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.

A161478 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+113)^2 = y^2.

Original entry on oeis.org

0, 52, 175, 339, 615, 1312, 2260, 3864, 7923, 13447, 22795, 46452, 78648, 133132, 271015, 458667, 776223, 1579864, 2673580, 4524432, 9208395, 15583039, 26370595, 53670732, 90824880, 153699364, 312816223, 529366467, 895825815, 1823226832, 3085374148
Offset: 1

Views

Author

Klaus Brockhaus, Jun 13 2009

Keywords

Comments

Corresponding values y of solutions (x, y) are in A161479.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (129+44*sqrt(2))/113 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (16131+6970*sqrt(2))/113^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A161479, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A161480 (decimal expansion of (129+44*sqrt(2))/113), A161481 (decimal expansion of (16131+6970*sqrt(2))/113^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,52,175,339,615,1312,2260},72] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)
  • PARI
    {forstep(n=0, 100000000, [3, 1], if(issquare(2*n^2+226*n+12769), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+226 for n > 6; a(1)=0, a(2)=52, a(3)=175, a(4)=339, a(5)=615, a(6)=1312.
G.f.: x*(52+123*x+164*x^2-36*x^3-41*x^4-36*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 113*A001652(k) for k >= 0.

A161480 Decimal expansion of (129 +44*sqrt(2))/113.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Jun 13 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A161478.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A161479.

Examples

			(129 +44*sqrt(2))/113 = 1.69226014818067417829...

Links

Crossrefs

Cf. A161478, A161479, A002193 (decimal expansion of sqrt(2)), A161481 (decimal expansion of (16131+6970*sqrt(2))/113^2).

Programs

  • Magma
    (129 +44*Sqrt(2))/113; // G. C. Greubel, Apr 07 2018
  • Maple
    with(MmaTranslator[Mma]): Digits:=150:
RealDigits(evalf((129+44*sqrt(2))/113))[1]; # Muniru A Asiru, Apr 08 2018
  • Mathematica
    RealDigits[(129 +44*Sqrt[2])/113, 10, 100][[1]] (* G. C. Greubel, Apr 07 2018 *)
  • PARI
    (129 +44*sqrt(2))/113 \\ G. C. Greubel, Apr 07 2018

Formula

Equals (11 +2*sqrt(2))/(11 -2*sqrt(2)).

A161479 Positive numbers y such that y^2 is of the form x^2+(x+113)^2 with integer x.

Original entry on oeis.org

85, 113, 173, 337, 565, 953, 1937, 3277, 5545, 11285, 19097, 32317, 65773, 111305, 188357, 383353, 648733, 1097825, 2234345, 3781093, 6398593, 13022717, 22037825, 37293733, 75901957, 128445857, 217363805, 442389025, 748637317, 1266889097
Offset: 1

Views

Author

Klaus Brockhaus, Jun 13 2009

Keywords

Comments

(-36, a(1)) and (A161478(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+113)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (129+44*sqrt(2))/113 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (16131+6970*sqrt(2))/113^2 for n mod 3 = 1.

Examples

			(-36, a(1)) = (-36, 85) is a solution: (-36)^2+(-36+113)^2 = 1296+5929 = 7225 = 85^2.
(A161478(1), a(2)) = (0, 113) is a solution: 0^2+(0+113)^2 = 12769 = 113^2.
(A161478(3), a(4)) = (175, 337) is a solution: 175^2+(175+113)^2 = 30625+82944 = 113569 = 337^2.

Crossrefs

Cf. A161478, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A161480 (decimal expansion of (129+44*sqrt(2))/113), A161481 (decimal expansion of (16131+6970*sqrt(2))/113^2).

Programs

  • PARI
    {forstep(n=-36, 10000000, [3, 1], if(issquare(2*n^2+226*n+12769, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=85, a(2)=113, a(3)=173, a(4)=337, a(5)=565, a(6)=953.
G.f.: (1-x)*(85+198*x+371*x^2+198*x^3+85*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 113*A001653(k) for k >= 1.
Showing 1-3 of 3 results.

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