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%I #51 Sep 05 2022 09:11:01
%S 1,11,13,23,25,35,37,47,49,59,61,71,73,83,85,95,97,107,109,119,121,
%T 131,133,143,145,155,157,167,169,179,181,191,193,203,205,215,217,227,
%U 229,239,241,251,253,263,265,275,277,287,289,299,301,311,313,323,325,335
%N Numbers that are congruent to {1, 11} mod 12.
%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 and n in A000027), then ((2*h*n + (h-4)*(-1)^n - h)/4)^2 - 1 == 0 (mod h); in our case, a(n)^2 - 1 == 0 (mod 12). Also a(n)^2 - 1 == 0 (mod 24).
%H Reinhard Zumkeller, <a href="/A091998/b091998.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 a(n) = 12*n - a(n-1) - 12 (with a(1)=1). - _Vincenzo Librandi_, Nov 16 2010
%F a(n) = 6*n + 2*(-1)^n - 3.
%F G.f.: x*(1+10*x+x^2)/((1+x)*(1-x)^2).
%F a(n) - a(n-1) - a(n-2) + a(n-3) = 0 for n > 3.
%F a(n) = a(n-2) + 12 for n > 2.
%F a(n) = 12*A000217(n-1) + 1 - 2*Sum_{i=1..n-1} a(i) for n > 1.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (2 + sqrt(3))*Pi/12. - _Amiram Eldar_, Dec 04 2021
%F E.g.f.: 1 + (6*x - 3)*exp(x) + 2*exp(-x). - _David Lovler_, Sep 04 2022
%t LinearRecurrence[{1,1,-1},{1,11,13},100] (* _Harvey P. Dale_, Jul 26 2017 *)
%o (Magma) [ n: n in [1..350] | n mod 12 eq 1 or n mod 12 eq 11 ];
%o (Haskell)
%o a091998 n = a091998_list !! (n-1)
%o a091998_list = 1 : 11 : map (+ 12) a091998_list
%o -- _Reinhard Zumkeller_, Jan 07 2012
%o (PARI) is(n)=n=n%12;n==11 || n==1 \\ _Charles R Greathouse IV_, Jul 02 2013
%Y First row of A092260.
%Y Cf. A175885 (n == 1 or 10 (mod 11)), A175886 (n == 1 or 12 (mod 13)).
%Y Cf. A097933 (primes), A195143 (partial sums).
%Y Cf. A000217, A005408, A047209, A007310, A047336, A047522, A056020, A090771, A113801, A175887.
%K nonn,easy
%O 1,2
%A _Ray Chandler_, Feb 21 2004
%E Formulae and comment added by _Bruno Berselli_, Nov 17 2010 - Nov 18 2010
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