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.

Previous Showing 11-13 of 13 results.

A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

Original entry on oeis.org

2, 4, 3, 6, 6, 5, 8, 9, 10, 7, 10, 12, 15, 14, 11, 12, 15, 20, 21, 22, 13, 14, 18, 25, 28, 33, 26, 17, 16, 21, 30, 35, 44, 39, 34, 19, 18, 24, 35, 42, 55, 52, 51, 38, 23, 20, 27, 40, 49, 66, 65, 68, 57, 46, 29, 22, 30, 45, 56, 77, 78, 85, 76, 69, 58, 31
Offset: 1

Views

Author

Stefano Spezia, Jul 14 2019

Keywords

Comments

T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019

Examples

			The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k|    0     1     2     3     4     5     6     7     8
---+-----------------------------------------------------
1  |    2
2  |    4     3
3  |    6     6     5
4  |    8     9    10     7
5  |   10    12    15    14    11
6  |   12    15    20    21    22    13
7  |   14    18    25    28    33    26    17
8  |   16    21    30    35    44    39    34    19
9  |   18    24    35    42    55    52    51    38    23
...
For n = 3 the matrix M(3) is
          2,         3,         5
    M_{2,1},         2,         3
    M_{3,1},   M_{3,2},         2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.

Links

Crossrefs

Cf. A025581, A318173, A325516.
Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal.

Programs

  • Magma
    [[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output
  • Maple
    a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);
  • Mathematica
    Flatten[Table[(n-k)*Prime[1+k],{n,1,11},{k,0,n-1}]]
  • PARI
    T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output
  • Sage
    [[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output

Formula

T(n, k) = A025581(n, k)*A000040(1 + k).

A323255 The permanent of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 2*n - 1, n - 1, ..., 1 and whose first column consists of 2*n - 1, 2*n - 2, ..., n.

Original entry on oeis.org

1, 1, 11, 248, 9968, 638772, 60061657, 7798036000, 1336715859150, 292406145227392, 79483340339739367, 26280500564448081664, 10386012861097225139356, 4834639222955142417477888, 2618110215141486526589786501, 1631888040186649673361825042432, 1159983453675106278249250918734938
Offset: 0

Views

Author

Stefano Spezia, Jan 09 2019

Keywords

Comments

The trace of the matrix M(n) is A000384(n).
The sum of the first row of the matrix M(n) is A034856(n).
The sum of the first column of the matrix M(n) is A000326(n).
For n > 1, the sum of the superdiagonal of the matrix M(n) is A000290(n-1).
For n > 1, the sum of the subdiagonal of the matrix M(n) is A001105(n-1).

Examples

			For n = 1 the matrix M(1) is
   1
with permanent a(1) = 1.
For n = 2 the matrix M(2) is
   3, 1
   2, 3
with permanent a(2) = 11.
For n = 3 the matrix M(3) is
   5, 2, 1
   4, 5, 2
   3, 4, 5
with permanent a(3) = 248.

Links

Crossrefs

Cf. A000290, A000326, A000384, A001105, A001792.
Cf. A034856, A204235, A318173, A322908, A322909.
Cf. A323254 (determinant of matrix M(n)).

Programs

  • Mathematica
    b[i_]:=i; a[n_]:=If[n==0, 1, Permanent[ToeplitzMatrix[Join[{b[2*n-1]}, Array[b, n-1, {2*n-2,n }]], Join[{b[2*n-1]},Array[b, n-1, {n-1,1}]]]]]; Array[a, 16, 0]
  • PARI
    tm(n) = {my(m = matrix(n, n, i, j, if (j==1, 2*n-i, n-j+1))); for (i=2, n, for (j=2, n, m[i, j] = m[i-1, j-1]; ); ); m;}
a(n) = matpermanent(tm(n)); \\ Stefano Spezia, Dec 11 2019

Extensions

a(0) = 1 prepended by Stefano Spezia, Dec 08 2019

A381514 a(n) is the hafnian of a symmetric Toeplitz matrix of order 2*n whose off-diagonal element (i,j) equals the |i-j|-th prime.

Original entry on oeis.org

1, 2, 23, 899, 85072, 15120411, 4439935299, 1989537541918, 1264044973158281, 1090056235155152713, 1227540523199054294506
Offset: 0

Views

Author

Stefano Spezia, Feb 25 2025

Keywords

Examples

			a(2) = 23 because the hafnian of
  [d  2  3  5]
  [2  d  2  3]
  [3  2  d  2]
  [5  3  2  d]
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 2*2 + 3*3 + 5*2 = 23. Here d denotes the generic element on the main diagonal of the matrix from which the hafnian does not depend.

Links

Crossrefs

Cf. A374067, A374068.
Cf. A071078, A071079, A085807, A306457, A318173, A356483.

Programs

  • Mathematica
    M[i_, j_]:=Prime[Abs[i-j]]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i]], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 5, 0]

Extensions

a(5)-a(10) from Pontus von Brömssen, Feb 26 2025
Previous Showing 11-13 of 13 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.