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.

A005744 Expansion of x*(1+x-x^2)/((1-x)^4*(1+x)).

Original entry on oeis.org

0, 1, 4, 9, 17, 28, 43, 62, 86, 115, 150, 191, 239, 294, 357, 428, 508, 597, 696, 805, 925, 1056, 1199, 1354, 1522, 1703, 1898, 2107, 2331, 2570, 2825, 3096, 3384, 3689, 4012, 4353, 4713, 5092, 5491, 5910, 6350, 6811, 7294, 7799, 8327, 8878, 9453, 10052
Offset: 0

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

Number of n-covers of a 2-set.
Boolean switching functions a(n,s) for s = 2.
Without the initial 0, this is row 1 of the convolution array A213778. - Clark Kimberling, Jun 21 2012
a(n) equals the second column of the triangle A355754. - Eric W. Weisstein, Mar 12 2024

References

  • R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

John W. Layman observes that A003453 appears to be the alternating sum transform (PSumSIGN) of A005744.
Cf. A002623, A005745, A005746, A005747, A005748, A005771, A003180, A008778, A052265.
Cf. A181971, A213778.
Cf. A355754.

Programs

  • Mathematica
    CoefficientList[Series[x (1+x-x^2)/((1-x)^4(1+x)),{x,0,50}],x] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{0,1,4,9,17},50] (* Harvey P. Dale, Apr 10 2012 *)
  • PARI
    a(n)=([0,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; -1,3,-2,-2,3]^n*[0;1;4;9;17])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

Formula

a(n) = A002623(n) - (n+1).
a(n) = n*(n-1)/2 + Sum_{j=1..floor((n+1)/2)} (n-2*j+1)*(n-2*j)/2. - N. J. A. Sloane, Nov 28 2003
From R. J. Mathar, Apr 01 2010: (Start)
a(n) = 5*n/12 - 1/16 + 5*n^2/8 + n^3/12 + (-1)^n/16.
a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). (End)
a(n) = A181971(n+1, n-1) for n > 0. - Reinhard Zumkeller, Jul 09 2012
a(n) + a(n+1) = A008778(n). - R. J. Mathar, Mar 13 2021
E.g.f.: (x*(2*x^2 + 21*x + 27)*cosh(x) + (2*x^3 + 21*x^2 + 27*x - 3)*sinh(x))/24. - Stefano Spezia, Jul 27 2022

Extensions

Additional comments from Alford Arnold

A355755 Irregular triangle read by rows: T(n,k) is the number of unlabeled connected n-node graphs with intersection number (or edge clique cover number) k; n >= 1, 0 <= k <= floor(n^2/4).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 1, 0, 1, 4, 7, 6, 2, 1, 0, 1, 6, 22, 36, 27, 13, 4, 2, 1, 0, 1, 9, 53, 161, 242, 209, 111, 43, 17, 5, 1, 1, 0, 1, 12, 114, 611, 1766, 2903, 2793, 1723, 773, 284, 86, 36, 9, 3, 2, 1, 0, 1, 16, 221, 1987, 10517, 33078, 60639, 67379, 48035, 24628, 9715, 3349, 1049, 310, 105, 36, 9, 4, 1, 1
Offset: 1

Views

Author

Pontus von Brömssen, Jul 16 2022

Keywords

Examples

			Triangle begins:
  n\k | 0  1  2   3   4    5    6    7    8   9  10 11 12 13 14 15 16
  ----+--------------------------------------------------------------
   1  | 1
   2  | 0  1
   3  | 0  1  1
   4  | 0  1  2   2   1
   5  | 0  1  4   7   6    2    1
   6  | 0  1  6  22  36   27   13    4    2   1
   7  | 0  1  9  53 161  242  209  111   43  17   5  1  1
   8  | 0  1 12 114 611 1766 2903 2793 1723 773 284 86 36  9  3  2  1

Links

Crossrefs

Cf. A001349 (row sums), A002620, A355754.

Formula

T(n,0) = 0 if n > 1.
T(n,1) = 1.
T(n,2) = floor((n-1)^2/4) = A002620(n-1).

A355756 Triangle read by rows: A(n,k) is the intersection number of the Turán graph T(n,k), 1 <= k <= n.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 0, 4, 2, 1, 0, 6, 4, 2, 1, 0, 9, 4, 4, 2, 1, 0, 12, 6, 4, 4, 2, 1, 0, 16, 9, 5, 4, 4, 2, 1, 0, 20, 9, 6, 5, 4, 4, 2, 1, 0, 25, 12, 9, 6, 5, 4, 4, 2, 1, 0, 30, 16, 9, 6, 6, 5, 4, 4, 2, 1
Offset: 1

Views

Author

Pontus von Brömssen, Jul 16 2022

Keywords

Examples

			Triangle begins:
  n\k | 1  2  3  4  5  6  7  8  9 10 11
  ----+--------------------------------
   1  | 0
   2  | 0  1
   3  | 0  2  1
   4  | 0  4  2  1
   5  | 0  6  4  2  1
   6  | 0  9  4  4  2  1
   7  | 0 12  6  4  4  2  1
   8  | 0 16  9  5  4  4  2  1
   9  | 0 20  9  6  5  4  4  2  1
  10  | 0 25 12  9  6  5  4  4  2  1
  11  | 0 30 16  9  6  6  5  4  4  2  1

Links

Crossrefs

Cf. A002620, A008133, A355754.

Programs

  • Python
    from networkx import find_cliques,turan_graph
from itertools import combinations,count
def A355756(n,k):
    if k==1: return 0
    G=turan_graph(n,k)
    cliques=[sorted(c) for c in find_cliques(G)]
    ne=G.number_of_edges()
    for r in count(1):
        for c0 in combinations(cliques[1:],r-1):
            c=(cliques[0],)+c0
            if len(set().union(e for i in range(r) for e in combinations(c[i],2)))==ne:
                return r

Formula

A(n,1) = 0.
A(n,2) = floor(n^2/4) = A002620(n).
A(n,3) = floor((n+1)/3)*floor((n+2)/3) = A008133(n+1).
A(n,n-k) = A(2*k,k) for 2 <= k <= n/2.
A(n,n-1) = 2 for n >= 3.
A(n,n) = 1 for n >= 2.
A(n,k) >= floor((n+k-1)/k)*floor((n+k-2)/k) for k >= 2.
Showing 1-3 of 3 results.

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

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