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

1, 2, 7, 25, 93, 346, 1291, 4817, 17977, 67090, 250383, 934441, 3487381, 13015082, 48572947, 181276705, 676533873, 2524858786, 9422901271, 35166746297, 131244083917, 489809589370, 1827994273563, 6822167504881, 25460675745961, 95020535478962

Offset: 0

Author

Oboifeng Dira, May 21 2014

Keywords

• nonn
• easy
• changed

Comments

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.

Examples

			Considering the pair a(1)=2 and a(2)=7, 2 divides 7^2+1 and 7 divides 2^2+2+1.
		

References

• T. Bier, Classifications of solutions of certain positive biquadratic division system, submitted May 12 2014.
• T. Bier and O. Dira, Construction of integer sequences, submitted May 12 2014.

Links

• Oboifeng Dira, Sequences solving division systems
• Index entries for linear recurrences with constant coefficients, signature (4,0,-4,1).

Crossrefs

A101368 gives a similar problem with x^2+x+1 mod y = 0 and y^2+y+1 mod x = 0.

Programs

Maple

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);

Mathematica

LinearRecurrence[{4,0,-4,1},{1,2,7,25},30] (* Harvey P. Dale, Sep 02 2025 *)

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

Formula

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.

a(n) = 4*a(n-1) - 4*a(n-3) + a(n-4). - Colin Barker, May 21 2014

G.f.: -(x^3-x^2-2*x+1) / ((x-1)*(x+1)*(x^2-4*x+1)). - Colin Barker, May 21 2014

a(n) = (1/12) * (2*A077136(n) + (-1)^n + 3). - Ralf Stephan, May 24 2014