A275599 Triangle read by rows: T(n,k) = number of right-skewed domino towers with n dominoes having a base of k dominoes placed end-to-end.

1, 3, 1, 7, 4, 1, 15, 12, 4, 1, 31, 27, 13, 4, 1, 63, 61, 34, 13, 4, 1, 127, 124, 77, 35, 13, 4, 1, 255, 258, 165, 86, 35, 13, 4, 1, 511, 513, 348, 185, 87, 35, 13, 4, 1, 1023, 1039, 698, 399, 196, 87, 35, 13, 4, 1, 2047, 2062, 1410, 811, 423, 197, 87, 35, 13, 4, 1

Offset: 2

Author

Tricia Muldoon Brown, Aug 03 2016

Keywords

• nonn
• tabl

Comments

Domino towers are created by stacking domino blocks horizontally on a convex base of k dominoes. A right-skewed domino tower is a parallelogram domino tower such that at least one column of the polyomino is to the right of the base.

Examples

			Triangle begins:
1;
3, 1;
7, 4, 1;
15, 12, 4, 1;
...
For n = 5 and k = 3, each tower has a convex base of three dominoes. The fourth domino may be placed directly above the rightmost domino of the base, in which case the fifth domino must be placed on the fourth domino so its right end is not above the base. Alternately, the fourth domino may be placed so its right end is not above the base, leaving three choices for the fifth domino: directly above, above and to the right, or directly to the left on the same level. Thus T(5,3) = 4.
		

Links

• T. M. Brown Convex domino towers, arXiv:1608.01562 [math.CO], (2016)
• Tricia Muldoon Brown, Examples of right- or left-skewed domino towers of 10 dominoes having a base of 4 dominoes

Crossrefs

Column 1: A000225, n>=1.

Cf. A275204.

Formula

T(n,k) = Sum_{i=1..k} 2*T(n-k,i)+A(n-k,i) where A(n,k) is given by A275204 and with initial conditions T(n+1,n)=1 and T(n,k)=0 if n<2 and k<1, or n

G.f.: x^k/(1-2x^k) Sum_{i=1..k}*A_k(x)*(Sum_{Subsets S of {i,i+1,..,k-1}} (Product_{j in S} 2x^j/(1-2x^k)) where A_k(x) is the generating function in A275204.

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