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
Examples
a(2)=3, a(3)=10 satisfy: 10 divides 3^2+1 and 3 divides 10^2+10+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
Cf. A101368 (which considers a similar problem with x^2+x+1 mod y = 0 and y^2+y+1 mod x = 0).
Programs
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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);
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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 *)
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]
Extensions
Edited by Bruno Berselli, May 25 2014
Comments