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A159777 Positive numbers y such that y^2 is of the form x^2+(x+167)^2 with integer x. 4
145, 167, 197, 673, 835, 1037, 3893, 4843, 6025, 22685, 28223, 35113, 132217, 164495, 204653, 770617, 958747, 1192805, 4491485, 5587987, 6952177, 26178293, 32569175, 40520257, 152578273, 189827063, 236169365, 889291345, 1106393203 (list; graph; refs; listen; history; text; internal format)
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

(-24, a(1)) = (-24, 145) is a solution: (-24)^2 + (-24+167)^2 = 576 + 20449 = 21025 = 145^2.

(A130608(1), a(2)) = (0, 167) is a solution: 0^2 + (0+167)^2 = 27889 = 167^2.

(A130608(3), a(4)) = (385, 673) is a solution: 385^2 + (385+167)^2 = 148225 + 304704 = 452929 = 673^2.

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

Cf. A130608, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159778 (decimal expansion of (171+26*sqrt(2))/167), A159779 (decimal expansion of (56211+34510*sqrt(2))/167^2).

Sequence in context: A043652 A296889 A164770 * A326258 A051414 A176699

Adjacent sequences:  A159774 A159775 A159776 * A159778 A159779 A159780

KEYWORD

nonn,easy

AUTHOR

Klaus Brockhaus, Apr 30 2009

STATUS

approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)