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 1-10 of 11 results. Next

A028362 Total number of self-dual binary codes of length 2n. Totally isotropic spaces of index n in symplectic geometry of dimension 2n.

Original entry on oeis.org

1, 3, 15, 135, 2295, 75735, 4922775, 635037975, 163204759575, 83724041661975, 85817142703524375, 175839325399521444375, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375
Offset: 1

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Keywords

Comments

These numbers appear in the second column of A155103. - Mats Granvik, Jan 20 2009
a(n) = n terms in the sequence (1, 2, 4, 8, 16, ...) dot n terms in the sequence (1, 1, 3, 15, 135). Example: a(5) = 2295 = (1, 2, 4, 8, 16) dot (1, 1, 3, 15, 135) = (1 + 2 + 12 + 120 + 2160). - Gary W. Adamson, Aug 02 2010

Examples

			G.f. = x + 3*x^2 + 15*x^3 + 135*x^4 + 2295*x^5 + 75735*x^6 + 4922775*x^7 + ...
		

References

  • F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 630.

Crossrefs

Cf. A155103. - Mats Granvik, Jan 20 2009
Cf. A005329, A006088. - Paul D. Hanna, Sep 16 2009

Programs

  • Magma
    [1] cat [&*[ 2^k+1: k in [1..n] ]: n in [1..16]]; // Vincenzo Librandi, Dec 24 2015
    
  • Maple
    seq(mul(1 + 2^j, j = 1..n-1), n = 1..20); # G. C. Greubel, Jun 06 2020
  • Mathematica
    Table[Product[2^i+1,{i,n-1}],{n,15}] (* or *) FoldList[Times,1, 2^Range[15]+1] (* Harvey P. Dale, Nov 21 2011 *)
    Table[QPochhammer[-2, 2, n - 1], {n, 15}] (* Arkadiusz Wesolowski, Oct 29 2012 *)
  • PARI
    {a(n)=polcoeff(sum(m=0,n,2^(m*(m-1)/2)*x^m/prod(k=0,m-1,1-2^k*x+x*O(x^n))),n)} \\ Paul D. Hanna, Sep 16 2009
    
  • PARI
    {a(n) = if( n<1, 0 , prod(k=1, n-1, 2^k + 1))}; /* Michael Somos, Jan 28 2018 */
    
  • PARI
    {a(n) = sum(k=0, n-1, 2^(k*(k+1)/2) * prod(j=1, k, (2^(n-j) - 1) / (2^j - 1)))}; /* Michael Somos, Jan 28 2018 */
    
  • Python
    for n in range(2,40,2):
      product = 1
      for i in range(1,n//2-1 + 1):
        product *= (2**i+1)
      print(product)
    # Nathan J. Russell, Mar 01 2016
    
  • Python
    from math import prod
    def A028362(n): return prod((1<Chai Wah Wu, Jun 20 2022
    
  • Sage
    from ore_algebra import *
    R. = QQ['x']
    A. = OreAlgebra(R, 'Qx', q=2)
    print((Qx - x - 1).to_list([0,1], 10))  # Ralf Stephan, Apr 24 2014
    
  • Sage
    from sage.combinat.q_analogues import q_pochhammer
    [q_pochhammer(n-1,-2,2) for n in (1..20)] # G. C. Greubel, Jun 06 2020
    
  • Scheme
    ;; With memoization-macro definec.
    (define (A028362 n) (A028362off0 (- n 1)))
    (definec (A028362off0 n) (if (zero? n) 1 (+ (A028362off0 (- n 1)) (* (expt 2 n) (A028362off0 (- n 1))))))
    ;; Antti Karttunen, Apr 15 2017

Formula

a(n) = Product_{i=1..n-1} (2^i+1).
Letting a(0)=1, we have a(n) = Sum_{k=0..n-1} 2^k*a(k) for n>0. a(n) is asymptotic to c*sqrt(2)^(n^2-n) where c=2.384231029031371724149899288.... = A079555 = Product_{k>=1} (1 + 1/2^k). - Benoit Cloitre, Jan 25 2003
G.f.: Sum_{n>=1} 2^(n*(n-1)/2) * x^n/(Product_{k=0..n-1} (1-2^k*x)). - Paul D. Hanna, Sep 16 2009
a(n) = 2^(binomial(n,2) - 1)*(-1; 1/2){n}, where (a;q){n} is the q-Pochhammer symbol. - G. C. Greubel, Dec 23 2015
From Antti Karttunen, Apr 15 2017: (Start)
a(n) = A048675(A285101(n-1)).
a(n) = b(n-1), where b(0) = 1, and for n > 0, b(n) = b(n-1) + (2^n)*b(n-1).
a(n) = Sum_{i=1..A000124(n-1)} A053632(n-1,i-1)*(2^(i-1)) [where the indexing of both rows and columns of irregular table A053632(row,col) is considered to start from zero].
(End)
G.f. A(x) satisfies: A(x) = x * (1 + A(2*x)) / (1 - x). - Ilya Gutkovskiy, Jun 06 2020
Conjectural o.g.f. as a continued fraction of Stieltjes type (S-fraction):
1/(1 - 3*x/(1 - 2*x/(1 - 10*x/(1 - 12*x/(1 - 36*x/(1 - 56*x/(1 - 136*x/(1 - 240*x/(1 - ... - 2^(n-1)*(2^n + 1)*x/(1 - 2^n*(2^n - 1)*x/(1 - ... ))))))))))). - Peter Bala, Sep 27 2023

A003179 Number of self-dual binary codes of length 2n (up to column permutation equivalence).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 7, 9, 16, 25, 55, 103, 261, 731, 3295, 24147, 519492
Offset: 0

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Comments

The length 36 binary self dual codes have been classified. - Nathan J. Russell, Feb 14 2016
This is number of binary self-dual codes of length 2n up to column permutation equivalence. Sequence A028362 gives an actual count of all possible binary self-dual codes of length 2n. - Nathan J. Russell, Nov 25 2018

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Extensions

a(18) from Nathan J. Russell, Feb 14 2016
Name clarified by Nathan J. Russell, Nov 26 2018

A106162 Number of indecomposable Type II binary self-dual codes of length 8n.

Original entry on oeis.org

1, 1, 1, 7, 75, 94251
Offset: 0

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Author

N. J. A. Sloane, May 09 2005

Keywords

References

  • J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, J. Comb. Theory, A28 (1980), 26-53.
  • V. S. Pless, The children of the (32,16) doubly even codes, IEEE Trans. Inform. Theory, 24 (1978), 738-746.

Crossrefs

Extensions

a(4) corrected by John van Rees, Jul 21 2005. It was given as 76 by Conway and Pless and as 74 by Rains and Sloane.
a(5) = 94251 = 94343 - 75 - 7 - 7 - 1 - 1 - 1 (cf. A106163) from Koichi Betsumaya, Aug 11 2012

A028363 Total number of doubly-even self-dual binary codes of length 8n.

Original entry on oeis.org

1, 30, 9845550, 171634285407048750, 193419995622362136809061156168750, 14272693289804307141953423466197932293533748208968750
Offset: 0

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Keywords

References

  • F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 631.

Crossrefs

Programs

  • Mathematica
    Join[{1},Table[2*Product[2^i+1,{i,4n-2}],{n,6}]] (* Harvey P. Dale, May 08 2013 *)
    Table[Product[2^i + 1, {i, 0, n/2 - 2}], {n, 8, 40, 8}] (* Nathan J. Russell, Mar 04 2016 *)
  • Python
    for n in range(8, 50, 8):
        product = 1
        for i in range(n//2 - 1):
            product *= 2**i + 1
        print(product, end=", ")
    # Nathan J. Russell, Mar 01 2016

Formula

a(n) = 2*Product_{i=1..4n-2} (2^i + 1).

Extensions

There is an error in Eq. (75) of F. J. MacWilliams and N. J. A. Sloane, the lower subscript should be 1 not 0.
Formula corrected by N. J. A. Sloane, May 07 2013 following a suggestion from Harvey P. Dale

A106163 Total number of (indecomposable or decomposable) Type II binary self-dual codes of length 8n.

Original entry on oeis.org

1, 1, 2, 9, 85, 94343
Offset: 0

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Author

N. J. A. Sloane, May 09 2005

Keywords

Comments

"There are 94343 inequivalent doubly even self-dual codes of length 40, 16470 of which are extremal" [Betsumiya et al.] - Jonathan Vos Post, Aug 06 2012

Crossrefs

A269455 Number of Type I (singly-even) self-dual binary codes of length 2n.

Original entry on oeis.org

1, 3, 15, 105, 2295, 75735, 4922775, 625192425, 163204759575, 83724041661975, 85817142703524375, 175667691114114395625, 720413716161839357604375, 5902349576513949856852644375, 96709997811181068404530578084375, 3168896498278970068411253452090715625, 207692645973961964120828372930661061284375, 27222898185745116523209337325140537285726884375, 7136346644902153570976711733098966146766874104484375, 3741493773415815389266667264411257664189964123617799515625
Offset: 1

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Author

Nathan J. Russell, Feb 27 2016

Keywords

Comments

A self dual binary linear code is either Type I (singly even) or Type II (doubly even). A self dual binary linear code can only be Type II if the length of the code (2n) is a multiple of 8. The total number self dual binary linear codes (including equivalent codes) is equal to the number of Type I self dual binary linear codes (including equivalent codes) when the length (2n) is not a multiple of 8. If the length is a multiple of 8 ( 2n =0 mod 8 ) then the total number of Type I codes is the number of type II codes subtracted from the total number of self dual codes of length 2n.

References

  • W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, 2003, Page 366.
  • F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1977.

Crossrefs

Programs

  • Mathematica
    Table[
    If[Mod[2 n, 8] == 0,
      Product[2^i + 1, {i, 1, n - 1}] - Product[2^i + 1, {i, 0, n - 2}] ,
      Product[2^i + 1, {i, 1, n - 1}]],
    {n, 1, 10}] (* Nathan J. Russell, Mar 01 2016 *)
  • PARI
    a(n) = if (2*n%8==0, prod(i=1, n-1, 2^i+1)-prod(i=0, n-2, 2^i+1), prod(i=1, n-1, 2^i+1))
    vector(20, n, a(n)) \\ Colin Barker, Feb 28 2016
  • Python
    for n in range(1,10):
        product1 = 1
        for i in range(1,n-1 + 1):
            product1 *= (2**i+1)
        if (2*n)%8 == 0:
            product2 = 1
            for i in range(n-2 + 1):
                product2 *= (2**i+1)
            print(product1 - product2)
        else:
            print(product1)
    

Formula

From Nathan J. Russell, Mar 01 2016: (Start)
If 2n = 0 MOD 8 then a(n) = prod_(2^i+1, i=1,...,n-1) - prod_(2^i+1, i=0,...,n-2);
If 2n != 0 MOD 8 then a(n) = prod_(2^i+1, i=1,...,n-1).
If 2n = 0 MOD 8 then a(n) = A028362(n) - A028363( n/8);
If 2n != 0 MOD 8 then a(n) = A028362(n).
(End)

Extensions

a(20) corrected by Andrew Howroyd, Feb 22 2018

A106165 Number of inequivalent (indecomposable or decomposable) Type I but not Type II binary self-dual codes of length 2n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 3, 4, 5, 9, 16, 25, 46, 103, 261, 731, 3210, 24147
Offset: 0

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Author

N. J. A. Sloane, May 09 2005, Aug 23 2008

Keywords

Comments

The minimal distance of these codes is not constrained. A105685 gives the number with the highest minimal distance.

Crossrefs

A106164 Number of indecomposable Type I but not Type II binary self-dual codes of length 2n.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 1, 1, 1, 2, 6, 8, 19, 45, 148, 457, 2448, 20786
Offset: 0

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Author

N. J. A. Sloane, May 09 2005

Keywords

References

  • R. T. Bilous, Enumeration of binary self-dual codes of length 34, Preprint, 2005.
  • R. T. Bilous and G. H. J. van Rees, An enumeration of binary self-dual codes of length 32, Designs, Codes Crypt., 26 (2002), 61-86.
  • J. H. Conway and V. S. Pless, On the enumeration of self-dual codes, J. Comb. Theory, A28 (1980), 26-53.
  • V. S. Pless, The children of the (32,16) doubly even codes, IEEE Trans. Inform. Theory, 24 (1978), 738-746.

Crossrefs

Extensions

a(34) computed by N. J. A. Sloane, based on data in Bilous's paper, Sep 06 2005

A215219 Number of (indecomposable or decomposable) Type II binary self-dual codes of length 8n with the highest minimal distance.

Original entry on oeis.org

1, 1, 2, 1, 5, 16470, 1
Offset: 0

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Author

N. J. A. Sloane, Aug 08 2012

Keywords

Comments

It is important to distinguish between "extremal" (meaning having the highest possible minimal distance permitted by Gleason's theorem) and "optimal" (meaning having the highest minimal distance that can actually be achieved). This sequence enumerates optimal codes. Extremal codes do not exist when n is sufficiently large. For lengths up to at least 64, "extremal" and "optimal" coincide.
"There are 94343 inequivalent doubly even self-dual codes of length 40, 16470 of which are extremal." [Betsumiya et al.] - Jonathan Vos Post, Aug 06 2012

Crossrefs

Extensions

a(6) = 1 (due to Houghten et al.) from Akihiro Munemasa, Aug 08 2012

A322429 Number of decomposable binary self-dual codes of length 2n (up to permutation equivalence).

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 5, 7, 10, 17, 29, 58, 113, 274, 772, 3361
Offset: 1

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Author

Nathan J. Russell, Dec 07 2018

Keywords

Comments

Every binary self-dual code is either indecomposable or decomposable. A decomposable binary self-dual code is the direct sum of a set of indecomposable binary self-dual codes of smaller length.

Examples

			There are A003179(17) = 24147 binary self-dual codes of length 2*17 = 34 up to permutation equivalence.  There are A003178(17) = 2523 binary self-dual codes of length 2*17 = 34 that are indecomposable.  This means that there are A003179(17) - A003178(17) = a(17) = 3361 binary self-dual codes of length 2*17=34 that are decomposable.
		

Crossrefs

Formula

a(n) = A003179(n) - A003178(n).
Showing 1-10 of 11 results. Next