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.

User: H. Tracy Hall

H. Tracy Hall's wiki page.

H. Tracy Hall has authored 6 sequences.

A338772 The number of different probabilities p for which a coin that lands heads with probability p can, using n flips, perfectly model one flip of a fair coin.

Original entry on oeis.org

1, 3, 19, 271, 8635, 623533
Offset: 1

Views

Author

H. Tracy Hall, Nov 08 2020

Keywords

Comments

This counts the distinct roots in the range 0 to 1 occurring among a set of degree-n polynomials the number of which is given by A055612. The 2^n possible outcomes of n coin flips are divided into n + 1 classes depending on how many times heads comes up, and there is one polynomial for each way of deciding how many of each class goes on which side of the partition of outcomes that will model a fair coin flip.

Examples

			For n = 2 the a(2) = 3 different values of p are, in increasing order:
1 - sqrt(1/2), which can model a fair flip with the partition (HH, HT, TH), (TT);
1/2, which can model a fair flip with the partition (HH, HT), (TH, TT) (i.e., by ignoring the second flip); and
sqrt(1/2), which can model a fair flip with the partition (HH), (HT, TH, TT).

Links

Programs

  • SageMath
    P.
     = QQ[]
def polystream(nn, pol=P(0), kk=0):
    if kk >= nn:
        yield pol - 1
    else:
        for ii in sxrange(binomial(nn, kk) + 1):
            for xx in polystream(nn, pol + 2 * ii * p^kk * (1-p)^(nn-kk), kk + 1):
                yield xx
def calculate(nn):
    solutions = Set()
    for pol in polystream(nn):
        rootlist = [xx[0] for xx in pol.roots(ring=QQbar)]
        for root in rootlist:
            if root.real() == root and 0 <= root <= 1:
                solutions += Set([root])
    return len(solutions)

A111214 Score for an n-letter word in the game of Boggle.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11
Offset: 1

Views

Author

H. Tracy Hall (h.tracy(AT)gmail.com), Oct 25 2005

Keywords

Comments

Words of one or two letters are not counted, which is equivalent to a score of 0. There are sixteen cubes in the game, which may not be reused within a single word, but 17-letter words involving the Qu tile are possible. Boggle is a registered trademark of Parker Brothers.

Examples

			The word "sulfonate" is worth 11 points in a round of Boggle (provided no other player has found it) because a(9) = 11.

Links

A051655 Continued fraction for imaginary part of first zero (in critical strip) of Riemann zeta function.

Original entry on oeis.org

14, 7, 2, 2, 1, 2, 1, 1, 1, 12, 23, 1, 1, 1, 4, 1, 15, 4, 3, 4, 1, 1, 23, 2, 66, 1, 1, 1, 1, 1, 8, 2, 1, 2, 17, 2, 4, 1, 3, 5, 3, 1, 6, 1, 1, 1, 7, 18, 1, 1, 1, 1, 7, 5, 2, 1, 1, 12, 4, 4, 1, 1, 1, 8, 48, 1, 1, 2, 4, 44, 26, 1, 7, 2, 6, 1, 3, 4, 4, 1, 1, 2, 8, 4, 1, 5, 4, 1, 2, 1, 3, 1, 53
Offset: 0

Views

Author

H. Tracy Hall (hthall(AT)math.berkeley.edu)

Keywords

Links

Crossrefs

Cf. A013629, A002410.

Programs

A051484 a(n) is the next natural number (besides 1) which is not congruent to a(i) mod a(j) for any i < j < n.

Original entry on oeis.org

0, 2, 3, 7, 13, 19, 25, 43, 61, 109, 139, 151, 181, 187, 229, 295, 337, 487, 505, 517, 565, 571, 643, 655, 685, 823, 883, 901, 985, 1189, 1243, 1279, 1285, 1429, 1441, 1597, 1621, 1639, 1699, 1735, 1741, 1867, 1915, 1933, 2101, 2143, 2155, 2167, 2371
Offset: 1

Views

Author

H. Tracy Hall (hthall(AT)math.berkeley.edu)

Keywords

Comments

What is the asymptotic distribution of these numbers?
All elements from 7 onward seem to be either 1 or 7 modulo 12. - Walter Kehowski, Oct 08 2005
The initial 3 terms force all subsequent terms to be congruent to 1 modulo 6. - Charlie Neder, Oct 07 2018

Examples

			5 is congruent to 2 (mod 3), so 5 cannot be in the sequence. 25 mod 2 (resp. 3, 7, 13, 19) gives 1 (resp. 1, 4, 12, 6), which is not in the sequence.

Links

Programs

  • Maple
    M:=[0,2]: for z to 1 do for n from 3 to 5000 do b:=true; for j from 1 to nops(M)-1 do for k from j+1 to nops(M) do if M[j] = n mod M[k] then b:=false; break; fi od od; if b then M:=[op(M),n] fi; od; od; M; # Walter Kehowski, Oct 08 2005
  • Mathematica
    a[1] = 0; a[2] = 2; a[n_] := a[n] = Block[{k = a[n - 1] + 1, t = a[ # ] & /@ Range[n - 1]}, While[ Intersection[t, Union[ Mod[k, Rest[ t]]]] != {}, k++ ]; k]; Table[ a[n], {n, 50}] (* Robert G. Wilson v, Oct 19 2005 *)

Extensions

More terms from Walter Kehowski, Oct 08 2005

A051482 Run lengths in binary expansion of e.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 6, 4, 1, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 6, 1, 3, 3, 1, 1, 1, 1, 2, 3, 1, 7, 1, 2, 3, 2, 4, 1, 1, 2, 4, 2, 4, 3, 3, 1, 2, 3, 1, 1, 3, 2, 3, 3, 1, 1, 2, 5, 4, 2, 3, 3, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2
Offset: 1

Views

Author

H. Tracy Hall (hthall(AT)math.berkeley.edu)

Keywords

Examples

			The binary expansion of e is: 1 0. 1 0 11 0 111111 0000 1 ...

Crossrefs

A004593.

Programs

  • Mathematica
    Length/@Split[RealDigits[E,2,200][[1]]] (* Harvey P. Dale, Nov 15 2018 *)

A051480 Run lengths in binary expansion of Pi.

Original entry on oeis.org

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

Views

Author

H. Tracy Hall (hthall(AT)math.berkeley.edu)

Keywords

Examples

			The binary expansion of Pi is: 11. 00 1 00 1 0000 111111 0 ...

Links

Crossrefs

Cf. A000796, A004601.

Programs

  • Mathematica
    Map[Length, Most[Split[First[RealDigits[Pi, 2, 300]]]]] (* Paolo Xausa, Jul 30 2024 *)

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