A059116 -id:A059116 - cp's OEIS frontend

A058809 The sequence lambda(3,n), where lambda is defined in A055203. Number of ways of placing n identifiable positive intervals with a total of exactly three starting and/or finishing points.

0, 0, 6, 24, 78, 240, 726, 2184, 6558, 19680, 59046, 177144, 531438, 1594320, 4782966, 14348904, 43046718, 129140160, 387420486, 1162261464, 3486784398, 10460353200, 31381059606, 94143178824, 282429536478, 847288609440

Offset: 0

N. J. A. Sloane, Jan 03 2001

For all n, a(n)=1*3^n-3*1^n+3*0^n-1*0^n [with 0^0=1] where powers are taken of triangular numbers and multiplied by binomial coefficients with alternating signs. - Henry Bottomley, Jan 05 2001
For n>=1, a(n) is the number of facets of the harmonic polytope. See Ardila and Escobar. - Michel Marcus, Jun 08 2020
For n >= 3, this is the number of acyclic orientations of the wheel graph of order n+1. - Peter Kagey, Oct 13 2020
Number of ternary strings of length n with at least 2 different digits. - Enrique Navarrete, Nov 20 2020
A level 1 Hanoi graph is a triangle. Level n+1 is formed from three copies of level n by adding edges between pairs of corner vertices of each pair of triangles.  This graph represents the allowable moves in the Towers of Hanoi problem with n disks.  a(n) is the number of degree 3 vertices in the level n Hanoi graph. - Allan Bickle, Aug 07 2024

			a(2)=6 since intervals a-a and b-b can be combined as a-ab-b, a-b-ab, ab-a-b, b-ab-a, b-a-ab, or ab-b-a.
The level 2 Hanoi graph has 9 vertices, 6 with degree 3, so a(2) = 6.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Federico Ardila and Laura Escobar, The harmonic polytope, arXiv:2006.03078 [math.CO], 2020.
Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
Eric Weisstein's World of Mathematics, Hanoi Graph
Eric Weisstein's World of Mathematics, Wheel Graph
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A059116, A059117.

Cf. A000225, A029858, A058809, A375256 (Hanoi graphs).

Join[{0},NestList[3#+6&,0,30]] (* or *) Join[{0},LinearRecurrence[{4,-3},{0,6},30]] (* Harvey P. Dale, Sep 29 2013 *)

concat([0,0], Vec(6*x^2 / ((1 - x)*(1 - 3*x)) + O(x^30))) \\ Colin Barker, Oct 14 2020

For n>0, a(n) = 3^n-3 = 3*a(n-1)+6.
a(0)=0, a(1)=0, a(2)=6, a(n) = 4*a(n-1)-3*a(n-2). - Harvey P. Dale, Sep 29 2013
G.f.: 6*x^2 / ((1 - x)*(1 - 3*x)). - Colin Barker, Oct 14 2020

A059117 Square array of lambda(k,n), where lambda is defined in A055203. Number of ways of placing n identifiable positive intervals with a total of exactly k starting and/or finishing points.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 6, 24, 1, 0, 0, 0, 0, 0, 114, 78, 1, 0, 0, 0, 0, 0, 180, 978, 240, 1, 0, 0, 0, 0, 0, 90, 4320, 6810, 726, 1, 0, 0, 0, 0, 0, 0, 8460, 63540, 43746, 2184, 1, 0, 0, 0, 0, 0, 0, 7560, 271170, 774000, 271194, 6558, 1

Offset: 0

Henry Bottomley, Jan 05 2001

			Rows are: 1,0,0,0,0,0,....; 0,0,1,0,0,0,....; 0,0,1,6,6,0,....; 0,0,1,24,114,180,.... etc.

Sum of rows gives A055203. Columns include A000007, A057427, A058809, A059116. Final positive number in each row is A000680.

A[ n_, k_] := If[n < 1 || k < 1, Boole[n == 0 && k == 0], n! k! Coefficient[ Normal[ Series[ Sum[ Exp[-x z] (x z)^m/m! Exp[y z m (m - 1)/2], {m, 0, n}], {z, 0, n + k}]], x^n y^k z^(n + k)]]; (* Michael Somos, Jul 17 2019 *)

lambda(k, n) = (lambda(k - 2, n - 1) + 2*lambda(k - 2, n - 1) + lambda(k - 2, n - 1))*k*(k - 1)/2 starting with lambda(k, 0) = 1 if k = 0 but = 0 otherwise. lambda(k, n) = sum_{j=0..k} (-1)^(k + j) * C(k, j) * ((j - 1)*j/2)^n.

A059517 The sequence A059515(3,n). Number of ways of placing n identifiable nonnegative intervals with a total of exactly three starting and/or finishing points.

Original entry on oeis.org

0, 0, 12, 138, 1056, 7050, 44472, 273378, 1659936, 10018650, 60289032, 362265618, 2175188016, 13055911050, 78349815192, 470141937858, 2820980767296, 16926272024250, 101558794406952, 609356253226098, 3656147979709776, 21936919259318250, 131621609699088312

Offset: 0

Henry Bottomley, Jan 19 2001

			a(2)=12 since if aA indicates a zero length interval and a-A one of positive length the possibilities are: aA-b-B, b-aA-B, b-B-aA, bB-a-A, a-bB-A, a-A-bB, ab-A-B, ab-B-A, a-b-AB, b-a-AB, a-bA-B, b-a-AB.

IBM Ponder This, Jan 01 2001
Index entries for linear recurrences with constant coefficients, signature (10,-27,18).

Cf. A059516.

concat([0,0], Vec(-6*x^2*(3*x+2)/((x-1)*(3*x-1)*(6*x-1)) + O(x^100))) \\ Colin Barker, Sep 13 2014

a(n) = A058809(n)+A059116(n) = 6^n-3*3^n+3 (for n>0).
a(n) = 10*a(n-1)-27*a(n-2)+18*a(n-3) for n>3. - Colin Barker, Sep 13 2014
G.f.: -6*x^2*(3*x+2) / ((x-1)*(3*x-1)*(6*x-1)). - Colin Barker, Sep 13 2014

More terms from Colin Barker, Sep 13 2014

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A058809 The sequence lambda(3,n), where lambda is defined in A055203. Number of ways of placing n identifiable positive intervals with a total of exactly three starting and/or finishing points.

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A059117 Square array of lambda(k,n), where lambda is defined in A055203. Number of ways of placing n identifiable positive intervals with a total of exactly k starting and/or finishing points.

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A059517 The sequence A059515(3,n). Number of ways of placing n identifiable nonnegative intervals with a total of exactly three starting and/or finishing points.

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