A177720 -id:A177720 - cp's OEIS frontend

A331774 a(n) = Sum_{-n

0, 0, 0, 20, 56, 108, 208, 340, 504, 788, 1136, 1548, 2136, 2820, 3600, 4748, 6056, 7524, 9312, 11292, 13464, 16380, 19584, 23076, 27272, 31820, 36720, 42668, 49064, 55908, 63712, 72028, 80856, 91596, 103008, 115092, 128456, 142556, 157392, 174932, 193400

Offset: 1

N. J. A. Sloane, Feb 08 2020

nonn

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. See p. 158.

This is twice A177720 and 4 times A331775.

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;
[seq(VR(n,n,3),n=1..50)];

A331775 a(n) = Sum_{-n

Original entry on oeis.org

0, 0, 0, 5, 14, 27, 52, 85, 126, 197, 284, 387, 534, 705, 900, 1187, 1514, 1881, 2328, 2823, 3366, 4095, 4896, 5769, 6818, 7955, 9180, 10667, 12266, 13977, 15928, 18007, 20214, 22899, 25752, 28773, 32114, 35639, 39348, 43733, 48350, 53199, 58544

Offset: 1

N. J. A. Sloane, Feb 08 2020

nonn

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. See p. 158.

This is A177720/2 and A331774/4.

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;
[seq(VR(n,n,3)/4,n=1..50)];

cp's OEIS Frontend

A331774 a(n) = Sum_{-n

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