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A028361
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Number of totally isotropic spaces of index n in orthogonal geometry of dimension 2n.
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17
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1, 2, 6, 30, 270, 4590, 151470, 9845550, 1270075950, 326409519150, 167448083323950, 171634285407048750, 351678650799042888750, 1440827432323678715208750, 11804699153027899713705288750, 193419995622362136809061156168750, 6338179836549184861096125026493768750
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refs;
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history;
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internal format)
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OFFSET
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0,2
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COMMENTS
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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
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
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
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REFERENCES
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DeTemple, Duane W. "Carlyle circles and the Lemoine simplicity of polygon constructions." The American Mathematical Monthly 98.2 (1991): 97-108.
W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Page 366. - Nathan J. Russell, Mar 04 2016
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LINKS
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Davide Gaiotto and Justin Kulp, Orbifold groupoids, arXiv:2008.05960 [hep-th], 2020, page 42.
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FORMULA
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a(n) = Product_{i=0..n-1} ( 2^i + 1 ).
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
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
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
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
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
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
G.f. A(x) satisfies: A(x) = (1 + x * A(2*x)) / (1 - x). - Ilya Gutkovskiy, Jun 06 2020
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MAPLE
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seq( mul((1+2^j), j=0..n-1), n = 0..20); # G. C. Greubel, Jun 06 2020
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MATHEMATICA
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Table[Product[2^i + 1, {i, 0, n/2 - 2}], {n, 2, 32, 2}] (* Nathan J. Russell, Mar 04 2016 *)
Table[Product[2^i + 1, {i, 0, n - 1}], {n, 0, 15}] (* Nathan J. Russell, Mar 04 2016 *)
FoldList[Times, 1, 2^Range[0, 20]+1] (* Harvey P. Dale, Apr 11 2016 *)
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PROG
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(PARI) {a(n) = prod(k=0, n-1, 2^k + 1)};
(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 */
(Python)
for n in range(2, 50, 2):
product = 1
for i in range(0, n//2-2 + 1):
product *= (2**i+1)
print(product)
(Magma) [1] cat [ (&*[1+2^j: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Jun 06 2020
(Sage) [product( 1+2^j for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Jun 06 2020
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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