A106612 a(n) = numerator(n/(n+11)).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 2, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 3, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 4, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 5, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 6, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 0

Author

N. J. A. Sloane, May 15 2005

Keywords

• nonn
• frac
• mult
• easy

Comments

In general, the numerators of n/(n+p) for prime p and n >= 0, form a sequence with the g.f.: x/(1-x)^2 - (p-1)*x^p/(1-x^p)^2. - Paul D. Hanna, Jul 27 2005

a(n) <> n iff n = 11 * k, in this case, a(n) = k. - Bernard Schott, Feb 19 2019

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,-1).

Crossrefs

Cf. A109052, A137564 (differs, e.g., for n=100).

Cf. Sequences given by the formula numerator(n/(n + k)): A026741 (k = 2), A051176 (k = 3), A060819 (k = 4), A060791 (k = 5), A060789 (k = 6), A106608 thru A106611 (k = 7 thru 10), A051724 (k = 12), A106614 thru A106621 (k = 13 thru 20).

Programs

GAP

List([0..80],n->NumeratorRat(n/(n+11))); # Muniru A Asiru, Feb 19 2019

Magma

[Numerator(n/(n+11)): n in [0..100]]; // Vincenzo Librandi, Apr 18 2011

Maple

seq(numer(n/(n+11)),n=0..80); # Muniru A Asiru, Feb 19 2019

Mathematica

f[n_]:=Numerator[n/(n+11)];Array[f,100,0] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2011 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,-1},{0,1,2,3,4,5,6,7,8,9,10,1,12,13,14,15,16,17,18,19,20,21},80] (* Harvey P. Dale, Jul 05 2021 *)

PARI

vector(100, n, n--; numerator(n/(n+11))) \\ G. C. Greubel, Feb 19 2019

Sage

[lcm(n,11)/11 for n in range(0, 54)] # Zerinvary Lajos, Jun 09 2009

Formula

G.f.: x/(1-x)^2 - 10*x^11/(1-x^11)^2. - Paul D. Hanna, Jul 27 2005

a(n) = lcm(n,11)/11.

From R. J. Mathar, Apr 18 2011: (Start)

a(n) = A109052(n)/11.

Dirichlet g.f.: zeta(s-1)*(1-10/11^s). (End)

a(n) = 2*a(n-11) - a(n-22). - G. C. Greubel, Feb 19 2019

From Amiram Eldar, Nov 25 2022: (Start)

Multiplicative with a(11^e) = 11^(e-1), and a(p^e) = p^e if p != 11.

Sum_{k=1..n} a(k) ~ (111/242) * n^2. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = 21*log(2)/11. - Amiram Eldar, Sep 08 2023