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A092259 Numbers that are congruent to {4, 8} mod 12.

%I A092259 #30 Nov 24 2024 01:51:31
%S A092259 4,8,16,20,28,32,40,44,52,56,64,68,76,80,88,92,100,104,112,116,124,
%T A092259 128,136,140,148,152,160,164,172,176,184,188,196,200,208,212,220,224,
%U A092259 232,236,244,248,256,260,268,272,280,284,292,296,304,308,316,320,328,332
%N A092259 Numbers that are congruent to {4, 8} mod 12.
%H A092259 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A092259 G.f.: 4*x*(1+x+x^2) / ( (1+x)*(x-1)^2 ).
%F A092259 a(n) = 4 * A001651(n).
%F A092259 Iff phi(n) = phi(3n/2), then n is in A069587. - _Labos Elemer_, Feb 25 2004
%F A092259 a(n) = 12*(n-1)-a(n-1) (with a(1)=4). - _Vincenzo Librandi_, Nov 16 2010
%F A092259 From _Wesley Ivan Hurt_, May 21 2016: (Start)
%F A092259 a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
%F A092259 a(n) = 6n - 3 - (-1)^n.
%F A092259 a(2n) = A017617(n-1) for n>1, a(2n-1) = A017569(n-1) for n>1.
%F A092259 a(n) = -a(1-n), a(n) = A092899(n) + 1 for n>0. (End)
%F A092259 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi*sqrt(3)/36. - _Amiram Eldar_, Dec 30 2021
%F A092259 From _Amiram Eldar_, Nov 24 2024: (Start)
%F A092259 Product_{n>=1} (1 - (-1)^n/a(n)) = 1/sqrt(2) + 1/sqrt(6) (A145439).
%F A092259 Product_{n>=1} (1 + (-1)^n/a(n)) = sqrt(2/3) (A157697). (End)
%p A092259 A092259:=n->6*n-3-(-1)^n: seq(A092259(n), n=1..100); # _Wesley Ivan Hurt_, May 21 2016
%t A092259 Table[6n-3-(-1)^n, {n, 80}] (* _Wesley Ivan Hurt_, May 21 2016 *)
%t A092259 LinearRecurrence[{1,1,-1},{4,8,16},60] (* _Harvey P. Dale_, Oct 07 2021 *)
%o A092259 (Magma) [n : n in [0..400] | n mod 12 in [4, 8]]; // _Wesley Ivan Hurt_, May 21 2016
%Y A092259 Cf. A001651, A017617, A017569, A069587, A092899, A145439, A157697.
%Y A092259 Fourth row of A092260.
%K A092259 nonn,easy
%O A092259 1,1
%A A092259 _Giovanni Teofilatto_, Feb 19 2004
%E A092259 Edited and extended by _Ray Chandler_, Feb 21 2004