A281098 a(n) is the GCD of the sequence d(n) = A261327(k+n) - A261327(k) for all k.

0, 1, 1, 3, 4, 1, 3, 1, 8, 3, 5, 1, 12, 1, 7, 3, 16, 1, 9, 1, 20, 3, 11, 1, 24, 1, 13, 3, 28, 1, 15, 1, 32, 3, 17, 1, 36, 1, 19, 3, 40, 1, 21, 1, 44, 3, 23, 1, 48, 1, 25, 3, 52, 1, 27, 1, 56, 3, 29, 1, 60, 1, 31, 3, 64, 1, 33, 1, 68, 3, 35, 1, 72, 1, 37, 3, 76, 1, 39, 1

Offset: 0

Author

Paul Curtz, Jan 14 2017

Keywords

• nonn
• easy

Comments

Successive sequences:

0: 0, 0, 0, 0, ... = 0 * ( )

1: 4, -3, 11, -8, ... = 1 * ( )

2: 1, 8, 3, 16, ... = 1 * ( ) A195161

3: 12, 0, 27, -3, ... = 3 * (4, 0, 9, -1, ...)

4: 4, 24, 8, 40, ... = 4 * (1, 6, 2, 10, ...) A064680

5; 28, 5, 51, 4, ... = 1 * ( )

6: 9, 48, 15, 72, ... = 3 * (3, 16, 5, 24, ...) A195161

7: 52, 12, 83, 13, ... = 1 * ( )

8: 16, 80, 24, 112, ... = 8 * (2, 10, 3, 14, ...) A064080

9: 84 21, 123, 24, ... = 3 * (28, 7, 41, 8, ...)

10: 25, 120, 35, 160, ... = 5 * (5, 24, 7, 32, ...) A195161

Links

• Antti Karttunen, Table of n, a(n) for n = 0..16384
• Index entries for linear recurrences with constant coefficients, signature (0,-1,0,1,0,2,0,1,0,-1,0,-1).

Crossrefs

Cf. A064680, A144433 or A195161.

Cf. A000012, A005408, A008586, A010701, A109007 (bisection), A016825, A165988 (via A022998), A166138, A166304, A280579.

Programs

Mathematica

CoefficientList[Series[(-x (-1 - x - 4 x^2 - 5 x^3 - 3 x^4 - 6 x^5 + 3 x^6 - 5 x^7 + 4 x^8 - x^9 + x^10))/((x^2 - x + 1) (1 + x + x^2) (x - 1)^2*(1 + x)^2*(1 + x^2)^2), {x, 0, 79}], x] (* Michael De Vlieger, Feb 02 2017 *)

PARI

f(n) = numerator((4 + n^2)/4);
a(n) = gcd(vector(1000, k, f(k+n) - f(k))); \\ Michel Marcus, Jan 15 2017

PARI

A281098(n) = if(n%2, gcd((n\2)-1,3), n>>(bitand(n,2)/2)); \\ Antti Karttunen, Feb 15 2023

Formula

G.f.: -x*( -1 - x - 4*x^2 - 5*x^3 - 3*x^4 - 6*x^5 + 3*x^6 - 5*x^7 + 4*x^8 - x^9 + x^10 )/( (x^2 - x + 1)*(1 + x + x^2)*(x - 1)^2*(1 + x)^2*(1 + x^2)^2 ). - R. J. Mathar, Jan 31 2017

a(2*k) = A022998(k).

a(2*k+1) = A109007(k-1).

a(3*k) = interleave 3*k*(3 +(-1)^k)/2, 3.

a(3*k+1) = interleave 1, A166304(k).

a(3*k+2) = interleave A166138(k), 1.

a(4*k) = 4*k.

a(4*k+1) = period 3: repeat [1, 1, 3].

a(4*k+2) = 1 + 2*k.

a(4*k+3) = period 3: repeat [3, 1, 1].

a(n+12) - a(n) = 6*A131743(n+3).

a(n) = (18*n + 40 - 16*cos(n*Pi/3) + 9*n*cos(n*Pi/2) + 32*cos(2*n*Pi/3) + (18*n - 40)*cos(n*Pi) + 3*n*cos(3*n*Pi/2) - 16*cos(5*n*Pi/3))/48. - Wesley Ivan Hurt, Oct 04 2018

Extensions

Corrected and extended by Michel Marcus, Jan 15 2017