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%I #62 Jun 24 2023 13:12:44
%S 1,7,9,15,17,23,25,31,33,39,41,47,49,55,57,63,65,71,73,79,81,87,89,95,
%T 97,103,105,111,113,119,121,127,129,135,137,143,145,151,153,159,161,
%U 167,169,175,177,183,185,191,193,199,201,207,209,215,217,223,225,231,233
%N Numbers that are congruent to {1, 7} mod 8.
%C Also n such that Kronecker(2,n) = mu(gcd(2,n)). - _Jon Perry_ and _T. D. Noe_, Jun 13 2003
%C Also n such that x^2 == 2 (mod n) has a solution. The primes are given in sequence A001132. - _T. D. Noe_, Jun 13 2003
%C As indicated in the formula, a(n) is related to the even triangular numbers. - Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 17 2004
%C Cf. property described by _Gary Detlefs_ in A113801: more generally, these a(n) are of the form (2*h*n + (h-4)*(-1)^n-h)/4 (h,n natural numbers). Therefore a(n)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 8). Also a(n)^2 - 1 == 0 (mod 16). - _Bruno Berselli_, Nov 17 2010
%C A089911(3*a(n)) = 2. - _Reinhard Zumkeller_, Jul 05 2013
%C S(a(n+1)/2, 0) = (1/2)*(S(a(n+1), sqrt(2)) - S(a(n+1) - 2, sqrt(2))) = T(a(n+1), sqrt(2)/2) = cos(a(n+1)*Pi/4) = sqrt(2)/2 = A010503, identically for n >= 0, where S is the Chebyshev polynomial (A049310) here extended to fractional n, evaluated at x = 0. (For T see A053120.) - _Wolfdieter Lang_, Jun 04 2023
%D L. B. W. Jolley, "Summation of Series", Dover Publications, p. 16.
%H Reinhard Zumkeller, <a href="/A047522/b047522.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F a(n) = sqrt(8*A014494(n)+1) = sqrt(16*ceiling(n/2)*(2*n+1)+1) = sqrt(8*A056575(n)-8*(2n+1)*(-1)^n+1). - Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 17 2004
%F 1 - 1/7 + 1/9 - 1/15 + 1/17 - ... = (Pi/8)*(1 + sqrt(2)). [Jolley] - _Gary W. Adamson_, Dec 16 2006
%F From _R. J. Mathar_, Feb 19 2009: (Start)
%F a(n) = 4n - 2 + (-1)^n = a(n-2) + 8.
%F G.f.: x(1+6x+x^2)/((1+x)(1-x)^2). (End)
%F a(n) = 8*n - a(n-1) - 8. - _Vincenzo Librandi_, Aug 06 2010
%F From _Bruno Berselli_, Nov 17 2010: (Start)
%F a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
%F a(n) = 8*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
%F E.g.f.: 1 + (4*x - 1)*cosh(x) + (4*x - 3)*sinh(x). - _Stefano Spezia_, May 13 2021
%F E.g.f.: 1 + (4*x - 3)*exp(x) + 2*cosh(x). - _David Lovler_, Jul 16 2022
%t Select[Range[1, 191, 2], JacobiSymbol[2, # ]==1&]
%o (Haskell)
%o a047522 n = a047522_list !! (n-1)
%o a047522_list = 1 : 7 : map (+ 8) a047522_list
%o -- _Reinhard Zumkeller_, Jan 07 2012
%o (PARI) a(n)=4*n-2+(-1)^n \\ _Charles R Greathouse IV_, Sep 24 2015
%Y Cf. A001132, A014494, A056575, A010709, A074378, A047336, A056020, A005408, A047209, A007310, A090771, A175885, A091998, A175886, A175887, A058529, A047621.
%Y Cf. A077221 (partial sums).
%Y Cf. A000217, A089911, A113801.
%Y Cf. A010503, A049310, A053120.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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