

A111588


Crazy Dice: number of ways to design a pair of nsided dice with positive integers on their faces, so that the sums when they are tossed occur with the same probabilities as if a pair of standard nsided dice were tossed.


3



1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 8, 1, 2, 2, 10, 1, 8, 1, 8, 2, 2, 1, 33, 2, 2, 4, 8, 1, 13, 1, 26, 2, 2, 2, 57, 1, 2, 2, 33, 1, 13, 1, 8, 8, 2, 1, 140, 2, 8, 2, 8, 1, 33, 2, 33, 2, 2, 1, 125, 1, 2, 8, 71, 2, 13, 1, 8, 2, 13, 1, 348, 1, 2, 8, 8, 2, 13, 1, 140, 10, 2, 1, 122, 2, 2, 2, 33, 1, 118, 2
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OFFSET

1,4


COMMENTS

It is not required that the two dice be identical, it is not required that the entries be bounded by n and we do not ask that the entries be distinct from one another on each cube.
We pretend for the purpose of this sequence that regular nsided dice exist for all n.
In other words, how many (unordered) pairs of polynomials B(x) = x^b_1 + x^b_2 + ... + x^b_n, C(x) = x^c_1 + x^c_2 + ... + x^c_n, are there with all exponents positive integers, such that B(x)*C(x) = (x+x^2+x^3+...+x^n)^2?
a(n) = 1 means that the only way two nsided dice can have the same probability distribution as two normal nsided dice (each side numbered 1 through n) is if they are normal. a(6) = 2 corresponds to normal dice and Sicherman dice (one labeled 1,2,2,3,3,4 and the other 1,3,4,5,6,8).  Charles R Greathouse IV, Jan 19 2017
Records are: 1, 2, 4, 8, 10, 33, 57, 140, 348, 583, 956, 2036, 2393, 3050, ... and they seem to occur at positions given by A033833.  Antti Karttunen, Aug 28 2017


REFERENCES

M. Gardner, "Penrose Tiles to Trapdoor Ciphers", p. 266.


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..239
Donald J. Newman, A Problem Seminar, Springer; see Problem #88.


EXAMPLE

The first nontrivial example is for n=4: {1,2,2,3} and {1,3,3,5} together have the same sum probabilities as a pair of {1,2,3,4}. That is, (x + 2x^2 + x^3)(x + 2x^3 + x^5)=(x + x^2 + x^3 + x^4)^2.


PROG

(PARI) ok(p, e, n)=my(v=Vec(factorback(p, e))); vecmin(v)>=0 && vecsum(v)==n
a(n)=if(n<4, return(1)); my(x='x, f=factor((x^n1)/(x1)), p=f[, 1], e=2*f[, 2]~, u=vector(#e, i, [0, e[i]]), s, t); t=vecmax(e); for(i=1, #e, if(e[i]==t, u[i][2]\=2; break)); forvec(v=u, t=ev; if(cmp(v, t)<=0 && ok(p, v, n) && ok(p, t, n), s++)); s \\ Charles R Greathouse IV, Jan 19 2017


CROSSREFS

Cf. A033833.
Sequence in context: A292504 A281118 A284289 * A070972 A180229 A304576
Adjacent sequences: A111585 A111586 A111587 * A111589 A111590 A111591


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Nov 17 2005


EXTENSIONS

Edited and extended by Matthew Conroy, Jan 16 2006
Correction to some terms, thanks to Adam Chalcraft.  Matthew Conroy, Apr 04 2010


STATUS

approved



