A175885 Numbers that are congruent to {1, 10} mod 11.

1, 10, 12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 100, 109, 111, 120, 122, 131, 133, 142, 144, 153, 155, 164, 166, 175, 177, 186, 188, 197, 199, 208, 210, 219, 221, 230, 232, 241, 243, 252, 254, 263, 265, 274, 276, 285, 287, 296, 298

Offset: 1

Author

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Keywords

• nonn
• easy

Comments

Cf. property described by Gary Detlefs in A113801: more generally, these numbers are of the form (2*h*n + (h-4)*(-1)^n - h)/4 (h, n natural numbers), therefore ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in this case, a(n)^2 - 1 == 0 (mod 11).

Links

• Bruno Berselli, Table of n, a(n) for n = 1..10000.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A090771 (n==1 or 9 mod 10), A091998 (n==1 or 11 mod 12).

Cf. A195043 (partial sums).

Cf. A000217, A005408, A047209, A007310, A047336, A047522, A056020, A113801, A175886, A175887, A195312, A195313.

Programs

Haskell

a175885 n = a175885_list !! (n-1)
a175885_list = 1 : 10 : map (+ 11) a175885_list
-- Reinhard Zumkeller, Jan 07 2012

Magma

[(22*n+7*(-1)^n-11)/4: n in [1..60]]; // Vincenzo Librandi, Sep 19 2011

Mathematica

Rest[Flatten[{#-1,#+1}&/@(11 Range[0,50])]] (* Harvey P. Dale, Nov 05 2010 *)

PARI

a(n)=n%2*9 + 1 \\ Charles R Greathouse IV, Aug 01 2016

Formula

G.f.: x*(1+9*x+x^2)/((1+x)*(1-x)^2).

a(n) = (22*n + 7*(-1)^n - 11)/4.

a(n) = -a(-n+1) = a(n-2) + 11 = a(n-1) + a(n-2) - a(n-3).

a(n) = 11*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1.

a(n) = A195312(n) + A195312(n-1) = A195313(n) - A195313(n-2). - Bruno Berselli, Sep 18 2011

Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/11)*cot(Pi/11). - Amiram Eldar, Dec 04 2021

E.g.f.: 1 + ((22*x - 11)*exp(x) + 7*exp(-x))/4. - David Lovler, Sep 04 2022

From Amiram Eldar, Nov 23 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = 2*cos(Pi/11).

Product_{n>=2} (1 + (-1)^n/a(n)) = (Pi/11)*cosec(Pi/11). (End)