A155097 Numbers k such that k^2 == -1 (mod 37).

6, 31, 43, 68, 80, 105, 117, 142, 154, 179, 191, 216, 228, 253, 265, 290, 302, 327, 339, 364, 376, 401, 413, 438, 450, 475, 487, 512, 524, 549, 561, 586, 598, 623, 635, 660, 672, 697, 709, 734, 746, 771, 783, 808, 820, 845, 857, 882, 894, 919, 931, 956, 968

Offset: 1

Author

Vincenzo Librandi, Jan 20 2009

Keywords

• nonn
• easy

Comments

Numbers k such that k == 6 or 31 (mod 37). - Charles R Greathouse IV, Dec 27 2011

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. A002144, A155086, A155095, A155096, A155098.

Programs

Mathematica

LinearRecurrence[{1,1,-1},{6,31,43},100] (* Vincenzo Librandi, Feb 29 2012 *)
Select[Range[1000],PowerMod[#,2,37]==36&] (* Harvey P. Dale, May 06 2012 *)
CoefficientList[Series[(6 + 25 x + 6 x^2)/((1 + x) (1 - x)^2), {x, 0, 30}], x] (* Vincenzo Librandi, May 03 2014 *)

PARI

A155097(n)=n\2*37-6*(-1)^n /* M. F. Hasler, Jun 16 2010 */

Formula

From M. F. Hasler, Jun 16 2010: (Start)

a(n) = 6*(-1)^(n+1) + 37*floor(n/2).

a(2k+1) = 37*k + a(1), a(2k) = 37*k - a(1), with a(1) = A002314(5) since 37 = A002144(5).

a(n) = a(n-2) + 37 for all n > 2. (End)

G.f.: x*(6 + 25*x + 6*x^2)/((1 + x)*(1 - x)^2). - Vincenzo Librandi, May 03 2014

Sum_{n>=1} (-1)^(n+1)/a(n) = cot(6*Pi/37)*Pi/37. - Amiram Eldar, Feb 26 2023

Extensions

Terms checked, a(28) corrected, and minor edits by M. F. Hasler, Jun 16 2010