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 1-3 of 3 results.

A309938 Triangle read by rows: T(n,k) is the number of compositions of n with k parts and differences all equal to 1 or -1.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 0, 1, 0, 1, 2, 1, 0, 0, 1, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 0, 0, 1, 0, 1, 4, 1, 0, 0, 0, 1, 2, 2, 0, 3, 2, 0, 0, 0, 1, 0, 1, 4, 2, 0, 1, 0, 0, 0, 1, 2, 1, 0, 3, 6, 1, 0, 0, 0, 0, 1, 0, 2, 4, 3, 0, 4, 2, 0, 0, 0, 0, 1, 2, 1, 0, 3, 8, 3, 0, 1, 0, 0, 0, 0
Offset: 1

Views

Author

Andrew Howroyd, Aug 23 2019

Keywords

Comments

Parts will alternate between being odd and even. For even k, a composition cannot be the same as its reversal and therefore for even k, T(n,k) is even.

Examples

			Triangle begins:
  1;
  1, 0;
  1, 2, 0;
  1, 0, 1, 0;
  1, 2, 1, 0, 0;
  1, 0, 2, 2, 0,  0;
  1, 2, 1, 0, 1,  0, 0;
  1, 0, 1, 4, 1,  0, 0, 0;
  1, 2, 2, 0, 3,  2, 0, 0, 0;
  1, 0, 1, 4, 2,  0, 1, 0, 0, 0;
  1, 2, 1, 0, 3,  6, 1, 0, 0, 0, 0;
  1, 0, 2, 4, 3,  0, 4, 2, 0, 0, 0, 0;
  1, 2, 1, 0, 3,  8, 3, 0, 1, 0, 0, 0, 0;
  1, 0, 1, 4, 3,  0, 6, 8, 1, 0, 0, 0, 0, 0;
  1, 2, 2, 0, 4, 10, 5, 0, 5, 2, 0, 0, 0, 0, 0;
  ...
For n = 6 there are a total of 5 compositions:
  k = 1: (6)
  k = 3: (123), (321)
  k = 4: (2121), (1212)

Links

Crossrefs

Row sums are A173258.
T(2n,n) gives A364529.
Cf. A309931, A309937, A309939, A325557.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n<1 or i<1, 0,
     `if`(n=i, x, add(expand(x*b(n-i, i+j)), j=[-1, 1])))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(add(b(n, j), j=1..n)):
seq(T(n), n=1..14);  # Alois P. Heinz, Jul 22 2023
  • Mathematica
    b[n_, i_] := b[n, i] = If[n < 1 || i < 1, 0, If[n == i, x, Sum[Expand[x*b[n - i, i + j]], {j, {-1, 1}}]]];
T[n_] :=  CoefficientList[Sum[b[n, j], {j, 1, n}], x] // Rest // PadRight[#, n]&;
Table[T[n], {n, 1, 13}] // Flatten (* Jean-François Alcover, Sep 06 2023, after Alois P. Heinz *)
  • PARI
    step(R,n)={matrix(n, n, i, j, if(i>j, if(j>1, R[i-j, j-1]) + if(j+1<=n, R[i-j, j+1])) )}
T(n)={my(v=vector(n), R=matid(n), m=0); while(R, m++; v[m]+=vecsum(R[n,]); R=step(R,n)); v}
for(n=1, 15, print(T(n)))

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).

A309939 Triangle read by rows: T(n,k) is the number of compositions of n with k parts and 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, 3, 6, 5, 1, 1, 2, 3, 6, 10, 6, 1, 1, 1, 3, 7, 12, 15, 7, 1, 1, 2, 3, 6, 14, 22, 21, 8, 1, 1, 1, 3, 8, 15, 27, 37, 28, 9, 1, 1, 2, 3, 6, 16, 32, 50, 58, 36, 10, 1, 1, 1, 3, 7, 16, 35, 63, 88, 86, 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, 3, 6,  5,  1;
  1, 2, 3, 6, 10,  6,  1;
  1, 1, 3, 7, 12, 15,  7,   1;
  1, 2, 3, 6, 14, 22, 21,   8,   1;
  1, 1, 3, 8, 15, 27, 37,  28,   9,   1;
  1, 2, 3, 6, 16, 32, 50,  58,  36,  10,   1;
  1, 1, 3, 7, 16, 35, 63,  88,  86,  45,  11,   1;
  1, 2, 3, 6, 16, 38, 74, 118, 147, 122,  55,  12,  1;
  1, 1, 3, 8, 16, 37, 83, 148, 212, 234, 167,  66, 13,  1;
  1, 2, 3, 6, 17, 40, 88, 174, 282, 366, 357, 222, 78, 14, 1;
  ...
For n = 6 there are a total of 17 compositions:
  k = 1: (6)
  k = 2: (33)
  k = 3: (123), (222), (321)
  k = 4: (1122), (1212), (1221), (2112), (2121), (2211)
  k = 5: (11112), (11121), (11211), (12111), (21111)
  k = 6: (111111)

Crossrefs

Row sums are A034297.
Cf. A309931, A309937, A309938.

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), R=matid(n), m=0); while(R, m++; v[m]+=vecsum(R[n,]); 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 for n > 1.
T(n, 3) = 3 for n > 3.
T(n, n - 1) = binomial(n-1, 1) = n - 1.
T(n, n - 2) = binomial(n-2, 2).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.