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.

A118674 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31)^2 = y^2.

Original entry on oeis.org

0, 9, 60, 93, 140, 429, 620, 893, 2576, 3689, 5280, 15089, 21576, 30849, 88020, 125829, 179876, 513093, 733460, 1048469, 2990600, 4274993, 6111000, 17430569, 24916560, 35617593, 101592876, 145224429, 207594620, 592126749, 846430076
Offset: 1

Views

Author

Mohamed Bouhamida, May 19 2006

Keywords

Comments

Also values x of Pythagorean triples (x, x+31, y).
Corresponding values y of solutions (x, y) are in A157646.
For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (33 + 8*sqrt(2))/31 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1539 + 850*sqrt(2))/31^2 for n mod 3 = 0.

Links

Crossrefs

cf. A157646, A066436 (primes of the form 2*n^2-1), A118673, A129836, A001652, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3 + 2*sqrt(2)), A157647 (decimal expansion of (33 + 8*sqrt(2))/31), A157648 (decimal expansion of (1539 + 850*sqrt(2))/31^2).

Programs

  • Magma
    I:=[0,9,60,93,140,429,620]; [n le 7 select I[n] else Self(n-1) - 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // G. C. Greubel, Mar 31 2018
  • Mathematica
    ClearAll[a]; Evaluate[Array[a, 6]] = {0, 9, 60, 93, 140, 429}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 62; Table[a[n], {n, 1, 31}] (* Jean-François Alcover, Dec 27 2011, after given formula *)
LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,9,60,93,140,429,620}, 50] (* G. C. Greubel, Mar 31 2018 *)
  • PARI
    {forstep(n=0, 850000000, [1, 3], if(issquare(2*n^2+62*n+961), print1(n, ",")))};

Formula

a(n) = 6*a(n-3) - a(n-6) + 62 for n > 6; a(1)=0, a(2)=9, a(3)=60, a(4)=93, a(5)=140, a(6)=429.
G.f.: x*(9 + 51*x + 33*x^2 - 7*x^3 - 17*x^4 - 7*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 31*A001652(k) for k >= 0.

Extensions

Edited by Klaus Brockhaus, Mar 11 2009

A157646 Positive numbers y such that y^2 is of the form x^2 + (x+31)^2 with integer x.

Original entry on oeis.org

25, 31, 41, 109, 155, 221, 629, 899, 1285, 3665, 5239, 7489, 21361, 30535, 43649, 124501, 177971, 254405, 725645, 1037291, 1482781, 4229369, 6045775, 8642281, 24650569, 35237359, 50370905, 143674045, 205378379, 293583149, 837393701
Offset: 1

Views

Author

Klaus Brockhaus, Mar 11 2009

Keywords

Comments

(-7,a(1)) and (A118674(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+31)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (33+8*sqrt(2))/31 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (1539+850*sqrt(2))/31^2 for n mod 3 = 1.
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, 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+1, a(3)=6m^2-10m+4, a(4)=3p, a(5)=6m^2+10m+4, a(6)=40m^2-58m+21.Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=2m^2+2m+1, b(3)=10m^2-14m+5, b(4)=5p, b(5)=10m^2+14m+5, b(6)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

Examples

			(-7, a(1)) = (-7, 25) is a solution: (-7)^2+(-7+31)^2 = 49+576 = 625 = 25^2.
(A118674(1), a(2)) = (0, 31) is a solution: 0^2+(0+31)^2 = 961 = 31^2.
(A118674(3), a(4)) = (60, 109) is a solution: 60^2+(60+31)^2 = 3600+8281 = 11881 = 109^2.

Links

Crossrefs

Cf. A118674, A001653, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157647 (decimal expansion of (33+8*sqrt(2))/31), A157648 (decimal expansion of (1539+850*sqrt(2))/31^2).

Programs

  • Magma
    I:=[25,31,41,109,155,221]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..50]]; // G. C. Greubel, Mar 31 2018
  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1},{25,31,41,109,155,221},40] (* Harvey P. Dale, Oct 12 2017 *)
  • PARI
    {forstep(n=-8, 840000000, [1, 3], if(issquare(2*n^2+62*n+961, &k), print1(k, ",")))};

Formula

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=25, a(2)=31, a(3)=41, a(4)=109, a(5)=155, a(6)=221.
G.f.: (1-x)*(25 + 56*x + 97*x^2 + 56*x^3 + 25*x^4)/(1 - 6*x^3 + x^6).
a(3*k-1) = 31*A001653(k) for k >= 1.

A157648 Decimal expansion of (1539+850*sqrt(2))/31^2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Mar 11 2009

Keywords

Comments

Limit_{n -> oo} b(n)/b(n-1) = (1539+850*sqrt(2))/31^2 for n mod 3 = 0, b = A118674.
Limit_{n -> oo} b(n)/b(n-1) = (1539+850*sqrt(2))/31^2 for n mod 3 = 1, b = A157646.

Examples

			2.85232208950794046980...

Links

Crossrefs

Cf. A118674, A157646, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3+2*sqrt(2)), A157647 (decimal expansion of (33+8*sqrt(2))/31).

Programs

  • Magma
    (1539+850*Sqrt(2))/31^2; // G. C. Greubel, Mar 30 2018
  • Maple
    with(MmaTranslator[Mma]): Digits:=100:
RealDigits(evalf((1539+850*sqrt(2))/31^2))[1]; # Muniru A Asiru, Mar 31 2018
  • Mathematica
    RealDigits[(1539+850*Sqrt[2])/31^2, 10, 100][[1]] (* G. C. Greubel, Mar 30 2018 *)
  • PARI
    (1539+850*sqrt(2))/31^2 \\ G. C. Greubel, Mar 30 2018

Formula

Equals (50+17*sqrt(2))/(50-17*sqrt(2)).
Equals (3+2*sqrt(2))*(8-sqrt(2))^2/(8+sqrt(2))^2.
Showing 1-3 of 3 results.

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