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.

A129641 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.

Original entry on oeis.org

0, 200, 611, 1227, 2291, 4620, 8180, 14364, 27927, 48671, 84711, 163760, 284664, 494720, 955451, 1660131, 2884427, 5569764, 9676940, 16812660, 32463951, 56402327, 97992351, 189214760, 328737840, 571142264, 1102825427, 1916025531, 3328862051, 6427738620
Offset: 1

Views

Author

Mohamed Bouhamida, May 31 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+409, y).
Corresponding values y of solutions (x, y) are in A160577.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (473+168*sqrt(2))/409 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (204819+83570*sqrt(2))/409^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A160577, A001652, A129640, A156035 (decimal expansion of 3+2*sqrt(2)), A160578 (decimal expansion of (473+168*sqrt(2))/409), A160579 (decimal expansion of (204819+83570*sqrt(2))/409^2).

Programs

  • Mathematica
    LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 200, 611, 1227, 2291, 4620, 8180}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+818*n+167281), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+818 for n > 6; a(1)=0, a(2)=200, a(3)=611, a(4)=1227, a(5)=2291, a(6)=4620.
G.f.: x*(200+411*x+616*x^2-136*x^3-137*x^4-136*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 409*A001652(k) for k >= 0.

Extensions

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

A160579 Decimal expansion of (204819+83570*sqrt(2))/409^2.

Original entry on oeis.org

1, 9, 3, 0, 9, 1, 1, 6, 2, 4, 1, 9, 8, 3, 2, 2, 3, 0, 3, 3, 5, 9, 9, 5, 7, 9, 5, 5, 0, 7, 0, 8, 3, 5, 7, 1, 0, 1, 2, 8, 8, 6, 9, 8, 5, 7, 4, 5, 6, 9, 2, 3, 0, 8, 4, 3, 6, 3, 2, 8, 7, 6, 7, 5, 8, 5, 0, 1, 2, 9, 5, 0, 2, 0, 0, 6, 0, 7, 5, 3, 1, 7, 7, 9, 8, 9, 5, 4, 3, 7, 1, 6, 4, 8, 6, 6, 9, 0, 4, 3, 3, 0, 7, 1, 6
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 = A129641.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160577.

Examples

			(204819+83570*sqrt(2))/409^2 = 1.93091162419832230335...

Links

Crossrefs

Cf. A129641, A160577, A002193 (decimal expansion of sqrt(2)), A160578 (decimal expansion of (473+168*sqrt(2))/409).

Programs

  • Magma
    (204819 +83570*Sqrt(2))/409^2; // G. C. Greubel, Apr 08 2018
  • Mathematica
    RealDigits[(204819+83570*Sqrt[2])/409^2,10,120][[1]] (* Harvey P. Dale, Jul 16 2013 *)
  • PARI
    (204819 +83570*sqrt(2))/409^2 \\ G. C. Greubel, Apr 08 2018

Formula

Equals (610+137*sqrt(2))/(610-137*sqrt(2)).
Equals (3+2*sqrt(2))*(42-8*sqrt(2))^2/(42+8*sqrt(2))^2.

A160577 Positive numbers y such that y^2 is of the form x^2+(x+409)^2 with integer x.

Original entry on oeis.org

305, 409, 641, 1189, 2045, 3541, 6829, 11861, 20605, 39785, 69121, 120089, 231881, 402865, 699929, 1351501, 2348069, 4079485, 7877125, 13685549, 23776981, 45911249, 79765225, 138582401, 267590369, 464905801, 807717425, 1559630965
Offset: 1

Views

Author

Klaus Brockhaus, Jun 08 2009

Keywords

Comments

(-136, a(1)) and (A129641(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+409)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (473+168*sqrt(2))/409 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (204819+83570*sqrt(2))/409^2 for n mod 3 = 1.

Examples

			(-136, a(1)) = (-136, 305) is a solution: (-136)^2+(-136+409)^2 = 18496+74529 = 93025 = 305^2.
(A129641(1), a(2)) = (0, 409) is a solution: 0^2+(0+409)^2 = 167281 = 409^2.
(A129641(3), a(4)) = (611, 1189) is a solution: 611^2+(611+409)^2 = 373321+1040400 = 1413721 = 1189^2.

Links

Crossrefs

Cf. A129641, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160578 (decimal expansion of (473+168*sqrt(2))/409), A160579 (decimal expansion of (204819+83570*sqrt(2))/409^2).

Programs

  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1},{305,409,641,1189,2045,3541},50] (* or *) Select[Table[Sqrt[x^2+(x+409)^2],{x,-140,10^6}],IntegerQ] (* The second program generates the first 16 terms of the sequence. To generate more, increase the x constant but the program may take a long time to run. *) (* Harvey P. Dale, Mar 14 2022 *)
  • PARI
    {forstep(n=-136, 10000000, [3, 1], if(issquare(2*n^2+818*n+167281, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=305, a(2)=409, a(3)=641, a(4)=1189, a(5)=2045, a(6)=3541.
G.f.: (1-x)*(305+714*x+1355*x^2+714*x^3+305*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 409*A001653(k) for k >= 1.
Showing 1-3 of 3 results.

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