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

1, 1, 3, 10, 37, 137, 511, 1906, 7113, 26545, 99067, 369722, 1379821, 5149561, 19218423, 71724130, 267678097, 998988257, 3728274931, 13914111466, 51928170933, 193798572265, 723266118127, 2699265900242, 10073797482841, 37595924031121, 140309898641643

Offset

0,3

Comments

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.

The negative part b(n) = a(-n) is given in A242728.

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

Table of n, a(n) for n=0..26.

Oboifeng Dira, Sequences solving division systems

Index entries for linear recurrences with constant coefficients, signature (4,0,-4,1).

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.

G.f.: (1 - 3*x - x^2 + 2*x^3)/((1 - x)*(1 + x)*(1 - 4*x + x^2)). [Bruno Berselli, May 22 2014]

a(n) = 4*a(n-1) - 4*a(n-3) + a(n-4) for n>3. [Bruno Berselli, May 22 2014]

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]

Example

a(2)=3, a(3)=10 satisfy: 10 divides 3^2+1 and 3 divides 10^2+10+1.

Maple

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);

Mathematica

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 *)

Crossrefs

Cf. A101368 (which considers a similar problem with x^2+x+1 mod y = 0 and y^2+y+1 mod x = 0).

Sequence in context: A149043 A296000 A360586 * A151315 A164048 A180717

Adjacent sequences: A242722 A242723 A242724 * A242726 A242727 A242728

Keyword

nonn,easy

Author

Oboifeng Dira, May 21 2014

Extensions

Edited by Bruno Berselli, May 25 2014

Status

approved