OFFSET
1,1
COMMENTS
(-24, a(1)) and (A130608(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+167)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (171+26*sqrt(2))/167 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (56211+34510*sqrt(2))/167^2 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..3900
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)=145, a(2)=167, a(3)=197, a(4)=673, a(5)=835, a(6)=1037.
G.f.: (1-x)*(145+312*x+509*x^2+312*x^3+145*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 167*A001653(k) for k >= 1.
EXAMPLE
MATHEMATICA
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {145, 167, 197, 673, 835, 1037}, 50] (* G. C. Greubel, May 21 2018 *)
PROG
(PARI) {forstep(n=-24, 10000000, [1, 3], if(issquare(2*n^2+334*n+27889, &k), print1(k, ", ")))};
(Magma) I:=[145, 167, 197, 673, 835, 1037]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 21 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Apr 30 2009
STATUS
approved