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.

A129836 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 97)^2 = y^2.

Original entry on oeis.org

0, 15, 228, 291, 368, 1575, 1940, 2387, 9416, 11543, 14148, 55115, 67512, 82695, 321468, 393723, 482216, 1873887, 2295020, 2810795, 10922048, 13376591, 16382748, 63658595, 77964720, 95485887, 371029716, 454411923, 556532768
Offset: 1

Views

Author

Mohamed Bouhamida, May 21 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x + 97, y).
Corresponding values y of solutions (x, y) are in A157469.
For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436, 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 (cf. A118673).
Pairs (p, m) are (7, 2), (17, 3), (31, 4), (71, 6), (97, 7), (127, 8), (199, 10), (241, 11), (337, 13), (449, 15), (577, 17), (647, 18), (881, 21), (967, 22), ...
lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (99 + 14*sqrt(2))/97 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (19491 + 12070*sqrt(2))/97^2 for n mod 3 = 0.
For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a prime number in A066436, m>=2, 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

Links

Crossrefs

Cf. A157469, A066436 (primes of the form 2*n^2 - 1), A001652, A118673, A118674, A156035 (decimal expansion of 3 + 2*sqrt(2)), A157470 (decimal expansion of (99 + 14*sqrt(2))/97), A157471 (decimal expansion of (19491 + 12070*sqrt(2))/97^2).

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(15+213*x+63*x^2-13*x^3-71*x^4-13*x^5)/((1-x)*(1-6*x^3 + x^6)))); // G. C. Greubel, May 07 2018
  • Mathematica
    ClearAll[a]; Evaluate[Array[a, 6]] = {0, 15, 228, 291, 368, 1575}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 194; Table[a[n], {n, 1, 29}] (* Jean-François Alcover, Dec 27 2011, after given formula *)
LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,15,228,291,368,1575,1940}, 50] (* G. C. Greubel, May 07 2018 *)
  • PARI
    forstep(n=0, 600000000, [3, 1], if(issquare(2*n^2+194*n+9409), print1(n, ",")))

Formula

a(n) = 6*a(n-3) - a(n-6) + 194 for n > 6; a(1)=0, a(2)=15, a(3)=228, a(4)=291, a(5)=368, a(6)=1575.
G.f.: x*(15 + 213*x + 63*x^2 - 13*x^3 - 71*x^4 - 13*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 97*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Mar 12 2009

A157469 Positive numbers y such that y^2 is of the form x^2 + (x+97)^2 with integer x.

Original entry on oeis.org

85, 97, 113, 397, 485, 593, 2297, 2813, 3445, 13385, 16393, 20077, 78013, 95545, 117017, 454693, 556877, 682025, 2650145, 3245717, 3975133, 15446177, 18917425, 23168773, 90026917, 110258833, 135037505, 524715325, 642635573, 787056257
Offset: 1

Views

Author

Klaus Brockhaus, Mar 12 2009

Keywords

Comments

(-13,a(1)) and (A129836(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2 + (x+97)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (99+14*sqrt(2))/97 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (19491+12070*sqrt(2))/97^2 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

			(-13, a(1)) = (-13, 85) is a solution: (-13)^2+(-13+97)^2 = 169+7056 = 7225 = 85^2.
(A129836(1), a(2)) = (0, 97) is a solution: 0^2+(0+97)^2 = 9409 = 97^2.
(A129836(3), a(4)) = (228, 397) is a solution: 228^2+(228+97)^2 = 51984+105625 = 157609 = 397^2.

Links

Crossrefs

Cf. A129836, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157470 (decimal expansion of (99+14*sqrt(2))/97), A157471 (decimal expansion of (19491+12070*sqrt(2))/97^2).

Programs

  • Magma
    I:=[85,97,113,397,485,593]; [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},{85,97,113,397,485,593},30] (* Harvey P. Dale, Apr 04 2013 *)
  • PARI
    {forstep(n=-20, 800000000, [3, 1], if(issquare(2*n^2+194*n+9409, &k), print1(k, ",")))};

Formula

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=85, a(2)=97, a(3)=113, a(4)=397, a(5)=485, a(6)=593.
G.f.: (1-x)*(85 + 182*x + 295*x^2 + 182*x^3 + 85*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 97*A001653(k) for k >= 1.

A157470 Decimal expansion of (99+14*sqrt(2))/97.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Mar 12 2009

Keywords

Comments

lim_{n -> infinity} b(n)/b(n-1) = (99+14*sqrt(2))/97 for n mod 3 = {1, 2}, b = A129836.
lim_{n -> infinity} b(n)/b(n-1) = (99+14*sqrt(2))/97 for n mod 3 = {0, 2}, b = A157469.

Examples

			(99+14*sqrt(2))/97 = 1.22473185436312712044...

Links

Crossrefs

Cf. A129836, A157469, A002193 (decimal expansion of sqrt(2)), A157471 (decimal expansion of (19491+12070*sqrt(2))/97^2).

Programs

  • Magma
    (99+14*Sqrt(2))/97; // G. C. Greubel, Mar 30 2018
  • Maple
    with(MmaTranslator[Mma]): Digits:=100:
RealDigits(evalf((99+14*sqrt(2))/97))[1]; # Muniru A Asiru, Mar 31 2018
  • Mathematica
    RealDigits[(99+14*Sqrt[2])/97, 10, 100][[1]] (* G. C. Greubel, Mar 30 2018 *)
  • PARI
    (99+14*sqrt(2))/97 \\ G. C. Greubel, Mar 30 2018

Formula

Equals (14+sqrt(2))/(14-sqrt(2)).
Showing 1-3 of 3 results.

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