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A003185
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a(n) = (4*n+1)*(4*n+5).
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5
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5, 45, 117, 221, 357, 525, 725, 957, 1221, 1517, 1845, 2205, 2597, 3021, 3477, 3965, 4485, 5037, 5621, 6237, 6885, 7565, 8277, 9021, 9797, 10605, 11445, 12317, 13221, 14157, 15125, 16125, 17157, 18221
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OFFSET
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0,1
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COMMENTS
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Bisection of A078371. - Lambert Klasen (Lambert.Klasen(AT)gmx.net), Nov 19 2004
a(n) is the smallest number not in the sequence such that Sum_{k=0..n} 1/a(k) has a denominator 4*n+5. - Derek Orr, Jun 21 2015
a(n) is the number of 2 X 2 matrices with all elements in {-n,..,0,..,n} with permanent = determinant^n except for a(0), where a(0)=0, but A003185(0) = 5. - Indranil Ghosh, Jan 04 2017
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LINKS
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FORMULA
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1 = Sum_{n>=0} 4/a(n). Sum_{k=0..n} 4/a(k) = 4(n+1)/[4(n+1)+1]. Integral_{x=0..1} 1/(1 + x^4) = Sum_{n>=0} 4/a(2*n) = Sum_{n>=0} (-1)^n/(4n+1). - Gary W. Adamson, Jun 18 2003
1 = 1/5 + Sum_{n>=1} 16/a(n); with partial sums (4n+1)/(4n+5). - Gary W. Adamson, Jun 18 2003
O.g.f.: (-5-30*x+3*x^2)/(-1+x)^3.
Conjecture: a(n+1)-a(n) = A063164(n+2). (End)
a(0)=5, a(1)=45, a(2)=117, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Jan 27 2013
Sum_{n>=0} (-1)^n/a(n) = (log(2*sqrt(2)+3) + Pi)/(8*sqrt(2)) - 1/4. - Amiram Eldar, Oct 08 2023
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MATHEMATICA
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Table[(4n+1)(4n+5), {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {5, 45, 117}, 40] (* Harvey P. Dale, Jan 27 2013 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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