A008867 Triangle of truncated triangular numbers: k-th term in n-th row is number of dots in hexagon of sides k, n-k, k, n-k, k, n-k.

1, 3, 3, 6, 7, 6, 10, 12, 12, 10, 15, 18, 19, 18, 15, 21, 25, 27, 27, 25, 21, 28, 33, 36, 37, 36, 33, 28, 36, 42, 46, 48, 48, 46, 42, 36, 45, 52, 57, 60, 61, 60, 57, 52, 45, 55, 63, 69, 73, 75, 75, 73, 69, 63, 55, 66, 75, 82, 87, 90, 91, 90, 87, 82, 75, 66, 78, 88, 96, 102, 106, 108, 108, 106, 102, 96, 88, 78

Offset: 2

Author

N. J. A. Sloane, J. H. Conway and R. K. Guy

Keywords

• nonn
• tabl
• easy

Comments

Closely related to A109439. The current sequence is made of truncated triangular numbers, the latter gives the full description. Both can help to build a cube with layers perpendicular to the great diagonal. E.g.: 15,18,19,18,15 in A008867 is a truncation of the lesser triangular numbers of 1,3,6,10,15,18,19,18,15,10,6,3,1 in A109439. - M. Dauchez (mdzzdm(AT)yahoo.fr), Sep 02 2005

The sequence is a triangle read by rows where the n-th row is obtained by multiplying by (1/3)*(n+1)*(2*(n+1)^2+1) the first row of the limit as k approaches infinity of P(n)^k where P(n) is the stochastic matrix associated with a variant of the Ehrenfest model using n balls. The elements of the stochastic matrix P(n) we have considered are given by P(n)[i,j] = n+1-(max(i,j)-min(i,j)), where each row must be normalized using the L1 norm and where i,j belong to the set {0,1,2,...,n}. They are defined as the probabilities of arriving in a state j given the previous state i. In particular the sum of every row of a stochastic matrix must be 1, and so the sum of the terms of the n-th row of this triangle is (1/3)*(n+1)*(2*(n+1)^2+1) (since the limit of a stochastic matrix is again a stochastic matrix). Furthermore, by the properties of Markov chains, we can interpret P(n)^k as the k-step transition matrix of this variant of the Ehrenfest model using n balls. It is important to note that the rows of the limit of the stochastic matrix are identical and since we know the first we know all the others. - Luca Onnis, Oct 29 2023

Examples

			Triangle begins:
n = 0:  1;
n = 1:  3,  3;
n = 2:  6,  7,  6;
n = 3: 10, 12, 12, 10;
n = 4: 15, 18, 19, 18, 15;
n = 5: 21, 25, 27, 27, 25, 21;
n = 6: 28, 33, 36, 37, 36, 33, 28;
		

References

• Paul and Tatjana Ehrenfest, Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Physikalische Zeitschrift, vol. 8 (1907), pp. 311-314.

Links

• Hugo Pfoertner, Table of n, a(n) for n = 2..1226, rows 1..50, flattened.
• J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
• Luca Onnis, Animation of this variant of the Ehrenfest model using n = 15 balls.
• Wikipedia, Ehrenfest model.

Crossrefs

Row sums are A005900(n-1).

Cf. A109439.

Programs

Maple

T:= (n, k)-> n*(n-3)/2 - k^2+k*n+1:
seq(seq(T(n,k), k=1..n-1), n=2..14);

Mathematica

T[n_,k_] := n*(n-3)/2 - k^2 + k*n + 1; Table[T[n,k], {n,3,20}, {k,n,2,-1}] // Flatten (* Amiram Eldar, Dec 12 2018 *)

Formula

T(n,k) = n*(n-3)/2 - k^2 + k*n + 1.