A145911 -id:A145911 - cp's OEIS frontend

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

Paul Curtz, Feb 05 2011

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

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

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

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 *)

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

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

A178978 a(n) = A144448(n+1)/8.

Original entry on oeis.org

0, 2, 5, 1, 14, 20, 1, 35, 44, 2, 65, 77, 10, 104, 119, 5, 152, 170, 7, 209, 230, 28, 275, 299, 4, 350, 377, 5, 434, 464, 55, 527, 560, 22, 629, 665, 26, 740, 779, 91, 860, 902, 35, 989, 1034, 40, 1127, 1175, 136, 1274, 1325, 17

Offset: 0

Paul Curtz, Jan 02 2011

Differs from A178971 for indices n > 23.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A061039, A145910, A145911, A178971, A178977, A178978, A144448.

A061039 := proc(n) numer(1/9-1/n^2) ; end proc:
A144448 := proc(n) A061039(1+2*n) ; end proc:
A178978 := proc(n) A144448(n+1)/8 ; end proc:
seq(A178978(n),n=0..80) ; # R. J. Mathar, Jan 06 2011

Table[Numerator[1/9 -1/(2*n+3)^2]/8, {n, 0, 75}] (* G. C. Greubel, Mar 06 2022 *)

[numerator(1/9 -1/(2*n+3)^2)/8 for n in (0..75)] # G. C. Greubel, Mar 06 2022

Trisections:
  a(3*n)   = A145911(n);
  a(3*n+1) = A145910(n);
  a(3*n+2) = A178977(n).
a(n) = 3*a(n-27) - 3*a(n-54) + a(n-81). - G. C. Greubel, Mar 06 2022

cp's OEIS Frontend

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

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