A174571 a(4n)=n, a(4n+1)=4, a(4n+2)=1, a(4n+3)=4.

0, 4, 1, 4, 1, 4, 1, 4, 2, 4, 1, 4, 3, 4, 1, 4, 4, 4, 1, 4, 5, 4, 1, 4, 6, 4, 1, 4, 7, 4, 1, 4, 8, 4, 1, 4, 9, 4, 1, 4, 10, 4, 1, 4, 11, 4, 1, 4, 12, 4, 1, 4, 13, 4, 1, 4, 14, 4, 1, 4, 15, 4, 1, 4, 16, 4, 1, 4, 17, 4, 1, 4, 18, 4, 1, 4, 19, 4, 1, 4, 20, 4, 1, 4

Offset: 0

Author

Paul Curtz, Nov 29 2010

Keywords

• nonn
• easy

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16383
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Crossrefs

Cf. A010685.

Programs

Magma

[(n mod 4) eq 0 select n/4 else Modexp(4,n,5): n in [0..90]]; // G. C. Greubel, Nov 23 2021

Mathematica

Array[Which[OddQ@ Mod[#, 4], 4, Mod[#, 4] == 0, #/4, True, 1] &, 84, 0] (* or *)
CoefficientList[Series[x*(4 +x +4*x^2 +x^3 -4*x^4 -x^5 -4*x^6)/(1-x^4)^2, {x, 0, 83}], x] (* Michael De Vlieger, Nov 06 2018 *)
LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,4,1,4,1,4,1,4},100] (* Harvey P. Dale, Dec 21 2024 *)

PARI

A174571(n) = if(!(n%4),n/4,if(2==(n%4),1,4)); \\ Antti Karttunen, Nov 06 2018

Sage

def A174571(n): return n/4 if (n%4==0) else power_mod(4,n,5)
[A174571(n) for n in (0..90)] # G. C. Greubel, Nov 23 2021

Formula

a(n) = A010685(n) if 4 does not divide n.

a(n) = 2*a(n-4) - a(n-8).

G.f.: x*(4 + x + 4*x^2 + x^3 - 4*x^4 - x^5 - 4*x^6)/( (1-x)*(1+x)*(1+x^2) )^2.

a(n) = (36 +n +(n-28)*(-1)^n +2*(n -5 +(-1)^n)*cos(n*Pi/2) +(1+(-1)^n)*sin(n*Pi/2) )/16. - Wesley Ivan Hurt, May 07 2021

E.g.f.: (1/8)*(4*cosh(x) + (x+32)*sinh(x) - 4*cos(x) - x*sin(x)). - G. C. Greubel, Nov 23 2021