cp's OEIS Frontend

A028361 Number of totally isotropic spaces of index n in orthogonal geometry of dimension 2n.

%I A028361 #120 Aug 11 2025 05:17:35
%S A028361 1,2,6,30,270,4590,151470,9845550,1270075950,326409519150,
%T A028361 167448083323950,171634285407048750,351678650799042888750,
%U A028361 1440827432323678715208750,11804699153027899713705288750,193419995622362136809061156168750,6338179836549184861096125026493768750
%N A028361 Number of totally isotropic spaces of index n in orthogonal geometry of dimension 2n.
%C A028361 These numbers appear in first column of A155103. - _Mats Granvik_, Jan 20 2009
%C A028361 Equals row sums of unsigned triangle A158474. - _Gary W. Adamson_, Mar 20 2009
%C A028361 a(n) = (n+2) terms in the sequence (1, 1, 2, 4, 8, 16, ...) dot (n+2) terms in the sequence (1, 1, 2, 6, 30, 270, ...). Example: a(4) = 4590 = (1, 2, 4, 8, 16) dot (1, 1, 2, 6, 30, 270) = (1 + 1 + 4 + 24 + 240 + 4320). - _Gary W. Adamson_, Aug 02 2010
%C A028361 a(n) is the right-hand side of the mass formula used to classify Type II Self Dual Binary Linear Codes of length 2(n+1). a(n) is the number of distinct Type II Self Dual Binary Linear codes of length 2(n+1) when 2(n+1) = 0 MOD 8.  It is important to note that Type II codes are only possible when the length is a multiple of 8. In short, this sequence only applies to Type II codes when 2(n+1) = 0 MOD 8, else the right hand side of the mass formula is zero. - _Nathan J. Russell_, Mar 04 2016
%C A028361 This is almost certainly the sequence of number of Carlyle circles needed for the construction of regular polygons using straightedge and compass mentioned on page 107 of DeTemple (1991). - _N. J. A. Sloane_, Aug 05 2021
%C A028361 a(n) is also the number of Sp(oo, F2)-orbits of V^n, where V is the countable-dimensional symplectic vector space over the two-element field. - _Jingjie Yang_, Jul 30 2025
%D A028361 W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Page 366. - _Nathan J. Russell_, Mar 04 2016
%H A028361 T. D. Noe, <a href="/A028361/b028361.txt">Table of n, a(n) for n=0..50</a>
%H A028361 C. Bachoc and P. Gaborit, <a href="http://www.numdam.org/item?id=JTNB_2000__12_2_255_0">On extremal additive F_4 codes of length 10 to 18</a>, J. Théorie Nombres Bordeaux, 12 (2000), 255-271.
%H A028361 Julien Clément and Antoine Genitrini, <a href="https://arxiv.org/abs/1907.06743">Binary Decision Diagrams: from Tree Compaction to Sampling</a>, arXiv:1907.06743 [cs.DS], 2019.
%H A028361 John P. D'Angelo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Dangelo/dan3.html">Counting Tournament Brackets</a>, J. Int. Seq., Vol. 25 (2022), Article 22.6.8.
%H A028361 Duane W. DeTemple, <a href="https://sharingthesoul.wordpress.com/wp-content/uploads/2021/03/carlyle-and-lemoine-polygon-constructions-detemple1991.pdf">Carlyle circles and the Lemoine simplicity of polygon constructions</a>, The American Mathematical Monthly 98.2 (1991): 97-108.
%H A028361 Davide Gaiotto and Justin Kulp, <a href="https://arxiv.org/abs/2008.05960">Orbifold groupoids</a>, arXiv:2008.05960 [hep-th], 2020, page 42.
%F A028361 a(n) = Product_{i=0..n-1} ( 2^i + 1 ).
%F A028361 Asymptotic to C*2^(n*(n-1)/2) where C = A081845 = 4.76846205806274344829979857... = Product_{k>=0} (1 + 1/2^k). - _Benoit Cloitre_, Apr 09 2003
%F A028361 It appears that a(n) = 2^((1/2)*(n - 1)*n) * Product_{k>=0} (1 + 1/(2^k)) / Product_{k>=0} (1 + 1/(2^(n + k))). - Peter Moxey (pmoxey(AT)live.com), Mar 21 2010
%F A028361 G.f.: Sum_{n>=0} 2^(n*(n-1)/2) * x^n / Product_{k=0..n} (1 - 2^k*x). - _Paul D. Hanna_, May 02 2012
%F A028361 a(n) = (a(n-2)^3 + a(n-1) * a(n-3) * (a(n-1) - 2 * a(n-2))) * a(n-1) / (a(n-2)^2 * (a(n-2) - a(n-3))) if n>2. - _Michael Somos_, Aug 21 2012
%F A028361 0 = a(n)*(+a(n+1) + a(n+2)) + a(n+1)*(-2*a(n+1)) for all n>=0. - _Michael Somos_, Oct 10 2014
%F A028361 Sum_{k=0..n} 2^k/a(k) = 3-2/a(n) and Sum_{k=0..n} 4^k/a(k) = 9-(4*(1+2^n))/a(n) for n >= 0. - _Werner Schulte_, Dec 25 2016
%F A028361 G.f. A(x) satisfies: A(x) = (1 + x * A(2*x)) / (1 - x). - _Ilya Gutkovskiy_, Jun 06 2020
%F A028361 a(n) = Sum_{k=0..n} q_binomial(n, k, q=2) * 2^(k*(k-1)/2). - _Jingjie Yang_, Jul 30 2025
%p A028361 seq( mul((1+2^j), j=0..n-1), n = 0..20); # _G. C. Greubel_, Jun 06 2020
%t A028361 Table[QPochhammer[-1, 2, n], {n, 0, 15}] (* _Arkadiusz Wesolowski_, Oct 29 2012 *)
%t A028361 Table[Product[2^i + 1, {i, 0, n/2 - 2}], {n, 2, 32, 2}] (* _Nathan J. Russell_, Mar 04 2016 *)
%t A028361 Table[Product[2^i + 1, {i, 0, n - 1}], {n, 0, 15}] (* _Nathan J. Russell_, Mar 04 2016 *)
%t A028361 FoldList[Times,1,2^Range[0,20]+1] (* _Harvey P. Dale_, Apr 11 2016 *)
%o A028361 (PARI) {a(n) = prod(k=0, n-1, 2^k + 1)};
%o A028361 (PARI) {a(n)=polcoeff(sum(m=0,n,2^(m*(m-1)/2)*x^m/prod(k=0,m,1-2^k*x+x*O(x^n))),n)} /* _Paul D. Hanna_, May 02 2012 */
%o A028361 (Python)
%o A028361 for n in range(2,50,2):
%o A028361   product = 1
%o A028361   for i in range(0,n//2-2 + 1):
%o A028361     product *= (2**i+1)
%o A028361   print(product)
%o A028361 # _Nathan J. Russell_, Mar 01 2016
%o A028361 (Magma) [1] cat [ (&*[1+2^j: j in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Jun 06 2020
%o A028361 (Sage) [product( 1+2^j for j in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Jun 06 2020
%Y A028361 Cf. A006125, A028362, A155103, A158474, A323716 (product of 3^i+1).
%K A028361 nonn,nice,easy
%O A028361 0,2
%A A028361 _N. J. A. Sloane_