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

1,2

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).

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

Mathematica

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

Prog

(Magma) [(22*n+7*(-1)^n-11)/4: n in [1..60]]; // Vincenzo Librandi, Sep 19 2011
(Haskell)
a175885 n = a175885_list !! (n-1)
a175885_list = 1 : 10 : map (+ 11) a175885_list
-- Reinhard Zumkeller, Jan 07 2012
(PARI) a(n)=n%2*9 + 1 \\ Charles R Greathouse IV, Aug 01 2016

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.

Sequence in context: A108965 A366958 A341002 * A061870 A348056 A120001

Adjacent sequences: A175882 A175883 A175884 * A175886 A175887 A175888

Keyword

nonn,easy

Author

Bruno Berselli, Oct 08 2010 - Nov 17 2010

Status

approved