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.

A129642 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+457)^2 = y^2.

Original entry on oeis.org

0, 348, 495, 1371, 3255, 4088, 9140, 20096, 24947, 54383, 118235, 146508, 318072, 690228, 855015, 1854963, 4024047, 4984496, 10812620, 23454968, 29052875, 63021671, 136706675, 169333668, 367318320, 796785996, 986950047, 2140889163, 4644010215, 5752367528
Offset: 1

Views

Author

Mohamed Bouhamida, May 31 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+457, y).
Corresponding values y of solutions (x, y) are in A160580.
Limit_{n->oo} a(n)/a(n-3) = 3+2*sqrt(2).
Limit_{n->oo} a(n)/a(n-1) = (601+276*sqrt(2))/457 for n mod 3 = {1, 2}.
Limit_{n->oo} a(n)/a(n-1) = (213651+31850*sqrt(2))/457^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A160580, A001652, A129641, A156035 (decimal expansion of 3+2*sqrt(2)), A160581 (decimal expansion of (601+276*sqrt(2))/457), A160582 (decimal expansion of (213651+31850*sqrt(2))/457^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,348,495,1371,3255,4088,9140},30] (* Harvey P. Dale, May 13 2012 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+914*n+208849), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+914 for n > 6; a(1)=0, a(2)=348, a(3)=495, a(4)=1371, a(5)=3255, a(6)=4088.
G.f.: x^2*(348+147*x+876*x^2-204*x^3-49*x^4-204*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 457*A001652(k) for k >= 0.
a(1)=0, a(2)=348, a(3)=495, a(4)=1371, a(5)=3255, a(6)=4088, a(7)=9140, a(n) = a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, May 13 2012

Extensions

Edited and two terms added by Klaus Brockhaus, Jun 08 2009

A160582 Decimal expansion of (213651 +31850*sqrt(2))/457^2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Jun 08 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A129642.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160580.
A quadratic number with minimal polynomial 208849*x^2 - 427302*x + 208849. - Charles R Greathouse IV, Dec 06 2016

Examples

			(213651 +31850*sqrt(2))/457^2 = 1.23866382870678373994...

Links

Crossrefs

Cf. A129642, A160580, A002193 (decimal expansion of sqrt(2)), A160581 (decimal expansion of (601+276*sqrt(2))/457).

Programs

  • Magma
    (213651 +31850*Sqrt(2))/457^2; // G. C. Greubel, Apr 07 2018
  • Maple
    with(MmaTranslator[Mma]): Digits:=150:
RealDigits(evalf((213651+31850*sqrt(2))/457^2))[1]; # Muniru A Asiru, Apr 08 2018
  • Mathematica
    RealDigits[(213651+31850Sqrt[2])/457^2,10,120][[1]] (* Harvey P. Dale, Jan 06 2013 *)
  • PARI
    (213651+31850*sqrt(2))/457^2 \\ Charles R Greathouse IV, Dec 06 2016

Formula

Equals (650 +49*sqrt(2))/(650 -49*sqrt(2)).
Equals (3 +2*sqrt(2))*(23 -6*sqrt(2))^2/(23 +6*sqrt(2))^2.

A160580 Positive numbers y such that y^2 is of the form x^2+(x+457)^2 with integer x.

Original entry on oeis.org

325, 457, 877, 1073, 2285, 4937, 6113, 13253, 28745, 35605, 77233, 167533, 207517, 450145, 976453, 1209497, 2623637, 5691185, 7049465, 15291677, 33170657, 41087293, 89126425, 193332757, 239474293, 519466873, 1126825885, 1395758465
Offset: 1

Views

Author

Klaus Brockhaus, Jun 08 2009

Keywords

Comments

(-204, a(1)) and (A129642(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+457)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (601+276*sqrt(2))/457 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (213651+31850*sqrt(2))/457^2 for n mod 3 = 1.

Examples

			(-204, a(1)) = (-204, 325) is a solution: (-204)^2+(-204+457)^2 = 41616+64009 = 105625 = 325^2.
(A129642(1), a(2)) = (0, 457) is a solution: 0^2+(0+457)^2 = 208849 = 457^2.
(A129642(3), a(4)) = (495, 1073) is a solution: 495^2+(495+457)^2 = 245025+906304 = 1151329 = 1073^2.

Crossrefs

Cf. A129642, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160581 (decimal expansion of (601+276*sqrt(2))/457), A160582 (decimal expansion of (213651+31850*sqrt(2))/457^2).

Programs

  • PARI
    {forstep(n=-204, 10000000, [3, 1], if(issquare(2*n^2+914*n+208849, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=325, a(2)=457, a(3)=877, a(4)=1073, a(5)=2285, a(6)=4937.
G.f.: (1-x)*(325+782*x+1659*x^2+782*x^3+325*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 457*A001653(k) for k >= 1.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.