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 A047522 Numbers that are congruent to {1, 7} mod 8. 32
 1, 7, 9, 15, 17, 23, 25, 31, 33, 39, 41, 47, 49, 55, 57, 63, 65, 71, 73, 79, 81, 87, 89, 95, 97, 103, 105, 111, 113, 119, 121, 127, 129, 135, 137, 143, 145, 151, 153, 159, 161, 167, 169, 175, 177, 183, 185, 191, 193, 199, 201, 207, 209, 215, 217, 223, 225, 231, 233 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also n such that Kronecker(2,n) = mu(gcd(2,n)). - Jon Perry and T. D. Noe, Jun 13 2003 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 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 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 A089911(3*a(n)) = 2. - Reinhard Zumkeller, Jul 05 2013 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 REFERENCES L. B. W. Jolley, "Summation of Series", Dover Publications, p. 16. LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 Index entries for linear recurrences with constant coefficients, signature (1,1,-1). FORMULA 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 1 - 1/7 + 1/9 - 1/15 + 1/17 - ... = (Pi/8)*(1 + sqrt(2)). [Jolley] - Gary W. Adamson, Dec 16 2006 From R. J. Mathar, Feb 19 2009: (Start) a(n) = 4n - 2 + (-1)^n = a(n-2) + 8. G.f.: x(1+6x+x^2)/((1+x)(1-x)^2). (End) a(n) = 8*n - a(n-1) - 8. - Vincenzo Librandi, Aug 06 2010 From Bruno Berselli, Nov 17 2010: (Start) a(n) = -a(-n+1) = a(n-1) + a(n-2) - a(n-3). a(n) = 8*A000217(n-1)+1 - 2*Sum_{i=1..n-1} a(i) for n > 1. (End) E.g.f.: 1 + (4*x - 1)*cosh(x) + (4*x - 3)*sinh(x). - Stefano Spezia, May 13 2021 E.g.f.: 1 + (4*x - 3)*exp(x) + 2*cosh(x). - David Lovler, Jul 16 2022 MATHEMATICA Select[Range[1, 191, 2], JacobiSymbol[2, # ]==1&] PROG (Haskell) a047522 n = a047522_list !! (n-1) a047522_list = 1 : 7 : map (+ 8) a047522_list -- Reinhard Zumkeller, Jan 07 2012 (PARI) a(n)=4*n-2+(-1)^n \\ Charles R Greathouse IV, Sep 24 2015 CROSSREFS Cf. A001132, A014494, A056575, A010709, A074378, A047336, A056020, A005408, A047209, A007310, A090771, A175885, A091998, A175886, A175887, A058529, A047621. Cf. A077221 (partial sums). Cf. A000217, A089911, A113801. Cf. A010503, A049310, A053120. Sequence in context: A067873 A217460 A347436 * A112072 A024902 A111312 Adjacent sequences: A047519 A047520 A047521 * A047523 A047524 A047525 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified December 8 14:30 EST 2023. Contains 367679 sequences. (Running on oeis4.)