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%I #42 Sep 05 2022 09:10:52
%S 1,10,12,21,23,32,34,43,45,54,56,65,67,76,78,87,89,98,100,109,111,120,
%T 122,131,133,142,144,153,155,164,166,175,177,186,188,197,199,208,210,
%U 219,221,230,232,241,243,252,254,263,265,274,276,285,287,296,298
%N Numbers that are congruent to {1, 10} mod 11.
%C 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).
%H Bruno Berselli, <a href="/A175885/b175885.txt">Table of n, a(n) for n = 1..10000</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F G.f.: x*(1+9*x+x^2)/((1+x)*(1-x)^2).
%F a(n) = (22*n + 7*(-1)^n - 11)/4.
%F a(n) = -a(-n+1) = a(n-2) + 11 = a(n-1) + a(n-2) - a(n-3).
%F a(n) = 11*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1.
%F a(n) = A195312(n) + A195312(n-1) = A195313(n) - A195313(n-2). - _Bruno Berselli_, Sep 18 2011
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (Pi/11)*cot(Pi/11). - _Amiram Eldar_, Dec 04 2021
%F E.g.f.: 1 + ((22*x - 11)*exp(x) + 7*exp(-x))/4. - _David Lovler_, Sep 04 2022
%t Rest[Flatten[{#-1,#+1}&/@(11 Range[0,50])]] (* _Harvey P. Dale_, Nov 05 2010 *)
%o (Magma) [(22*n+7*(-1)^n-11)/4: n in [1..60]]; // _Vincenzo Librandi_, Sep 19 2011
%o (Haskell)
%o a175885 n = a175885_list !! (n-1)
%o a175885_list = 1 : 10 : map (+ 11) a175885_list
%o -- _Reinhard Zumkeller_, Jan 07 2012
%o (PARI) a(n)=n%2*9 + 1 \\ _Charles R Greathouse IV_, Aug 01 2016
%Y Cf. A090771 (n==1 or 9 mod 10), A091998 (n==1 or 11 mod 12).
%Y Cf. A195043 (partial sums).
%Y Cf. A000217, A005408, A047209, A007310, A047336, A047522, A056020, A113801, A175886, A175887, A195312, A195313.
%K nonn,easy
%O 1,2
%A _Bruno Berselli_, Oct 08 2010 - Nov 17 2010
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