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A073577
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a(n) = 4*n^2 + 4*n - 1.
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10
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7, 23, 47, 79, 119, 167, 223, 287, 359, 439, 527, 623, 727, 839, 959, 1087, 1223, 1367, 1519, 1679, 1847, 2023, 2207, 2399, 2599, 2807, 3023, 3247, 3479, 3719, 3967, 4223, 4487, 4759, 5039, 5327, 5623, 5927, 6239, 6559, 6887, 7223, 7567, 7919, 8279, 8647
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OFFSET
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1,1
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COMMENTS
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The sum of the squares of two consecutive terms multiplied (or divided) by 2 is always a perfect square. In general, numbers represented by the quadratic form a(n) = (2*i*n + j)^2 - 2*i^2 for any i and j have 2(a(n)^2 + a(n+1)^2)) and (a(n)^2 + a(n+1)^2)/2 as perfect squares: in this case, i=j=1.
The terms of this sequence may be seen to be 2 less than the odd squares. As such they run parallel to those in the square spiral as well as the Ulam square spiral. - Stuart M. Ellerstein (ellerstein(AT)aol.com), Oct 01 2002
The continued fraction expansion of sqrt(4*a(n)) is [4n+1; {1, n-1, 2, 2n, 2, n-1, 1, 8n+2}]. For n=1, this collapses to [5; {3, 2, 3, 10}]. - Magus K. Chu, Sep 12 2022
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LINKS
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Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, Expositiones Mathematicae, Vol. 38, No. 4 (2020), pp. 430-479; arXiv preprint, arXiv:1807.08899 [math.NT], 2018-2019. See 6.6.7, p. 36 (p. 35 in the preprint).
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FORMULA
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a(n) = FrobeniusNumber(2*n+1, 2*n+3). - Darrell Minor, Jul 29 2008
a(n) = (2*n+1)+(2*n-1) + (2*n+1)*(2*n-1). - J. M. Bergot, Apr 17 2016
L.g.f.: 4*x*(2+x)/(1+x)^2-log(1+x).
L.h.g.f.: -4*(-2+x)*x/(-1+x)^2+log(1-x).
(End)
Sum_{n>=1} 1/a(n) = 1 + sqrt(2)*Pi*tan(Pi/sqrt(2))/8. - Amiram Eldar, Jan 03 2021
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EXAMPLE
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a(2) = 8*2 + 7 = 23;
a(3) = 8*3 + 23 = 47;
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MAPLE
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MATHEMATICA
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LinearRecurrence[{3, -3, 1}, {7, 23, 47}, 50] (* Harvey P. Dale, Dec 04 2018 *)
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PROG
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(PARI) vector(50, n, 4*n^2 + 4*n - 1) \\ Michel Marcus, Jan 14 2015
(Python) for n in range(1, 50): print(4*n**2+4*n-1, end=', ') # Stefano Spezia, Nov 01 2018
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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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M. N. Deshpande (dpratap(AT)nagpur.dot.net.in), Aug 27 2002
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EXTENSIONS
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STATUS
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approved
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