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.

A010497 Decimal expansion of square root of 43.

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

Continued fraction expansion is 6 followed by {1, 1, 3, 1, 5, 1, 3, 1, 1, 12} repeated. - Harry J. Smith, Jun 05 2009

Examples

			6.557438524302000652344109997636001627926966319883789769865460105585659....

Links

Crossrefs

Cf. A010134 (continued fraction).

Programs

  • Mathematica
    RealDigits[N[Sqrt[43],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 24 2011 *)
  • PARI
    default(realprecision, 20080); x=sqrt(43); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010497.txt", n, " ", d));  \\ Harry J. Smith, Jun 05 2009

A228932 Optimal ascending continued fraction expansion of sqrt(43) - 6.

Original entry on oeis.org

2, 9, 30, 60, 122, -878, 11429, 35241, -177141, 709582, -3123032, -1157723745, 3237738813, -16178936725, 33395053634, -71863018424, -153349368674, -386763022623, -8021033029400, 16314606875900, 52522689388692
Offset: 1

Views

Author

Giovanni Artico, Sep 10 2013

Keywords

Comments

See A228929 for the definition of "optimal ascending continued fraction".
In A228931 it is shown that many numbers of the type sqrt(x) seem to present in their expansion a recurrence relation a(n) = a(n-1)^2 - 2 between the terms, starting from some point onward; 43 is the first natural number whose terms don't respect this relation.
The numbers in range 1 .. 200 that exhibit this behavior are 43, 44, 46, 53, 58, 61, 67, 73, 76, 85, 86, 89, 91, 94, 97, 103, 106, 108, 109, 113, 115, 116, 118, 125, 127, 129, 131, 134, 137, 139, 149, 151, 153, 154, 157, 159, 160, 161, 163, 166, 172, 173, 176, 177, 179, 181, 184, 186, 190, 191, 193, 199.
Nevertheless, the expansions of 3*sqrt(43), 9*sqrt(43), and sqrt(43)/5 satisfy the recurrence relation.

Examples

			sqrt(43) = 6 + 1/2*(1 + 1/9*(1 + 1/30*(1 + 1/60*(1 + 1/122*(1 - 1/878*(1 + ...)))))).

References

Links

Crossrefs

Cf. A010134, A010497, A228929, A228931.

Programs

  • Maple
    ArticoExp := proc (n, q::posint)::list; local L, i, z; Digits := 50000; L := []; z := frac(evalf(n)); for i to q+1 do if z = 0 then break end if; L := [op(L), round(1/abs(z))*sign(z)]; z := abs(z)*round(1/abs(z))-1 end do; return L end proc
# List the first 8 terms of the expansion of sqrt(43)-6
ArticoExp(sqrt(43),20)
  • Mathematica
    ArticoExp[x_, n_] := Round[1/#] & /@ NestList[Round[1/Abs[#]]*Abs[#] - 1 &, FractionalPart[x], n]; Block[{$MaxExtraPrecision = 50000},
ArticoExp[Sqrt[43] - 6, 20]] (* G. C. Greubel, Dec 26 2016 *)

A041072 Numerators of continued fraction convergents to sqrt(43).

Original entry on oeis.org

6, 7, 13, 46, 59, 341, 400, 1541, 1941, 3482, 43725, 47207, 90932, 320003, 410935, 2374678, 2785613, 10731517, 13517130, 24248647, 304500894, 328749541, 633250435, 2228500846, 2861751281, 16537257251
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A041073, A010497, A010134.

Programs

  • Mathematica
    Numerator[Convergents[Sqrt[43], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

Formula

G.f.: -(x^19 -6*x^18 +7*x^17 -13*x^16 +46*x^15 -59*x^14 +341*x^13 -400*x^12 +1541*x^11 -1941*x^10 -3482*x^9 -1941*x^8 -1541*x^7 -400*x^6 -341*x^5 -59*x^4 -46*x^3 -13*x^2 -7*x -6) / (x^20 -6964*x^10 +1). - Colin Barker, Nov 04 2013
Showing 1-3 of 3 results.

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

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