Robert A. Beeler - cp's OEIS frontend

User: Robert A. Beeler

Robert A. Beeler has authored 3 sequences.

A243521 The number of states in a Tower of Hanoi puzzle with three pegs and n discs, where a larger disc can be placed directly on top of a smaller one at most once per peg.

1, 3, 12, 57, 300, 1701, 10206, 63825, 411096, 2702349, 17992506, 120543561, 808224372, 5400815829, 35868103734, 236354531841, 1544182760496, 10001335837725, 64233753928722, 409298268016761, 2589206145139596

Offset: 0

Robert A. Beeler, Jun 05 2014

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (40,-715,7522,-51583,240964,-776637,1705554,-2442744,2060640,-777600).
Index entries for sequences related to Towers of Hanoi

Terms in product are A000325.

Vec((1 - 37*x + 607*x^2 - 5800*x^3 + 35617*x^4 - 146023*x^5 + 400653*x^6 - 711780*x^7 + 746142*x^8 - 353412*x^9) / ((1 - 3*x)^4*(1 - 4*x)^3*(1 - 5*x)^2*(1 - 6*x)) + O(x^30)) \\ Colin Barker, Jul 18 2019

for n in range(11):
    t=0
    for k in range(n+1):
        for j in range(n-k+1):
            t=t+((Combinations(n,k).cardinality())*(Combinations(n-k,j).cardinality())*((2^k)-k)*((2^j)-j)*((2^(n-k-j))-n+k+j));
    print(t)

a(n) = Sum_{i+j+k=n, i >= 0, j >= 0, k>= 0} {n choose i, j, k}(2^i-i)(2^j-j)(2^k-k).
a(n) = 6^n-3*n*5^{n-1}+3*n*(n-1)*4^{n-2}-n*(n-1)*(n-2)3^{n-3}.
From Colin Barker, Jul 18 2019: (Start)
G.f.: (1 - 37*x + 607*x^2 - 5800*x^3 + 35617*x^4 - 146023*x^5 + 400653*x^6 - 711780*x^7 + 746142*x^8 - 353412*x^9) / ((1 - 3*x)^4*(1 - 4*x)^3*(1 - 5*x)^2*(1 - 6*x)).
a(n) = 40*a(n-1) - 715*a(n-2) + 7522*a(n-3) - 51583*a(n-4) + 240964*a(n-5) - 776637*a(n-6) + 1705554*a(n-7) - 2442744*a(n-8) + 2060640*a(n-9) - 777600*a(n-10) for n>9.
(End)

A228094 Triangle starting at row 3 read by rows of the number of permutations in the n-th Dihedral group which are the product of k disjoint cycles, d(n,k), n >= 3, 1 <= k <= n.

Original entry on oeis.org

2, 3, 1, 2, 3, 2, 1, 4, 0, 5, 0, 1, 2, 2, 4, 3, 0, 1, 6, 0, 0, 7, 0, 0, 1, 4, 2, 0, 5, 4, 0, 0, 1, 6, 0, 2, 0, 9, 0, 0, 0, 1, 4, 4, 0, 0, 6, 5, 0, 0, 0, 1, 10, 0, 0, 0, 0, 11, 0, 0, 0, 0, 1, 4, 2, 2, 2, 0, 7, 6, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 1

Offset: 3

Robert A. Beeler, Aug 09 2013

The multivariable row polynomials give n times the cycle index for the Dihedral group D_n, called Z(D_n) (see the MathWorld link with the Harary reference). For example, 12*Z(D_6) = 2*(y_6)^1 + 2*(y_3)^2 + 4*(y_2)^3+3*(y_1)^2*(y_2)^2 + 1*(y_1)^6.

			Triangle begins
   2, 3, 1;
   2, 3, 2, 1;
   4, 0, 5, 0, 1;
   2, 2, 4, 3, 0,  1;
   6, 0, 0, 7, 0,  0, 1;
   4, 2, 0, 5, 4,  0, 0, 1;
   6, 0, 2, 0, 9,  0, 0, 0, 1;
   4, 4, 0, 0, 6,  5, 0, 0, 0, 1;
  10, 0, 0, 0, 0, 11, 0, 0, 0, 0, 1;
   4, 2, 2, 2, 0,  7, 6, 0, 0, 0, 0, 1;
   ...

Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015. See Theorem 8.4.12 at pp. 246-247.
Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, 1973, p. 37.

Stefano Spezia, First 150 rows of the triangle, flattened
Eric Weisstein's World of Mathematics, Cycle Index.

Cf. A000010, A054523, A130534.

d[n_,k_]:=If[Divisible[n,k],EulerPhi[n/k],0]+If[OddQ[n]&&k==(n+1)/2,n,If[EvenQ[n]&&(k==n/2||k==(n+2)/2),n/2,0]]; Table[d[n,k],{n,3,12},{k,n}]//Flatten (* Stefano Spezia, Jun 26 2023 *)

d(n,k) = A054523(n,k) + d'(n,k), where: If n is odd, then d'(n,k)= n when k=(n+1)/2 and d'(n,k)=0 otherwise. If n is even, then d'(n,k)=n/2 when k=n/2, (n+2)/2 and d'(n,k)=0 otherwise.

Terms corrected by Stefano Spezia, Jun 30 2023

A181044 The number of ways to compute the determinant of an n X n matrix using cofactor expansion.

Original entry on oeis.org

1, 4, 384, 173946175488, 1592481597212922365761871004823571903636713118111555911680

Offset: 1

Robert A. Beeler, Sep 30 2010

nonn

Robert A. Beeler, How to Count: An Introduction to Combinatorics and Its Applications, Springer International Publishing, 2015. See Theorem 6.1.9 at p. 153.

Stefano Spezia, Table of n, a(n) for n = 1..6
Robert A. Beeler, A Note on the number of ways to compute a determinant using cofactor expansion, Bull. Inst. Combin. Appl., 63 (2011), 36-38. [ResearchGate link]

Cf. A363767.

a[1]=1; a[n_]:=2n a[n-1]^n; Array[a,5] (* Stefano Spezia, Jun 20 2023 *)

a(n) = if (n==1, 1, 2*n*a(n-1)^n); \\ Michel Marcus, Jun 21 2023

a(n) = 2*n*(a(n-1))^n.
a(n) = 2*2^n*2^(n*(n-1))*2^(n*(n-1)*(n-2))*...*2^(n*(n-1)*...*4*3)*n*(n-1)^n*(n-2)^(n*(n-1))*(n-3)^(n*(n-1)*(n-2))*...*2^(n*(n-1)*...*4*3).
From Robert A. Beeler, Oct 11 2010: (Start)
4^(n!*(e-2)) < a(n) < (2*e)^(n!*(e-2)).
a(n) ~ A363767^n!. (End)

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User: Robert A. Beeler

A243521 The number of states in a Tower of Hanoi puzzle with three pegs and n discs, where a larger disc can be placed directly on top of a smaller one at most once per peg.

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A228094 Triangle starting at row 3 read by rows of the number of permutations in the n-th Dihedral group which are the product of k disjoint cycles, d(n,k), n >= 3, 1 <= k <= n.

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A181044 The number of ways to compute the determinant of an n X n matrix using cofactor expansion.

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Mathematica

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