A295707
Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of lines through at least 2 points of an n X k grid of points.
Original entry on oeis.org
0, 1, 1, 1, 6, 1, 1, 11, 11, 1, 1, 18, 20, 18, 1, 1, 27, 35, 35, 27, 1, 1, 38, 52, 62, 52, 38, 1, 1, 51, 75, 93, 93, 75, 51, 1, 1, 66, 100, 136, 140, 136, 100, 66, 1, 1, 83, 131, 181, 207, 207, 181, 131, 83, 1, 1, 102, 164, 238, 274, 306, 274, 238, 164, 102, 1
Offset: 1
Square array begins:
0, 1, 1, 1, 1, ...
1, 6, 11, 18, 27, ...
1, 11, 20, 35, 52, ...
1, 18, 35, 62, 93, ...
1, 27, 52, 93, 140, ...
1, 38, 75, 136, 207, ...
Main diagonal gives
A018808. Reading up to the diagonal gives
A107348.
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A[n_, k_] := (1/2)(f[n, k, 1] - f[n, k, 2]);
f[n_, k_, m_] := Sum[If[GCD[mx/m, my/m] == 1, (n - Abs[mx])(k - Abs[my]), 0], {mx, -n, n}, {my, -k, k}];
Table[A[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 04 2023 *)
A107348
Triangle read by rows: T(m,n) = number of different lines in a rectangular m X n array of points with integer coordinates (x,y): 0 <= x <= m, 0 <= y <= n.
Original entry on oeis.org
0, 1, 6, 1, 11, 20, 1, 18, 35, 62, 1, 27, 52, 93, 140, 1, 38, 75, 136, 207, 306, 1, 51, 100, 181, 274, 405, 536, 1, 66, 131, 238, 361, 534, 709, 938, 1, 83, 164, 299, 454, 673, 894, 1183, 1492, 1, 102, 203, 370, 563, 836, 1111, 1470, 1855, 2306
Offset: 0
Triangle begins
0,
1, 6,
1, 11, 20,
1, 18, 35, 62,
1, 27, 52, 93, 140,
1, 38, 75, 136, 207, 306,
1, 51, 100, 181, 274, 405, 536,
1, 66, 131, 238, 361, 534, 709, 938,
1, 83, 164, 299, 454, 673, 894, 1183, 1492,
1, 102, 203, 370, 563, 836, 1111, 1470, 1855, 2306,
...
- M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh, On the minimal teaching sets of two-dimensional threshold functions, SIAM J. Disc. Math. 29(1), 2015, pp. 157-165.
- Les Reid, Problem #7: How Many Lines Does the Lattice of Points Generate?, Problems from the 04-05 academic year, Challenge Archive, Missouri State University's Problem Corner.
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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;
LL:=(m,n)->(VR(m,n,1)-VR(m,n,2))/2;
for m from 1 to 12 do lprint([seq(LL(m,n),n=1..m)]); od: # N. J. A. Sloane, Feb 10 2020
-
VR[m_, n_, q_] := Sum[If[GCD[i, j] == q, (m - Abs[i])(n - Abs[j]), 0], {i, -m + 1, m - 1}, {j, -n + 1, n - 1}];
LL[m_, n_] := (1/2)(VR[m, n, 1] - VR[m, n, 2]);
Table[LL[m, n], {m, 1, 10}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jun 04 2023, after N. J. A. Sloane *)
T(3,3) corrected and sequence extended by
R. J. Mathar, Dec 17 2017
A197985
a(n) = round((n+1/n)^2).
Original entry on oeis.org
4, 6, 11, 18, 27, 38, 51, 66, 83, 102, 123, 146, 171, 198, 227, 258, 291, 326, 363, 402, 443, 486, 531, 578, 627, 678, 731, 786, 843, 902, 963, 1026, 1091, 1158, 1227, 1298, 1371, 1446, 1523, 1602, 1683, 1766, 1851, 1938, 2027
Offset: 1
-
[Round((n+1/n)^2): n in [1..60]];
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Table[Floor[(n+1/n)^2+1/2],{n,50}] (* Harvey P. Dale, Aug 12 2012 *)
Join[{4}, 2+Range[2,50]^2] (* G. C. Greubel, Feb 04 2024 *)
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[4]+[n^2+2 for n in range(2,51)] # G. C. Greubel, Feb 04 2024
A010015
a(0) = 1, a(n) = 25*n^2 + 2 for n > 0.
Original entry on oeis.org
1, 27, 102, 227, 402, 627, 902, 1227, 1602, 2027, 2502, 3027, 3602, 4227, 4902, 5627, 6402, 7227, 8102, 9027, 10002, 11027, 12102, 13227, 14402, 15627, 16902, 18227, 19602, 21027, 22502, 24027, 25602, 27227, 28902, 30627, 32402, 34227, 36102, 38027, 40002
Offset: 0
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Join[{1}, 25 Range[40]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
Join[{1}, LinearRecurrence[{3, -3, 1}, {27, 102, 227}, 50]] (* Vincenzo Librandi, Feb 08 2012 *)
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A010015(n)=25*n^2+2-!n \\ M. F. Hasler, Feb 14 2012
Showing 1-4 of 4 results.
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