A281098 - OEIS

%I #41 Feb 15 2023 13:51:57

%S 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,

%T 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,

%U 3,29,1,60,1,31,3,64,1,33,1,68,3,35,1,72,1,37,3,76,1,39,1

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

%C Successive sequences:

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

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

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

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

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

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

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

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

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

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

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

%H Antti Karttunen, <a href="/A281098/b281098.txt">Table of n, a(n) for n = 0..16384</a>

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1,0,1,0,2,0,1,0,-1,0,-1).

%F 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

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

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

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

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

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

%F a(4*k)   = 4*k.

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

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

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

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

%F 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

%t 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 *)

%o (PARI) f(n) = numerator((4 + n^2)/4);

%o a(n) = gcd(vector(1000, k, f(k+n) - f(k))); \\ _Michel Marcus_, Jan 15 2017

%o (PARI) A281098(n) = if(n%2, gcd((n\2)-1,3), n>>(bitand(n,2)/2)); \\ _Antti Karttunen_, Feb 15 2023

%Y Cf. A064680, A144433 or A195161.

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

%K nonn,easy

%O 0,4

%A _Paul Curtz_, Jan 14 2017

%E Corrected and extended by _Michel Marcus_, Jan 15 2017