This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A047522 #66 Nov 22 2024 11:05:12
%S A047522 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 A047522 97,103,105,111,113,119,121,127,129,135,137,143,145,151,153,159,161,
%U A047522 167,169,175,177,183,185,191,193,199,201,207,209,215,217,223,225,231,233
%N A047522 Numbers that are congruent to {1, 7} mod 8.
%C A047522 Also n such that Kronecker(2,n) = mu(gcd(2,n)). - _Jon Perry_ and _T. D. Noe_, Jun 13 2003
%C A047522 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 A047522 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 A047522 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 A047522 A089911(3*a(n)) = 2. - _Reinhard Zumkeller_, Jul 05 2013
%C A047522 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 A047522 L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 16.
%H A047522 Reinhard Zumkeller, <a href="/A047522/b047522.txt">Table of n, a(n) for n = 1..10000</a>
%H A047522 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A047522 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 A047522 1 - 1/7 + 1/9 - 1/15 + 1/17 - ... = (Pi/8)*(1 + sqrt(2)). [Jolley] - _Gary W. Adamson_, Dec 16 2006
%F A047522 From _R. J. Mathar_, Feb 19 2009: (Start)
%F A047522 a(n) = 4n - 2 + (-1)^n = a(n-2) + 8.
%F A047522 G.f.: x(1+6x+x^2)/((1+x)(1-x)^2). (End)
%F A047522 a(n) = 8*n - a(n-1) - 8. - _Vincenzo Librandi_, Aug 06 2010
%F A047522 From _Bruno Berselli_, Nov 17 2010: (Start)
%F A047522 a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3).
%F A047522 a(n) = 8*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End)
%F A047522 E.g.f.: 1 + (4*x - 1)*cosh(x) + (4*x - 3)*sinh(x). - _Stefano Spezia_, May 13 2021
%F A047522 E.g.f.: 1 + (4*x - 3)*exp(x) + 2*cosh(x). - _David Lovler_, Jul 16 2022
%F A047522 From _Amiram Eldar_, Nov 22 2024: (Start)
%F A047522 Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2+sqrt(2)) (A179260).
%F A047522 Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/8)*cosec(Pi/8) (A352125). (End)
%t A047522 Select[Range[1, 191, 2], JacobiSymbol[2, # ]==1&]
%o A047522 (Haskell)
%o A047522 a047522 n = a047522_list !! (n-1)
%o A047522 a047522_list = 1 : 7 : map (+ 8) a047522_list
%o A047522 -- _Reinhard Zumkeller_, Jan 07 2012
%o A047522 (PARI) a(n)=4*n-2+(-1)^n \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A047522 Cf. A001132, A014494, A056575, A010709, A074378, A047336, A056020, A005408, A047209, A007310, A090771, A175885, A091998, A175886, A175887, A058529, A047621, A179260, A352125.
%Y A047522 Cf. A077221 (partial sums).
%Y A047522 Cf. A000217, A089911, A113801.
%Y A047522 Cf. A010503, A049310, A053120.
%K A047522 nonn,easy
%O A047522 1,2
%A A047522 _N. J. A. Sloane_