A332351
Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.
0, 1, 6, 2, 13, 28, 3, 22, 49, 86, 4, 33, 74, 131, 200, 5, 46, 105, 188, 289, 418, 6, 61, 140, 251, 386, 559, 748, 7, 78, 181, 326, 503, 730, 979, 1282, 8, 97, 226, 409, 632, 919, 1234, 1617, 2040, 9, 118, 277, 502, 777, 1132, 1521, 1994, 2517, 3106, 10, 141, 332, 603, 934, 1361, 1828, 2397, 3026, 3735, 4492
Offset: 1
Examples
Triangle begins: 0, 1, 6, 2, 13, 28, 3, 22, 49, 86, 4, 33, 74, 131, 200, 5, 46, 105, 188, 289, 418, 6, 61, 140, 251, 386, 559, 748, 7, 78, 181, 326, 503, 730, 979, 1282, 8, 97, 226, 409, 632, 919, 1234, 1617, 2040, 9, 118, 277, 502, 777, 1132, 1521, 1994, 2517, 3106, ...
References
- Jovisa Zunic, Note on the number of two-dimensional threshold functions, SIAM J. Discrete Math. Vol. 25 (2011), No. 3, pp. 1266-1268. See Equation (1.2).
Links
- Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)
- M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. This sequence is f_1(m,n)/2.
Crossrefs
Programs
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Maple
VR := proc(m,n,q) local a,i,j; a:=0; for i from -m+1 to m-1 do for j from -n+1 to n-1 do if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end; for m from 1 to 12 do lprint(seq(VR(m,n,1)/2,n=1..m),); od:
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Mathematica
A332351[m_,n_]:=Sum[If[CoprimeQ[i,j],2(m-i)(n-j),0],{i,m-1},{j,n-1}]+2m*n-m-n;Table[A332351[m,n],{m,15},{n,m}] (* Paolo Xausa, Oct 18 2023 *)
Comments