A159896 Positive numbers y such that y^2 is of the form x^2+(x+839)^2 with integer x.

785, 839, 901, 3809, 4195, 4621, 22069, 24331, 26825, 128605, 141791, 156329, 749561, 826415, 911149, 4368761, 4816699, 5310565, 25463005, 28073779, 30952241, 148409269, 163625975, 180402881, 864992609, 953682071, 1051465045

Offset

1,1

Comments

(-56, a(1)) and (A130647(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+839)^2 = y^2.

lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).

lim_{n -> infinity} a(n)/a(n-1) = (843+58*sqrt(2))/839 for n mod 3 = {0, 2}.

lim_{n -> infinity} a(n)/a(n-1) = (1760979+1141390*sqrt(2))/839^2 for n mod 3 = 1.

For the generic case x^2+(x+p)^2=y^2 with p=m^2-2 a prime number in A028871, m>=5, the x values are given by the sequence defined by: a(n) = 6*a(n-3) -a(n-6) +2*p with a(1)=0, a(2) = 2*m+2, a(3) = 3*m^2 -10*m +8, a(4) = 3*p, a(5) = 3*m^2 +10*m +8, a(6) = 20*m^2 -58*m +42. Y values are given by the sequence defined by: b(n) = 6*b(n-3) -b(n-6) with b(1)=p, b(2)= m^2 +2*m +2, b(3)= 5*m^2 -14*m +10, b(4)= 5*p, b(5)= 5*m^2 +14*m +10, b(6)= 29*m^2 -82*m +58. - Mohamed Bouhamida, Sep 09 2009

Links

G. C. Greubel, Table of n, a(n) for n = 1..3895

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Formula

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=785, a(2)=839, a(3)=901, a(4)=3809, a(5)=4195, a(6)=4621.

G.f.: (1-x)*(785+1624*x+2525*x^2+1624*x^3+785*x^4)/(1-6*x^3+x^6).

a(3*k-1) = 839*A001653(k) for k >= 1.

Example

(-56, a(1)) = (-56, 785) is a solution: (-56)^2+(-56+839)^2 = 3136+613089 = 616225 = 785^2.
(A130647(1), a(2)) = (0, 839) is a solution: 0^2+(0+839)^2 = 703921 = 839^2.
(A130647(3), a(4)) = (2241, 3809) is a solution: 2241^2+(2241+839)^2 = 5022081+9486400 = 14508481 = 3809^2.

Mathematica

LinearRecurrence[{0, 0, 6, 0, 0, -1}, {785, 839, 901, 3809, 4195, 4621}, 30] (* Harvey P. Dale, Mar 03 2013 *)

Prog

(PARI) {forstep(n=-56, 10000000, [1, 3], if(issquare(2*n^2+1678*n+703921, &k), print1(k, ", ")))}
(Magma) I:=[785, 839, 901, 3809, 4195, 4621]; [n le 6 select I[n] else 6*Self(n-3) -Self(n-6): n in [1..30]]; // G. C. Greubel, May 17 2018
(PARI) is(n, p=839)=for(m=sqrtint((max(n, 984)^2-p^2)\2)-p\2, n, m^2+(m+p)^2<n^2||return(m^2+(m+p)^2==n^2))
A159896(n)=(matrix(6, 6, i, j, if(i<6, i+1==j, j==4, 6, j==1, -1))^n*[785, 839, 901, 3809, 4195, 4621]~)[1] \\ M. F. Hasler, May 17 2018

Crossrefs

Cf. A130647, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159897 (decimal expansion of (843+58*sqrt(2))/839), A159898 (decimal expansion of (1760979+1141390*sqrt(2))/839^2).

Sequence in context: A231771 A366829 A252389 * A031734 A097776 A031526

Adjacent sequences: A159893 A159894 A159895 * A159897 A159898 A159899

Keyword

nonn,easy

Author

Klaus Brockhaus, Apr 30 2009

Status

approved