A159008 Positive numbers k such that k^2 == 2 (mod 89).

25, 64, 114, 153, 203, 242, 292, 331, 381, 420, 470, 509, 559, 598, 648, 687, 737, 776, 826, 865, 915, 954, 1004, 1043, 1093, 1132, 1182, 1221, 1271, 1310, 1360, 1399, 1449, 1488, 1538, 1577, 1627, 1666, 1716, 1755, 1805, 1844, 1894, 1933, 1983, 2022, 2072

Offset: 1

Author

Vincenzo Librandi, Jun 30 2009

Keywords

• nonn
• easy

Comments

Numbers congruent to {25, 64} mod 89. - Amiram Eldar, Feb 26 2023

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. A159007.

Programs

Magma

I:=[25, 64, 114]; [n le 3 select I[n] else Self(n-1)+Self(n-2)-Self(n-3): n in [1..60]]; // Vincenzo Librandi, Mar 02 2012

Mathematica

LinearRecurrence[{1, 1, -1}, {25, 64, 114}, 50] (* Vincenzo Librandi, Mar 02 2012 *)
Select[Range[2100],PowerMod[#,2,89]==2&] (* Harvey P. Dale, May 09 2019 *)

PARI

for(n=1, 60, print1((89+11*(-1)^(n-1)+178*(n-1))/4", ")); \\ Vincenzo Librandi, Mar 02 2012

Formula

a(n) = a(n-1) + a(n-2) - a(n-3).

G.f.: x*(25 + 39*x + 25*x^2)/((1+x)*(x-1)^2).

a(n) = (89 + 11*(-1)^(n-1) + 178*(n-1))/4.

Sum_{n>=1} (-1)^(n+1)/a(n) = tan(39*Pi/178)*Pi/89. - Amiram Eldar, Feb 26 2023

Extensions

Slightly edited by R. J. Mathar, Jul 26 2009