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A242725 Sequence with all (x, y) = (a(2m), a(2m+-1)) satisfying x|y^2+y+1 and y|x^2+1.

%I A242725 #30 Jun 13 2015 00:55:02
%S A242725 1,1,3,10,37,137,511,1906,7113,26545,99067,369722,1379821,5149561,
%T A242725 19218423,71724130,267678097,998988257,3728274931,13914111466,
%U A242725 51928170933,193798572265,723266118127,2699265900242,10073797482841,37595924031121,140309898641643
%N A242725 Sequence with all (x, y) = (a(2m), a(2m+-1)) satisfying x|y^2+y+1 and y|x^2+1.
%C A242725 a(n) with a(0)=1, a(1)=1, a(2)=3 is that two way sequence such that (a(n), a(n+1)) and (a(n), a(n-1)) for n even together with corresponding pairs of A242728 give all solutions of the two congruences x^2+1 mod y = 0 and y^2+y+1 mod x = 0.
%C A242725 The negative part b(n) = a(-n) is given in A242728.
%D A242725 T. Bier, Classifications of solutions of certain positive biquadratic division system, submitted May 12 2014.
%D A242725 T. Bier and O. Dira, Construction of integer sequences, submitted May 12 2014.
%H A242725 Oboifeng Dira, <a href="/A242725/a242725.pdf">Sequences solving division systems</a>
%H A242725 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-4,1).
%F A242725 a(n+1) = 4*a(n) - a(n-1) - p_n (n>0), where p_n = 0 if n is odd and p_n = 1 if n is even.
%F A242725 G.f.: (1 - 3*x - x^2 + 2*x^3)/((1 - x)*(1 + x)*(1 - 4*x + x^2)). [_Bruno Berselli_, May 22 2014]
%F A242725 a(n) = 4*a(n-1) - 4*a(n-3) + a(n-4) for n>3. [_Bruno Berselli_, May 22 2014]
%F A242725 a(n) = (3+(-1)^n+(4+sqrt(3))*(2-sqrt(3))^n+(4-sqrt(3))*(2+sqrt(3))^n)/12. [_Bruno Berselli_, May 25 2014]
%e A242725 a(2)=3, a(3)=10 satisfy: 10 divides 3^2+1 and 3 divides 10^2+10+1.
%p A242725 x0:=1: x1:=1: L:=[x0,x1]: for k from 1 to 30 do: if k mod 2 = 1 then z:=4*x1-x0: fi: if k mod 2 = 0 then z:=4*x1-x0-1: fi: L:=[op(L),z]: x0:=x1: x1:=z: od: print(L);
%t A242725 Table[(3 + (-1)^n + (4 + Sqrt[3]) (2 - Sqrt[3])^n + (4 - Sqrt[3]) (2 + Sqrt[3])^n)/12, {n, 0, 30}] (* _Bruno Berselli_, May 25 2014 *)
%Y A242725 Cf. A101368 (which considers a similar problem with x^2+x+1 mod y = 0 and y^2+y+1 mod x = 0).
%K A242725 nonn,easy
%O A242725 0,3
%A A242725 _Oboifeng Dira_, May 21 2014
%E A242725 Edited by _Bruno Berselli_, May 25 2014