      (3232)  (65)
                            (2321)  (121212)          (121232)
                            (3212)  (212121)          (123212)
                                                      (212123)
                                                      (212321)
                                                      (232121)
                                                      (321212)

Links

Crossrefs

Cf. A000079, A008965, A034297, A173258, A325553, A325558, A325590, A325591.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ[1,##]&@@Abs[Differences[Append[#,First[#]]]]&]],{n,15}]
  • PARI
    step(R,n,s)={matrix(n, n, i, j, if(i>j, if(j>s, R[i-j, j-s]) + if(j+s<=n, R[i-j, j+s])) )}
a(n)={sum(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)==1), t=0); while(R, R=step(R,n,1); t+=R[n,k]); t)} \\ Andrew Howroyd, Aug 23 2019

Extensions

Terms a(26) and beyond from Andrew Howroyd, Aug 23 2019

A325590 Number of necklace compositions of n with circular differences all equal to 1 or -1.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 3, 3, 4, 4, 5, 7, 6, 7, 10, 10, 11, 15, 16, 18, 23, 25, 32, 38, 43, 53, 64, 73, 89, 108, 131, 153, 188, 223, 272, 329, 395, 475, 583, 697, 848, 1027, 1247, 1506, 1837, 2223, 2708, 3282, 3993, 4848, 5913, 7175, 8745, 10640
Offset: 1

Views

Author

Gus Wiseman, May 12 2019

Keywords

Comments

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.
The circular differences of a sequence c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2).
Up to rotation, a(n) is the number of ways to arrange positive integers summing to n in a circle such that adjacent parts differ by 1 or -1.

Examples

			The first 16 terms count the following compositions:
   3: (12)
   5: (23)
   6: (1212)
   7: (34)
   8: (1232)
   9: (45)
   9: (121212)
  10: (2323)
  11: (56)
  11: (121232)
  12: (2343)
  12: (12121212)
  13: (67)
  13: (123232)
  14: (3434)
  14: (12121232)
  15: (78)
  15: (123432)
  15: (232323)
  15: (1212121212)
  16: (3454)
  16: (12321232)
  16: (12123232)
The a(21) = 7 necklace compositions:
  (10,11)
  (2,3,4,5,4,3)
  (3,4,3,4,3,4)
  (1,2,1,2,1,2,3,4,3,2)
  (1,2,3,2,1,2,3,2,3,2)
  (1,2,1,2,3,2,3,2,3,2)
  (1,2,1,2,1,2,1,2,1,2,1,2,1,2)

Links

Crossrefs

Cf. A000079, A000740, A008965, A034297, A173258, A325556, A325588, A325589, A325591.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],neckQ[#]&&(SameQ[1,##]&@@Abs[Differences[Append[#,First[#]]]])&]],{n,15}]
  • PARI
    step(R,n,s)={matrix(n,n,i,j, if(i>j, if(j>s, R[i-j, j-s]) + if(j+s<=n, R[i-j, j+s])) )}
a(n)={sum(k=1, n, my(R=matrix(n,n,i,j,i==j&&abs(i-k)==1), t=0, m=1); while(R, R=step(R,n,1); m++; t+=sumdiv(n, d, R[d,k]*d*eulerphi(n/d))/m ); t/n)} \\ Andrew Howroyd, Aug 23 2019

Extensions

a(26)-a(40) from Lars Blomberg, Jun 11 2019
Terms a(41) and beyond from Andrew Howroyd, Aug 23 2019

A325551 Number of compositions of n with distinct circular differences.

Original entry on oeis.org

1, 1, 3, 6, 11, 8, 26, 50, 79, 121, 195, 265, 478, 742, 1269, 1914, 2929, 4462, 6825, 10309, 16324, 24633, 37213, 56828, 84482
Offset: 1

Views

Author

Gus Wiseman, May 10 2019

Keywords

Comments

A composition of n is a finite sequence of positive integers summing to n.
The circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2), which are distinct, so (1,2,1,3) is counted under a(7).

Examples

			The a(1) = 1 through a(7) = 26 compositions:
  (1)  (2)  (3)   (4)    (5)    (6)    (7)
            (12)  (13)   (14)   (15)   (16)
            (21)  (31)   (23)   (24)   (25)
                  (112)  (32)   (42)   (34)
                  (121)  (41)   (51)   (43)
                  (211)  (113)  (114)  (52)
                         (122)  (141)  (61)
                         (131)  (411)  (115)
                         (212)         (124)
                         (221)         (133)
                         (311)         (142)
                                       (151)
                                       (214)
                                       (223)
                                       (232)
                                       (241)
                                       (313)
                                       (322)
                                       (331)
                                       (412)
                                       (421)
                                       (511)
                                       (1213)
                                       (1312)
                                       (2131)
                                       (3121)

Links

Crossrefs

Cf. A000079, A000740, A008965, A242882, A325545, A325549, A325553, A325558, A325589, A325591.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Differences[Append[#,First[#]]]&]],{n,15}]

A309931 Triangle read by rows: T(n,k) is the number of compositions of n with k parts and circular differences all equal to 1, 0, or -1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 3, 4, 1, 1, 1, 1, 6, 5, 1, 1, 2, 3, 4, 10, 6, 1, 1, 1, 3, 5, 10, 15, 7, 1, 1, 2, 1, 4, 10, 20, 21, 8, 1, 1, 1, 3, 6, 11, 21, 35, 28, 9, 1, 1, 2, 3, 4, 10, 24, 42, 56, 36, 10, 1, 1, 1, 1, 5, 10, 25, 49, 78, 84, 45, 11, 1
Offset: 1

Views

Author

Andrew Howroyd, Aug 23 2019

Keywords

Examples

			Triangle begins:
  1;
  1, 1;
  1, 2, 1;
  1, 1, 3, 1;
  1, 2, 3, 4,  1;
  1, 1, 1, 6,  5,  1;
  1, 2, 3, 4, 10,  6,  1;
  1, 1, 3, 5, 10, 15,  7,   1;
  1, 2, 1, 4, 10, 20, 21,   8,   1;
  1, 1, 3, 6, 11, 21, 35,  28,   9,   1;
  1, 2, 3, 4, 10, 24, 42,  56,  36,  10,   1;
  1, 1, 1, 5, 10, 25, 49,  78,  84,  45,  11,  1;
  1, 2, 3, 4, 10, 24, 56,  96, 135, 120,  55, 12,  1;
  1, 1, 3, 6, 10, 21, 57, 116, 180, 220, 165, 66, 13, 1;
  ...
For n = 6 there are a total of 15 compositions:
  k = 1: (6)
  k = 2: (33)
  k = 3: (222)
  k = 4: (1122), (1212), (1221), (2112), (2121), (2211)
  k = 5: (11112), (11121), (11211), (12111), (21111)
  k = 6: (111111)

Crossrefs

Row sums are A325591.
Cf. A309937, A309938, A309939.

Programs

  • PARI
    step(R,n)={matrix(n, n, i, j, if(i>j, if(j>1, R[i-j, j-1]) + R[i-j, j] + if(j+1<=n, R[i-j, j+1])) )}
T(n)={my(v=vector(n)); for(k=1, n, my(R=matrix(n, n, i, j, i==j&&abs(i-k)<=1), m=0); while(R, m++; v[m]+=R[n, k]; R=step(R, n))); v}
for(n=1, 12, print(T(n)));

Formula

T(n, 1) = T(n, n) = 1.
T(n, 2) = (3 - (-1)^n)/2.
T(n, n - 1) = binomial(n-1, 1) = n - 1.
T(n, n - 2) = binomial(n-2, 2).
Showing 1-4 of 4 results.

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

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