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

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, 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, 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

Offset

0,5

Links

Peter Luschny, Table of n, a(n) for n = 0..10000 (first 105 terms by Henry Bottomley).

Formula

a(n) = A056556(n) - A056557(n).

Example

First triangle: [0]; second triangle: [1; 0 0]; third triangle: [2; 1 1; 0 0 0]; ...

Prog

(Julia)
function a_list(N)
    a = Int[]
    for n in 1:N
        for j in ((k:-1:1) for k in 1:n)
            t = n - j[1]
            for m in j
                push!(a, t)
end end end; a end
A = a_list(10) # Peter Luschny, Feb 19 2020
(Python)
from math import isqrt, comb
from sympy import integer_nthroot
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

Crossrefs

Together with A056558 and A056560 might enable reading "by antidiagonals" of cube arrays as 3-dimensional analog of A002262 and A025581 with square arrays.

Bisection (y-coordinates) of A332662.

Sequence in context: A361509 A143809 A385240 * A117200 A107015 A015374

Adjacent sequences: A056556 A056557 A056558 * A056560 A056561 A056562

Keyword

nonn,look

Author

Henry Bottomley, Jun 26 2000

Status

approved