A155107 - OEIS

%I #31 Sep 08 2022 08:45:40

%S 23,30,76,83,129,136,182,189,235,242,288,295,341,348,394,401,447,454,

%T 500,507,553,560,606,613,659,666,712,719,765,772,818,825,871,878,924,

%U 931,977,984,1030,1037,1083,1090,1136,1143,1189,1196,1242,1249,1295

%N Numbers that are 23 or 30 (mod 53).

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

%C 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...

%H Vincenzo Librandi, <a href="/A155107/b155107.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

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

%F 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

%F a(n) = a(n-2) + 53 for all n > 2. - _M. F. Hasler_, Jun 16 2010

%F From _R. J. Mathar_, Feb 19 2009: (Start)

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

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

%t Select[Range[1300], PowerMod[#, 2, 53] == 52 &] (* or *) LinearRecurrence[ {1, 1, -1}, {23, 30, 76}, 50] (* _Harvey P. Dale_, Nov 30 2011 *)

%t CoefficientList[Series[(23 + 7 x + 23 x^2)/((1 + x) (1 - x)^2), {x, 0, 100}], x] (* _Vincenzo Librandi_, Apr 24 2014 *)

%o (PARI) A155107(n)=n\2*53-23*(-1)^n /* _M. F. Hasler_, Jun 16 2010 */

%o (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

%Y 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).

%K nonn,easy

%O 1,1

%A _Vincenzo Librandi_, Jan 20 2009

%E Terms checked & minor edits by _M. F. Hasler_, Jun 16 2010