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

1,1

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).

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)

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 *)

Prog

(PARI) A155107(n)=n\2*53-23*(-1)^n /* M. F. Hasler, Jun 16 2010 */
(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]] /* or */ [-23*(-1)^n+53*Floor(n/2): n in [1..50]]; // Vincenzo Librandi, Apr 24 2014

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).

Sequence in context: A036267 A107934 A356087 * A126719 A110677 A169640

Adjacent sequences: A155104 A155105 A155106 * A155108 A155109 A155110

Keyword

nonn,easy

Author

Vincenzo Librandi, Jan 20 2009

Extensions

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

Status

approved