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.

A000231 Number of inequivalent Boolean functions of n variables under action of complementing group.

Original entry on oeis.org

2, 3, 7, 46, 4336, 134281216, 288230380379570176, 2658455991569831764110243006194384896, 452312848583266388373324160190187140390789016525312000869601987902398529536
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

The next term has 152 digits. - Harvey P. Dale, Jun 21 2011

References

  • M. A. Harrison, Introduction to Switching and Automata Theory. McGraw Hill, NY, 1965, p. 143.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (Includes this sequence, correctly, although in the Preface on page viii 4336 is mis-typed as 4436).

Links

Crossrefs

Cf. A051502.
Row sums of A054724.

Programs

  • Maple
    a:= n-> (2^(2^n)+(2^n-1)*2^(2^(n-1)))/2^n:
seq(a(n), n=0..8);  # Alois P. Heinz, Jan 27 2023
  • Mathematica
    Table[(2^(2^n)+(2^n-1)*2^(2^(n-1)))/2^n,{n,10}] (* Harvey P. Dale, Jun 21 2011 *)
  • PARI
    a(n)=(2^(2^n-n)+(2^n-1)*2^(2^(n-1)-n)) \\ Charles R Greathouse IV, Jul 29 2016

Formula

a(n) = (2^(2^n)+(2^n-1)*2^(2^(n-1)))/2^n.

Extensions

More terms from Vladeta Jovovic, Apr 20 2000
a(0)=2 prepended by Alois P. Heinz, Jan 27 2023

A362905 Array read by antidiagonals: T(n,k) is the number of n element multisets of length k vectors over GF(2) that sum to zero.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 1, 8, 5, 3, 1, 1, 1, 16, 15, 11, 3, 1, 1, 1, 32, 51, 50, 14, 4, 1, 1, 1, 64, 187, 276, 99, 24, 4, 1, 1, 1, 128, 715, 1768, 969, 232, 30, 5, 1, 1, 1, 256, 2795, 12496, 11781, 3504, 429, 45, 5, 1, 1, 1, 512, 11051, 93600, 162877, 73440, 10659, 835, 55, 6, 1
Offset: 0

Views

Author

Andrew Howroyd, May 27 2023

Keywords

Comments

Equivalently, T(n,k) is the number multisets with n elements drawn from {0..2^k-1} such that the bitwise-xor of all the elements gives zero.
T(n,k) is the number of equivalence classes of n X k binary matrices with an even number of 1's in each column under permutation of rows.
T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and complementation of columns.

Examples

			Array begins:
=========================================
n/k| 0 1  2   3     4      5        6 ...
---+-------------------------------------
0  | 1 1  1   1     1      1        1 ...
1  | 1 1  1   1     1      1        1 ...
2  | 1 2  4   8    16     32       64 ...
3  | 1 2  5  15    51    187      715 ...
4  | 1 3 11  50   276   1768    12496 ...
5  | 1 3 14  99   969  11781   162877 ...
6  | 1 4 24 232  3504  73440  1878976 ...
7  | 1 4 30 429 10659 394383 18730855 ...
  ...

Links

Crossrefs

Columns k=0..4 are A000012, A004526(n+2), A053307, A362906, A363350.
Rows n=2..3 are A000079, A007581.
Main diagonal is A363351.
Cf. A054724, A340030, A340312, A363349.

Programs

  • Mathematica
    A362905[n_,k_]:=(Binomial[2^k+n-1,n]+If[EvenQ[n],(2^k-1)Binomial[2^(k-1)+n/2-1,n/2],0])/2^k;Table[A362905[n-k,k],{n,0,15},{k,n,0,-1}] (* Paolo Xausa, Nov 19 2023 *)
  • PARI
    T(n,k)={(binomial(2^k+n-1, n) + if(n%2==0, (2^k-1)*binomial(2^(k-1)+n/2-1,n/2)))/2^k}

Formula

T(n,k) = binomial(2^k+n-1, n)/2^k for odd n;
T(n,k) = (binomial(2^k+n-1, n) + (2^k-1)*binomial(2^(k-1)+n/2-1, n/2))/2^k for even n.
G.f. of column k: (1/(1-x)^(2^k) + (2^k-1)/(1-x^2)^(2^(k-1)))/2^k.

A022619 Triangle T(n,k)of numbers of asymmetric Boolean functions of n variables with exactly k = 0..2^n nonzero values (atoms) under action of complementing group C(n,2).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 7, 7, 7, 0, 1, 0, 0, 1, 0, 35, 105, 273, 448, 715, 750, 715, 448, 273, 105, 35, 0, 1, 0, 0, 1, 0, 155, 1085, 6293, 27776, 105183, 327050, 876525, 2011776, 4032015, 7048811, 10855425, 14721280, 17678835, 18771864
Offset: 1

Views

Author

Vladeta Jovovic, Jul 13 2000

Keywords

Examples

			Triangle begins:
  [0,1,0],
  [0,1,0,1,0],
  [0,1,0,7,7,7,0,1,0],
  ...;
T(5,k) = coefficient of x^k in (1/32)*((1+x)^32-31*(1+x^2)^16+310*(1+x^4)^8-1240*(1+x^8)^4+1984*(1+x^16)^2-1024*(1+x^32)),k = 0..32.

Links

Crossrefs

Row sums give A051502.
Cf. A054724.

Programs

  • Mathematica
    T[n_,0]:=0; T[n_, k_] := (1/2^n)*Coefficient[Sum[(-1)^j*2^(Binomial[j, 2])* QBinomial[n, j, 2]*(1 + x^(2^j))^(2^(n - j)), {j, 0, n}], x^k];
Table[T[n, k], {n, 1, 5}, {k, 0, 2^n}] // Flatten (* G. C. Greubel, Feb 15 2018 *)

Formula

T(n, k) = coefficient of x^k in (1/2^n)*Sum_{j = 0..n} (-1)^j*2^C(j, 2)*[n, j]*(1+x^(2^j))^(2^(n-j)), where [n, j] is Gaussian 2-binomial coefficient; k = 0..2^n.

A227724 T(n,k) = number of small equivalence classes of half full n-ary Boolean functions that contain 2^k functions.

Original entry on oeis.org

0, 0, 1, 0, 3, 0, 0, 7, 0, 7, 0, 15, 0, 105, 750
Offset: 0

Views

Author

Tilman Piesk, Jul 22 2013

Keywords

Comments

Like A227725, but counting only small equivalence classes (sec) of half full Boolean functions, i.e. those with an equal number of zeros and ones.

Examples

			Triangle begins:             Row sums (central numbers of A054724)
            0                        0
         0     1                     1
      0     3     0                  3
   0     7     0     7              14
0    15     0    105   750         870

Links

Crossrefs

Cf. A227725, A054724, A000225.
Showing 1-4 of 4 results.

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

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