cp's OEIS Frontend

A242728 Sequence a(n) with all (x,y)=(a(2m),a(2m+-1)) satisfying y|x^2+1 and x|y^2+y+1.

%I A242728 #29 Sep 02 2025 12:53:07
%S A242728 1,2,7,25,93,346,1291,4817,17977,67090,250383,934441,3487381,13015082,
%T A242728 48572947,181276705,676533873,2524858786,9422901271,35166746297,
%U A242728 131244083917,489809589370,1827994273563,6822167504881,25460675745961,95020535478962
%N A242728 Sequence a(n) with all (x,y)=(a(2m),a(2m+-1)) satisfying y|x^2+1 and x|y^2+y+1.
%C A242728 a(n) with a(1)=2, a(2)=7 is that two-way sequence such that (a(n),a(n+1)) and (a(n),a(n-1)) for n even together with the corresponding pairs of A242725 give all solutions of the two congruences x^2+1 mod y = 0 and y^2+y+1 mod x = 0.  The negative part b(n) = a(-n) is given in sequence A242725.
%D A242728 T. Bier, Classifications of solutions of certain positive biquadratic division system, submitted May 12 2014.
%D A242728 T. Bier and O. Dira, Construction of integer sequences, submitted May 12 2014.
%H A242728 Oboifeng Dira, <a href="/A242728/a242728.pdf">Sequences solving division systems</a>
%H A242728 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,0,-4,1).
%F A242728 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 A242728 a(n) = 4*a(n-1) - 4*a(n-3) + a(n-4). - _Colin Barker_, May 21 2014
%F A242728 G.f.: -(x^3-x^2-2*x+1) / ((x-1)*(x+1)*(x^2-4*x+1)). - _Colin Barker_, May 21 2014
%F A242728 a(n) = (1/12) * (2*A077136(n) + (-1)^n + 3). - _Ralf Stephan_, May 24 2014
%e A242728 Considering the pair a(1)=2 and a(2)=7, 2 divides 7^2+1 and 7 divides 2^2+2+1.
%p A242728 x0:=1: x1:=2: 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 A242728 LinearRecurrence[{4,0,-4,1},{1,2,7,25},30] (* _Harvey P. Dale_, Sep 02 2025 *)
%o A242728 (PARI) Vec(-x*(x^3-x^2-2*x+1)/((x-1)*(x+1)*(x^2-4*x+1)) + O(x^100)) \\ _Colin Barker_, May 21 2014
%Y A242728 A101368 gives a similar problem with x^2+x+1 mod y = 0 and y^2+y+1 mod x = 0.
%K A242728 nonn,easy,changed
%O A242728 0,2
%A A242728 _Oboifeng Dira_, May 21 2014