A156619 Numbers congruent to {7, 18} mod 25.

7, 18, 32, 43, 57, 68, 82, 93, 107, 118, 132, 143, 157, 168, 182, 193, 207, 218, 232, 243, 257, 268, 282, 293, 307, 318, 332, 343, 357, 368, 382, 393, 407, 418, 432, 443, 457, 468, 482, 493, 507, 518, 532, 543, 557, 568, 582, 593, 607, 618, 632, 643, 657, 668

Offset: 1

Author

Vincenzo Librandi, Feb 11 2009

Keywords

• nonn
• easy

Comments

Also, numbers k such that k^2 + 1 == 0 (mod 25).

Numbers of the form 25*k+7 or 25*k+18. Numbers b such that 25 is a base-b Euler pseudoprime. - Karsten Meyer, Jan 05 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).

Programs

Magma

[n: n in [1..700] | n mod 25 in [7, 18]]; // Vincenzo Librandi, Apr 08 2013

Mathematica

fQ[n_] := Mod[n^2 + 1, 25] == 0; Select[ Range@ 670, fQ]
Flatten[#+{7,18}&/@(25*Range[0,30])] (* Harvey P. Dale, Jan 24 2013 *)
Select[Range[1, 700], MemberQ[{7, 18}, Mod[#, 25]]&] (* Vincenzo Librandi, Apr 08 2013 *)

Formula

a(n) = 2*a(n-1)-a(n-2)-3, if n is even, and a(n) = 2*a(n-1)-a(n-2)+3, if n is odd, with a(1)=7, a(2)=18.

From R. J. Mathar, Feb 19 2009: (Start)

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

a(n) = 25*n/2-25/4-3*(-1)^n/4.

G.f.: x*(7+11*x+7*x^2)/((1+x)*(1-x)^2). (End)

E.g.f.: 7 + ((50*x - 25)*exp(x) - 3*exp(-x))/4. - David Lovler, Sep 08 2022

Sum_{n>=1} (-1)^(n+1)/a(n) = tan(11*Pi/50)*Pi/25. - Amiram Eldar, Feb 26 2023