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A155107 Numbers n that are 23 or 30 (mod 53). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

also, numbers n such that n^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

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). [R. J. Mathar, Feb 19 2009]

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: A068714 A036267 A107934 * 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

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Last modified August 18 15:00 EDT 2019. Contains 326106 sequences. (Running on oeis4.)