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A365904 Triangle read by rows T(n,k) = n^2 - binomial(k+1,2), n>=1, k

%I A365904 #28 Oct 23 2023 01:38:21
%S A365904 1,4,3,9,8,6,16,15,13,10,25,24,22,19,15,36,35,33,30,26,21,49,48,46,43,
%T A365904 39,34,28,64,63,61,58,54,49,43,36,81,80,78,75,71,66,60,53,45,100,99,
%U A365904 97,94,90,85,79,72,64,55,121,120,118,115,111,106,100,93,85,76,66
%N A365904 Triangle read by rows T(n,k) = n^2 - binomial(k+1,2), n>=1, k<n.
%C A365904 T(n,k) is the number of points in a rhombus that has 1 point in the first row, then 2 in the second, and following until the n-th row with n points, and then n-1 in the following row, n-2 in the following to end with a row with k+1 points.
%C A365904 T(n,0) are the perfect squares (A000290).
%C A365904 T(n,n-1) are the triangular numbers (A000217).
%H A365904 Joan Llobera Querol, <a href="/A365904/b365904.txt">Table of n, a(n) for n = 1..10000</a>
%H A365904 Zach Wissner-Gross, <a href="https://thefiddler.substack.com/p/can-you-shape-the-peloton">Can You Shape the Peloton?</a>, Fiddler on the Proof, Sep 22, 2023.
%F A365904 G.f.: x*(1 + x - 4*x^2*y + x^3*y^2 + x^4*y^2)/((1 - x)^3*(1 - x*y)^3). - _Stefano Spezia_, Oct 05 2023
%e A365904 Triangle begins:
%e A365904     1;
%e A365904     4,  3;
%e A365904     9,  8,  6;
%e A365904    16, 15, 13, 10;
%e A365904    25, 24, 22, 19, 15;
%e A365904    36, 35, 33, 30, 26, 21;
%e A365904    49, 48, 46, 43, 39, 34, 28;
%e A365904    64, 63, 61, 58, 54, 49, 43, 36;
%e A365904    81, 80, 78, 75, 71, 66, 60, 53, 45;
%e A365904   100, 99, 97, 94, 90, 85, 79, 72, 64, 55;
%e A365904   ...
%Y A365904 Row sums give A004068.
%Y A365904 Cf. A214859.
%K A365904 nonn,tabl
%O A365904 1,2
%A A365904 _Joan Llobera Querol_, Sep 22 2023