cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A118676 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+79)^2 = y^2.

Original entry on oeis.org

0, 20, 161, 237, 341, 1140, 1580, 2184, 6837, 9401, 12921, 40040, 54984, 75500, 233561, 320661, 440237, 1361484, 1869140, 2566080, 7935501, 10894337, 14956401, 46251680, 63497040, 87172484, 269574737, 370088061, 508078661, 1571196900, 2157031484, 2961299640
Offset: 1

Views

Author

Mohamed Bouhamida, May 19 2006

Keywords

Comments

Also values x of Pythagorean triples (x, x+79, y).
Corresponding values y of solutions (x, y) are in A159758.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (83+18*sqrt(2))/79 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (10659+6110*sqrt(2))/79^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159758, A028871, A118337, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159759 (decimal expansion of (83+18*sqrt(2))/79), A159760 (decimal expansion of (10659+6110*sqrt(2))/79^2).

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(20+141*x+76*x^2-16*x^3-47*x^4-16*x^5)/((1-x)*(1- 6*x^3+x^6)))); // G. C. Greubel, May 07 2018
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,20,161,237,341,1140,1580},75] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)
  • PARI
    forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+158*n+6241), print1(n, ",")))

Formula

a(n) = 6*a(n-3) -a(n-6) +158 for n > 6; a(1)=0, a(2)=20, a(3)=161, a(4)=237, a(5)=341, a(6)=1140.
G.f.: x*(20+141*x+76*x^2-16*x^3-47*x^4-16*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 79*A001652(k) for k >= 0.

Extensions

Edited by Klaus Brockhaus, Apr 30 2009

A159758 Positive numbers y such that y^2 is of the form x^2+(x+79)^2 with integer x.

Original entry on oeis.org

65, 79, 101, 289, 395, 541, 1669, 2291, 3145, 9725, 13351, 18329, 56681, 77815, 106829, 330361, 453539, 622645, 1925485, 2643419, 3629041, 11222549, 15406975, 21151601, 65409809, 89798431, 123280565, 381236305, 523383611, 718531789
Offset: 1

Views

Author

Klaus Brockhaus, Apr 30 2009

Keywords

Comments

(-16, a(1)) and (A118676(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+79)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (83+18*sqrt(2))/79 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (10659+6110*sqrt(2))/79^2 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

Examples

			(-16, a(1)) = (-16, 65) is a solution: (-16)^2 + (-16+79)^2 = 256+3969 = 4225 = 65^2.
(A118676(1), a(2)) = (0, 79) is a solution: 0^2 + (0+79)^2 = 6241 = 79^2.
(A118676(3), a(4)) = (161, 289) is a solution: 161^2 + (161+79)^2 = 25921 + 57600 = 83521 = 289^2.

Links

Crossrefs

Cf. A118676, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159759 (decimal expansion of (83+18*sqrt(2))/79), A159760 (decimal expansion of (10659+6110*sqrt(2))/79^2).

Programs

  • Magma
    I:=[65,79,101,289,395,541]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 22 2018
  • Mathematica
    RecurrenceTable[{a[1]==65,a[2]==79,a[3]==101,a[4]==289,a[5]==395, a[6]== 541, a[n]==6a[n-3]-a[n-6]},a[n],{n,30}] (* or *) LinearRecurrence[ {0,0,6,0,0,-1},{65,79,101,289,395,541},30] (* Harvey P. Dale, Oct 03 2011 *)
  • PARI
    {forstep(n=-16, 10000000, [1, 3], if(issquare(2*n^2+158*n+6241, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=65, a(2)=79, a(3)=101, a(4)=289, a(5)=395, a(6)=541.
G.f.: (1-x)*(65+144*x+245*x^2+144*x^3+65*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 79*A001653(k) for k >= 1.

A159760 Decimal expansion of (10659+6110*sqrt(2))/79^2.

Original entry on oeis.org

3, 0, 9, 2, 4, 2, 8, 2, 7, 5, 2, 9, 2, 3, 5, 8, 7, 1, 6, 2, 5, 9, 9, 4, 5, 2, 1, 7, 9, 0, 1, 1, 7, 8, 5, 3, 8, 7, 0, 5, 4, 4, 7, 0, 6, 9, 1, 4, 8, 0, 7, 7, 6, 6, 8, 1, 7, 9, 7, 0, 0, 8, 8, 1, 1, 2, 8, 5, 6, 5, 2, 0, 4, 8, 3, 9, 5, 2, 4, 7, 5, 6, 8, 2, 9, 9, 7, 4, 0, 1, 6, 1, 4, 1, 9, 4, 6, 8, 0, 4, 9, 0, 4, 8, 1
Offset: 1

Views

Author

Klaus Brockhaus, Apr 30 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A118676.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A159758.

Examples

			(10659+6110*sqrt(2))/79^2 = 3.09242827529235871625...

Links

Crossrefs

Cf. A118676, A159758, A002193 (decimal expansion of sqrt(2)), A159759 (decimal expansion of (83+18*sqrt(2))/79).

Programs

  • Magma
    (10659 +6110*Sqrt(2))/79^2 // G. C. Greubel, May 21 2018
  • Mathematica
    RealDigits[(10659 + 6110*Sqrt[2])/79^2, 10, 100][[1]] (* G. C. Greubel, May 21 2018 *)
  • PARI
    (10659+6110*sqrt(2))/79^2 \\ G. C. Greubel, May 21 2018

Formula

Equals (130 + 47*sqrt(2))/(130 - 47*sqrt(2)).
Equals (3 + 2*sqrt(2))*(9 - sqrt(2))^2/(9 + sqrt(2))^2.
Showing 1-3 of 3 results.

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