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.
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)
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
KEYWORD
nonn,easy,changed
AUTHOR
Bruno Berselli, Oct 08 2010 - Nov 17 2010
STATUS
approved