A185727 Integers of the form A145911(k)/(k+1) sorted along increasing k.

0, 2, 1, 5, 2, 8, 1, 11, 4, 14, 5, 17, 2, 20, 7, 23, 8, 26, 3, 29, 10, 32, 11, 35, 4, 38, 13, 41, 14, 44, 5, 47, 16, 50, 17, 53, 6, 56, 19, 59, 20, 62, 7, 65, 22, 68, 23, 71, 8, 74, 25, 77, 26, 80, 9, 83, 28, 86, 29, 89, 10, 92, 31, 95, 32, 98, 11, 101, 34, 104, 35, 107, 12, 110, 37, 113, 38, 116, 13, 119, 40, 122

Offset: 0

Author

Paul Curtz, Feb 05 2011

Keywords

• nonn
• easy
• less

Comments

The standard offset is changed to zero to simplify formulas related to the a(n).

The sequence of fractions A145911(k)/(k+1) is 0, 1/2, 1/3, 1/2, 2, 5/6, 1, 7/2, 4/9, 1/2, 5, 11/6, 2, 13/2, 7/3, 5/2, 8, 17/18, 1, 19/2, 10/3, 7/2, 11, 23/6, 4, 25/2, 13/9, 3/2, 14, 29/6, ....

Its numerators are A106619. Integer values appear at indices of the form 6*n and 4+6*n.

The sequence of denominators of the fractions appears to have a period of length 18.

a(n+18)-a(n) = 3*(a(n+6)-a(n)) = 3, 27, 9, 27, 9, 27, 3, 27, 9, ,... are multiples of 3, apparently with a period of length 6.

The recurrence a(n) = 2a(n-6)-a(n-12) shows that the sequence consists of 6 interleaved first-order polynomials: a(6*n)=n. a(1+6*n) = 2+9*n. a(2+6*n) = 1+3*n = A016777(n). a(3+6*n) = 5+9*n. a(4+6*n) = 2+3*n = A016789(n). a(5+6*n) = 8+9*n. - Paul Curtz, Feb 23 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Programs

Maple

A106619 := proc(n) numer(n/(n+18)) ; end proc:
A185727 := proc(n) if type(n,'even') then A106619(3*n) ; else A106619(3*n+1) ; end if; end proc:
seq(A185727(n),n=0..80) ; # R. J. Mathar, Feb 18 2011

Mathematica

CoefficientList[Series[x*(2 + x + 5*x^2 + 2*x^3 + 8*x^4 + x^5 + 7*x^6 + 2*x^7 + 4*x^8 + x^9 + x^10)/((x - 1)^2*(1 + x)^2*(1 + x + x^2)^2*(x^2 - x + 1)^2), {x,0,50}], x] (* G. C. Greubel, Jul 11 2017 *)

PARI

x='x+O('x^50); concat([0], Vec(x*(2 + x + 5*x^2 + 2*x^3 + 8*x^4 + x^5 + 7*x^6 + 2*x^7 + 4*x^8 + x^9 + x^10)/((x - 1)^2*(1 + x)^2*(1 + x + x^2)^2*(x^2 - x + 1)^2))) \\ G. C. Greubel, Jul 11 2017

Formula

a(2*n) = A051176(n) = A106619(6n).

a(1+2*n) = 2+3*n = A106619(4+6*n).

a(6*n) = n.

From R. J. Mathar, Feb 10 2011: (Start)

a(n)= +2*a(n-6) -a(n-12).

G.f.: x*(2+x +5*x^2 +2*x^3 +8*x^4 +x^5 +7*x ^6 +2*x^7 +4*x^8 +x^9 +x^10) / ( (x-1)^2*(1+x)^2*(1+x+x^2)^2*(x^2-x+1)^2 ). (End)

a(n) = A014682(n) if n is not a multiple of 6. - Paul Curtz, Feb 23 2011