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A056559 Tetrahedron with T(t,n,k) = t - n; succession of growing finite triangles with declining values per row.

%I A056559 #20 Dec 11 2024 15:41:41
%S A056559 0,1,0,0,2,1,1,0,0,0,3,2,2,1,1,1,0,0,0,0,4,3,3,2,2,2,1,1,1,1,0,0,0,0,
%T A056559 0,5,4,4,3,3,3,2,2,2,2,1,1,1,1,1,0,0,0,0,0,0,6,5,5,4,4,4,3,3,3,3,2,2,
%U A056559 2,2,2,1,1,1,1,1,1,0,0,0,0,0,0,0,7,6,6,5,5,5,4,4,4,4,3,3,3,3,3,2,2,2,2,2,2
%N A056559 Tetrahedron with T(t,n,k) = t - n; succession of growing finite triangles with declining values per row.
%H A056559 Peter Luschny, <a href="/A056559/b056559.txt">Table of n, a(n) for n = 0..10000</a> (first 105 terms by Henry Bottomley).
%F A056559 a(n) = A056556(n) - A056557(n).
%e A056559 First triangle: [0]; second triangle: [1; 0 0]; third triangle: [2; 1 1; 0 0 0]; ...
%o A056559 (Julia)
%o A056559 function a_list(N)
%o A056559     a = Int[]
%o A056559     for n in 1:N
%o A056559         for j in ((k:-1:1) for k in 1:n)
%o A056559             t = n - j[1]
%o A056559             for m in j
%o A056559                 push!(a, t)
%o A056559 end end end; a end
%o A056559 A = a_list(10) # _Peter Luschny_, Feb 19 2020
%o A056559 (Python)
%o A056559 from math import isqrt, comb
%o A056559 from sympy import integer_nthroot
%o A056559 def A056559(n): return (m:=integer_nthroot(6*(n+1),3)[0])-(a:=n<comb(m+2,3))-(k:=isqrt(r:=n+1-comb(m-a+2,3)<<1))+(r<<2<=(k<<2)*(k+1)+1) # _Chai Wah Wu_, Dec 11 2024
%Y A056559 Together with A056558 and A056560 might enable reading "by antidiagonals" of cube arrays as 3-dimensional analog of A002262 and A025581 with square arrays.
%Y A056559 Bisection (y-coordinates) of A332662.
%K A056559 nonn,look
%O A056559 0,5
%A A056559 _Henry Bottomley_, Jun 26 2000