A275518 Number of simplices in corner-cut triangulation of the n-cube.

1, 2, 5, 16, 67, 364, 2445, 19296, 173015, 1728604, 19011049, 228124384, 2965598547, 41518338684, 622774990133, 9964399645504, 169394793547567, 3049106282938684, 57933019373868897, 1158660387473183616, 24331868136927943019, 535301099012395872028

Offset

1,2

Comments

This corrects the value of a(10) in A239911 published by Sallee in Discr. Math. 40. The correct value is for example given by Lee.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Carl W. Lee, Triangulating the d-cube, Annals of the New York Academy of Sciences 440 (1985): 205-211.

John F. Sallee, A note on minimal triangulations of an n-cube, Discrete Appl. Math. 4 (1982), no. 3, 211-215. MR0675850 (84g:52019)

John F. Sallee, The middle-cut triangulations of the n-cube, SIAM J. Algebraic Discrete Methods 5 (1984), no. 3, 407-419. MR0752044 (86c:05054). See Table 2.

John F. Sallee, A triangulation of the n-cube, Discrete Math. 40 (1982), no. 1, 81-86. MR0676714 (84d:05065b)

Formula

a(n) = 1 + 2^(n-1) - n! + n!*Sum_{i=1..n} (2^(i-1)-1)/i!. - Andrew Howroyd, Sep 06 2023, after Maple program

Maple

p := proc(d, x)
    add( x^i/i!, i=0..d) ;
end proc:
A275518 := proc(d)
    d!*(p(d, 2)/2-p(d, 1))+2^(d-1)-d!/2+1 ;
end proc:
seq(A275518(d), d=1..18) ;

Mathematica

p[d_, x_] := Sum[x^i/i!, {i, 0, d}];
A275518[d_] := d!*(p[d, 2]/2 - p[d, 1]) + 2^(d - 1) - d!/2 + 1;
Table[A275518[d], {d, 1, 18}] (* Jean-François Alcover, Sep 06 2023, after Maple program *)

Prog

(PARI) a(n) = 1 + 2^(n-1) - n! + n!*sum(i=1, n, (2^(i-1)-1)/i!) \\ Andrew Howroyd, Sep 06 2023

Crossrefs

Cf. A239911, A019502, A019503, A019504, A166932.

Sequence in context: A019504 A239912 A239911 * A005163 A359986 A006116

Adjacent sequences: A275515 A275516 A275517 * A275519 A275520 A275521

Keyword

nonn,easy

Author

R. J. Mathar, Jul 31 2016

Extensions

Terms a(19) and beyond from Andrew Howroyd, Sep 06 2023

Status

approved