A156638 Numbers k such that k^2 + 2 == 0 (mod 9).

4, 5, 13, 14, 22, 23, 31, 32, 40, 41, 49, 50, 58, 59, 67, 68, 76, 77, 85, 86, 94, 95, 103, 104, 112, 113, 121, 122, 130, 131, 139, 140, 148, 149, 157, 158, 166, 167, 175, 176, 184, 185, 193, 194, 202, 203, 211, 212, 220, 221, 229, 230, 238, 239, 247, 248, 256

Offset: 1

Author

Vincenzo Librandi, Feb 12 2009

Keywords

• nonn
• easy

Comments

From Artur Jasinski, Apr 30 2010: (Start)

Numbers congruent to 4 or 5 mod 9.

Numbers which are not the sum of 3 cubes.

Complement to A060464. (End)

Numbers k such that A010888(k^2) = 7. - V.J. Pohjola, Aug 18 2012

References

• Henri Cohen, Number Theory Volume I: Tools and Diophantine Equations. Springer Verlag (2007) p. 380. - Artur Jasinski, Apr 30 2010

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Andrew Sutherland, Sums of three cubes, Slides of a talk given May 07 2020 on the Number Theory Web.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A060464, A010888, A332437, A334521, A334522.

Programs

Magma

[9*n/2 - 9/4 - 7*(-1)^n/4 : n in [1..80]]; // Wesley Ivan Hurt, Aug 16 2015

Maple

A156638:=n->9*n/2 - 9/4 - 7*(-1)^n/4: seq(A156638(n), n=1..80); # Wesley Ivan Hurt, Aug 16 2015

Mathematica

LinearRecurrence[{1, 1, -1}, {4, 5, 13}, 50] (* Vincenzo Librandi, Mar 01 2012 *)
Flatten[Table[9n - {5, 4}, {n, 30}]] (* Alonso del Arte, Aug 09 2015 *)
Select[Range[300],PowerMod[#,2,9]==7&] (* Harvey P. Dale, Jan 31 2023 *)

PARI

a(n) = (18*n - 9 - 7*(-1)^n)/4 \\ David Lovler, Aug 21 2022

Formula

For n > 2, a(n) = a(n-2) + 9.

G.f.: x*(4*x^2 + x + 4)/(x^3 - x^2 - x + 1). - Alexander R. Povolotsky, Feb 15 2009

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

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

a(n) = 9*n/2 - 9/4 - 7*(-1)^n/4.

G.f.: x*(4 + x + 4*x^2)/((1 + x)*(1 - x)^2). (End)

a(n) = -a(-n+1). - Bruno Berselli, Jan 08 2012

E.g.f.: 4 + ((18*x - 9)*exp(x) - 7*exp(-x))/4. - David Lovler, Aug 21 2022

Sum_{n>=1} (-1)^(n+1)/a(n) = tan(Pi/18)*Pi/9. - Amiram Eldar, Sep 26 2022

From Amiram Eldar, Nov 22 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = 1.

Product_{n>=1} (1 + (-1)^n/a(n)) = 2*cos(Pi/9) - 1 (= A332437 - 1). (End)