A047522 Numbers that are congruent to {1, 7} mod 8.

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

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, 1961, 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

From Amiram Eldar, Nov 22 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2+sqrt(2)) (A179260).

Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/8)*cosec(Pi/8) (A352125). (End)

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, A179260, A352125.

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