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A113801 Numbers that are congruent to {1, 13} mod 14. 24
1, 13, 15, 27, 29, 41, 43, 55, 57, 69, 71, 83, 85, 97, 99, 111, 113, 125, 127, 139, 141, 153, 155, 167, 169, 181, 183, 195, 197, 209, 211, 223, 225, 237, 239, 251, 253, 265, 267, 279, 281, 293, 295, 307, 309, 321, 323, 335, 337, 349, 351, 363, 365, 377, 379 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

If 14k+1 is a perfect square..(0,12,16,52,60,120..) then the square root of 14k+1 = a(n) - Gary Detlefs, Feb 22 2010

More generally, these numbers are of the form (2*h*n+(h-4)*(-1)^n-h)/4 (h, n natural numbers), therefore ((2*h*n+(h-4)*(-1)^n-h)/4)^2-1==0 (mod h); in our case, a(n)^2-1==0 (mod 14). Also a(n)^2-1==0 (mod 28). - Bruno Berselli, Oct 26 2010 - Nov 17 2010

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) = 14*(n-1)-a(n-1), n>1. - R. J. Mathar, Jan 30 2010

From Bruno Berselli, Oct 26 2010: (Start)

a(n) = -a(-n+1) = (14*n+5*(-1)^n-7)/2.

G.f.: x*(1+12*x+x^2)/((1+x)*(1-x)^2).

a(n) = a(n-2)+14 for n>2.

a(n) = 14*A000217(n-1)+1 - 2*sum[i=1..n-1] a(i) for n>1. (End)

a(0)=1, a(1)=13, a(2)=15, a(n)=a(n-1)+a(n-2)-a(n-3). - Harvey P. Dale, May 11 2011

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/14)*cot(Pi/14). - Amiram Eldar, Dec 04 2021

MATHEMATICA

LinearRecurrence[{1, 1, -1}, {1, 13, 15}, 60] (* or *) Select[Range[500], MemberQ[{1, 13}, Mod[#, 14]]&] (* Harvey P. Dale, May 11 2011 *)

PROG

(Haskell)

a113801 n = a113801_list !! (n-1)

a113801_list = 1 : 13 : map (+ 14) a113801_list

-- Reinhard Zumkeller, Jan 07 2012

(PARI) a(n)=n\2*14-(-1)^n \\ Charles R Greathouse IV, Sep 15 2015

CROSSREFS

Cf. A000217, A113802, A113803, A113804, A113805, A113806, A113807, A008589, A045472 (primes), A195145 (partial sums), A005408, A047209, A007310, A047336, A047522, A056020, A090771, A175885, A091998, A175886, A175887.

Sequence in context: A178724 A087814 A227449 * A108257 A217252 A299593

Adjacent sequences:  A113798 A113799 A113800 * A113802 A113803 A113804

KEYWORD

nonn,easy

AUTHOR

Giovanni Teofilatto, Jan 22 2006

EXTENSIONS

Corrected and extended by Giovanni Teofilatto, Nov 14 2008

Replaced the various formulas by a correct one - R. J. Mathar, Jan 30 2010

STATUS

approved

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Last modified August 17 19:38 EDT 2022. Contains 356189 sequences. (Running on oeis4.)