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A159844 Positive numbers y such that y^2 is of the form x^2+(x+359)^2 with integer x. 4
325, 359, 401, 1549, 1795, 2081, 8969, 10411, 12085, 52265, 60671, 70429, 304621, 353615, 410489, 1775461, 2061019, 2392505, 10348145, 12012499, 13944541, 60313409, 70013975, 81274741, 351532309, 408071351, 473703905, 2048880445 (list; graph; refs; listen; history; text; internal format)
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

(-36, a(1)) = (-36, 325) is a solution: (-36)^2+(-36+359)^2 = 1296+104329 = 105625 = 325^2.

(A130610(1), a(2)) = (0, 359) is a solution: 0^2+(0+359)^2 = 128881 = 359^2.

(A130610(3), a(4)) = (901, 1549) is a solution: 901^2+(901+359)^2 = 811801+1587600 = 2399401 = 1549^2.

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

Cf. A130610, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159845 (decimal expansion of (363+38*sqrt(2))/359), A159846 (decimal expansion of (293619+186550*sqrt(2))/359^2).

Sequence in context: A253440 A253157 A184036 * A000443 A097101 A025294

Adjacent sequences:  A159841 A159842 A159843 * A159845 A159846 A159847

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Apr 30 2009

STATUS

approved

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Last modified March 4 23:37 EST 2021. Contains 341812 sequences. (Running on oeis4.)