A155107 Numbers that are 23 or 30 (mod 53).

23, 30, 76, 83, 129, 136, 182, 189, 235, 242, 288, 295, 341, 348, 394, 401, 447, 454, 500, 507, 553, 560, 606, 613, 659, 666, 712, 719, 765, 772, 818, 825, 871, 878, 924, 931, 977, 984, 1030, 1037, 1083, 1090, 1136, 1143, 1189, 1196, 1242, 1249, 1295

Offset: 1

Author

Vincenzo Librandi, Jan 20 2009

Keywords

• nonn
• easy

Comments

Also, numbers k such that k^2 == -1 (mod 53).

The first pair (a,b) is such that a+b=p=53, a*b=p*h+1, with h<=(p-1)/4; subsequent pairs are given as (a+kp, b+kp), k=1,2,3...

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. numbers n such that n^2 == -1 (mod p), where p is a prime of the form 4k+1: A047221 (p=5), A155086 (p=13), A155095 (p=17), A155096 (p=29), A155097 (p=37), A155098 (p=41), this sequence (p=53), A241406 (p=61), A241407 (p=73), A241520 (p=89), A241521 (p=97).

Programs

Magma

I:=[23,30,76]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..50]]; // Vincenzo Librandi, Apr 24 2014

Magma

[-23*(-1)^n+53*Floor(n/2): n in [1..50]]; // Vincenzo Librandi, Apr 24 2014

Mathematica

Select[Range[1300], PowerMod[#, 2, 53] == 52 &] (* or *) LinearRecurrence[ {1, 1, -1}, {23, 30, 76}, 50] (* Harvey P. Dale, Nov 30 2011 *)
CoefficientList[Series[(23 + 7 x + 23 x^2)/((1 + x) (1 - x)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Apr 24 2014 *)

PARI

A155107(n)=n\2*53-23*(-1)^n /* M. F. Hasler, Jun 16 2010 */

Formula

a(n) = 23*(-1)^(n+1) + 53*floor(n/2). - M. F. Hasler, Jun 16 2010

a(2k+1) = 53 k + a(1), a(2k) = 53 k - a(1), with a(1) = 23 = A002314(7) since 53 = A002144(7). - M. F. Hasler, Jun 16 2010

a(n) = a(n-2) + 53 for all n > 2. - M. F. Hasler, Jun 16 2010

From R. J. Mathar, Feb 19 2009: (Start)

a(n) = a(n-1) + a(n-2) - a(n-3) = 53*n/2 - 53/4 - 39*(-1)^n/4.

G.f.: x*(23 + 7*x + 23*x^2)/((1+x)*(1-x)^2). (End)

Extensions

Terms checked & minor edits by M. F. Hasler, Jun 16 2010