OFFSET
1,1
COMMENTS
(-36, a(1)) and (A130610(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+359)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (363+38*sqrt(2))/359 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (293619+186550*sqrt(2))/359^2 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)+2p with a(1)=0, a(2)=2m+2, a(3)=3m^2-10m+8, a(4)=3p, a(5)=3m^2+10m+8, a(6)=20m^2-58m+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+2m+2, b(3)=5m^2-14m+10, b(4)=5p, b(5)=5m^2+14m+10, b(6)=29m^2-82m+58. [Mohamed Bouhamida, Sep 09 2009]
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
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)=325, a(2)=359, a(3)=401, a(4)=1549, a(5)=1795, a(6)=2081.
G.f.: (1-x)*(325+684*x+1085*x^2+684*x^3+325*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 359*A001653(k) for k >= 1.
EXAMPLE
MATHEMATICA
t={325, 359, 401, 1549, 1795, 2081}; Do[AppendTo[t, 6*t[[-3]]-t[[-6]]], {25}]; t
CoefficientList[Series[(325+359 x+401 x^2-401 x^3-359 x^4-325 x^5)/(1-6 x^3+x^6), {x, 0, 30}], x] (* Harvey P. Dale, Feb 16 2011 *)
LinearRecurrence[{0, 0, 6, 0, 0, -1}, {325, 359, 401, 1549, 1795, 2081}, 50] (* G. C. Greubel, May 19 2018 *)
PROG
(PARI) {forstep(n=-36, 10000000, [1, 3], if(issquare(2*n^2+718*n+128881, &k), print1(k, ", ")))}
(PARI) V=[]; v=[[-323, -325], [-323, 325], [0, -359], [-359, 359], [-399, -401], [399, 401]]; for(n=1, 100, u=[]; for(i=1, #v, if(v[i][2]>0, u=concat(u, v[i][2])); t=3*v[i][1]+2*v[i][2]+359; v[i][2]=4*v[i][1]+3*v[i][2]+718; v[i][1]=t); V=concat(V, u)); vecsort(V, , 8) \\ Charles R Greathouse IV, Feb 14 2011
(Magma) I:=[325, 359, 401, 1549, 1795, 2081]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 19 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Klaus Brockhaus, Apr 30 2009
STATUS
approved