A130793 Periodic sequence with period 3: 1, 3, 5.

1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1, 3, 5, 1

Offset: 0

Author

Paul Curtz, Jul 15 2007

Keywords

• nonn
• easy

Comments

Continued fraction expansion of (9+sqrt(145))/16. - Klaus Brockhaus, Apr 28 2010

Decimal expansion of 5/37. - Pontus von Brömssen, Dec 11 2024

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 277.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Crossrefs

Cf. A021078, A049347.

Cf. A176907 (decimal expansion of (9+sqrt(145))/16). - Klaus Brockhaus, Apr 28 2010

Programs

Magma

&cat [[1, 3, 5]^^35]; // Vincenzo Librandi, Jul 25 2018

Maple

A130793:=n->((2*n+1) mod 6); seq(A130793(n), n=0..100); # Wesley Ivan Hurt, Mar 30 2014

Mathematica

Table[Mod[2 n + 1, 6], {n, 0, 100}] (* Wesley Ivan Hurt, Mar 30 2014 *)
PadRight[{}, 105, {1, 3, 5}] (* After Harvey P. Dale *)
Nest[Flatten[# /. {1 -> {1, 3}, 3 -> {5, 1}, 5 -> {3, 5}}] &, {1}, 7] (* or *) CoefficientList[Series[-(5 x^2 + 3 x + 1)/(x^3 - 1), {x, 0, 105}], x] (* or *) LinearRecurrence[{0, 0, 1}, {1, 3, 5}, 105] (* Robert G. Wilson v, Jul 25 2018 *)

PARI

a(n)=[1,3,5][n%3+1] \\ Charles R Greathouse IV, Jun 02 2011

Python

def A130793(n): return (n%3<<1)+1 # Chai Wah Wu, Apr 02 2025

Formula

From R. J. Mathar, Jun 13 2008: (Start)

a(n) = 3+2*A049347(n+1).

O.g.f.: (1+3x+5x^2)/((1-x)(1+x+x^2)). (End)

a(n) = ((n+1)^6 - n^6) mod 6. - Gary Detlefs, Mar 25 2012

a(n) = (2n+1) mod 6. - Wesley Ivan Hurt, Mar 30 2014

a(n) = 2*(n mod 3) + 1. - Bruno Berselli, Jul 25 2018

a(n) = (2*r^n*(r-1)-2*r^(2*n)*(r+2)+9)/3 where r=(-1+i*sqrt(3))/2. - Ammar Khatab, Nov 28 2020

E.g.f.: 3*exp(x) - 2*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Jul 28 2025