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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.

%I A008867 #61 Dec 15 2023 23:55:32
%S A008867 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,
%T A008867 36,33,28,36,42,46,48,48,46,42,36,45,52,57,60,61,60,57,52,45,55,63,69,
%U A008867 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
%N 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.
%C A008867 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
%C A008867 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
%D A008867 Paul and Tatjana Ehrenfest, Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem, Physikalische Zeitschrift, vol. 8 (1907), pp. 311-314.
%H A008867 Hugo Pfoertner, <a href="/A008867/b008867.txt">Table of n, a(n) for n = 2..1226</a>, rows 1..50, flattened.
%H A008867 J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (<a href="http://neilsloane.com/doc/Me220.pdf">pdf</a>).
%H A008867 Luca Onnis, <a href="/A008867/a008867.gif">Animation of this variant of the Ehrenfest model using n = 15 balls</a>.
%H A008867 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ehrenfest_model">Ehrenfest model</a>.
%F A008867 T(n,k) = n*(n-3)/2 - k^2 + k*n + 1.
%e A008867 Triangle begins:
%e A008867 n = 0:  1;
%e A008867 n = 1:  3,  3;
%e A008867 n = 2:  6,  7,  6;
%e A008867 n = 3: 10, 12, 12, 10;
%e A008867 n = 4: 15, 18, 19, 18, 15;
%e A008867 n = 5: 21, 25, 27, 27, 25, 21;
%e A008867 n = 6: 28, 33, 36, 37, 36, 33, 28;
%p A008867 T:= (n, k)-> n*(n-3)/2 - k^2+k*n+1:
%p A008867 seq(seq(T(n,k), k=1..n-1), n=2..14);
%t A008867 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 *)
%Y A008867 Row sums are A005900(n-1).
%Y A008867 Cf. A109439.
%K A008867 nonn,tabl,easy
%O A008867 2,2
%A A008867 _N. J. A. Sloane_, _J. H. Conway_ and _R. K. Guy_