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 1-5 of 5 results.

A118452 Decimal expansion of onset of logistic map 5-bifurcation.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Apr 28 2006

Keywords

Comments

Algebraic of order 22.

Examples

			3.7381723752659623694...

Links

Crossrefs

Cf. A086178, A086180, A091517, A098587, A118453, A118454.

Programs

  • Mathematica
    RealDigits[Root[ -28629151 - 31399714*#1 - 14329471*#1^2 + 3613360*#1^3 + 13369804*#1^4 + 5836984*#1^5 - 1495060*#1^6 - 4002880*#1^7 - 1613676*#1^8 + 1484268*#1^9 + 808298*#1^10 - 234632*#1^11 - 294146*#1^12 + 8008*#1^13 + 103556*#1^14 - 17344*#1^15 - 18092*#1^16 + 7488*#1^17 + 116*#1^18 - 700*#1^19 + 189*#1^20 - 22*#1^21 + #1^22 &, 4],10,110][[1]]

A118453 Decimal expansion of onset of logistic map 6-bifurcation.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Apr 28 2006

Keywords

Comments

Algebraic of order 40.

Examples

			3.6265531616949737258...

Links

Crossrefs

Cf. A086178, A086180, A091517, A098587, A118454.

Programs

  • Mathematica
    RealDigits[Root[3063651608241 + 583552687284*#1 + 1847916843066*#1^2 - 195203691396*#1^3 - 266965430067*#1^4 - 930539982708*#1^5 - 32353949026*#1^6 - 20846657216*#1^7 + 268499651644*#1^8 + 203273817100*#1^9 - 70347416426*#1^10 - 23477492116*#1^11 - 148805533357*#1^12 + 63425976316*#1^13 + 29299283358*#1^14 + 1219753488*#1^15 + 10442067223*#1^16 - 25709485160*#1^17 + 5757007476*#1^18 + 945192256*#1^19 + 3195434424*#1^20 + 746246488*#1^21 - 3839476860*#1^22 + 1483852216*#1^23 + 347099271*#1^24 - 126361652*#1^25 - 100717898*#1^26 - 75365092*#1^27 + 137026587*#1^28 - 48885412*#1^29 - 12966642*#1^30 + 16770304*#1^31 - 5563348*#1^32 + 195228*#1^33 + 516582*#1^34 - 225060*#1^35 + 52181*#1^36 - 7676*#1^37 + 722*#1^38 - 40*#1^39 + #1^40 &, 5],10,110][[1]]

A118746 Decimal expansion of onset of logistic map 7-bifurcation.

Original entry on oeis.org

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

Views

Author

Eric W. Weisstein, Apr 28 2006

Keywords

Comments

Algebraic of order 114.

Examples

			3.7016407641603495818...

Links

Crossrefs

Cf. A086178, A086180, A091517, A098587, A118453, A118454.

A118454 Algebraic degree of the onset of the logistic map n-bifurcation.

Original entry on oeis.org

1, 1, 2, 2, 22, 40, 114, 12, 480, 944, 2026, 3918, 8166, 16104, 32630, 240, 131038, 260928, 524250, 1046418, 2096706, 4190168, 8388562, 16768200, 33554240, 67092432, 134216136, 268402446, 536870854, 1073672968, 2147483586, 65280, 8589928346, 17179606976, 34359737478
Offset: 1

Views

Author

Eric W. Weisstein, Apr 28 2006

Keywords

Comments

a(2^n) is A087046(n).

Examples

			The onsets begin at 1, 3, 1+2*sqrt(2), 1+sqrt(6), ...

Links

Crossrefs

Cf. A000740, A006876, A086178, A086180, A087046, A091517, A098587, A118452, A118453, A118746, A181906, A181907, A181909, A181910, A181911.

Programs

  • Mathematica
    degRp[n_] := Sum[MoebiusMu[n/d] 2^(d - 1), {d, Divisors[n]}]; degRo[n_] := degRp[n]*2 - Sum[EulerPhi[n/d] degRp[d], {d, Divisors[n]}]; Table[If[n <= 2, 1, 2 If[2^Round[Log2[n]] == n, degRp[n/2], degRo[n]]], {n, 1, 35}] (* Cheng Zhang, Apr 02 2012 *)

Extensions

More terms from Cheng Zhang, Apr 02 2012

A140357 a(1)=1; a(n)=floor(4*a(n-1)*(n-a(n-1)) / n) for n > 1.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 3, 7, 6, 9, 6, 12, 3, 9, 14, 7, 16, 7, 17, 10, 20, 7, 19, 15, 24, 7, 20, 22, 21, 25, 19, 30, 10, 28, 22, 34, 11, 31, 25, 37, 14, 37, 20, 43, 7, 23, 46, 7, 24, 49, 7, 24, 52, 7, 24, 54, 11, 35, 56, 14, 43, 52, 36, 63, 7, 25, 62, 21, 58, 39, 70, 7, 25, 66, 31, 73, 15, 48
Offset: 1

Views

Author

Franklin T. Adams-Watters, May 30 2008, May 31 2008

Keywords

Comments

a(n)/n approximates the behavior of the logistic map x(n+1) = r*x(n)*(1-x(n)) at the critical value r = 4 where its iterated behavior becomes chaotic.
Conjecture: starting with any given n and any 1 <= a(n) <= n and applying the rule for the sequence produces a sequence which eventually joins this one. For example, starting with a(9)=5, the sequence continues 10,3,9,11,9, at which point it has joined.
There is a number x(1) such that iterating the logistic map x(n+1) = 4*x(n)*(1-x(n)) approaches a(n)/n; in particular x(n) > 1/2 iff a(n)/n > 1/2 and lim_{n->infinity} x(n)-a(n)/n = 0. x(1) is approximately 0.74300456748016924159182578873962328734252790178266693834898117732270042549583799064232908893034253248. It appears that |x(n)-a(n)/n| < 1/sqrt(n) for all n.

Links

Crossrefs

Cf. A079271, A087089, A098587, A118454.

Programs

  • Mathematica
    a[1]=1;a[n_]:=a[n]=Floor[(4a[n-1](n-a[n-1]))/n];Table[a[n],{n,100}]  (* Harvey P. Dale, Mar 28 2011 *)
nxt[{n_,a_}]:={n+1,Floor[4a (n+1-a)/(n+1)]}; NestList[nxt,{1,1},80][[All,2]] (* Harvey P. Dale, Dec 22 2019 *)
Showing 1-5 of 5 results.

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