A049615 -id:A049615 - cp's OEIS frontend

A049627 Array T read by diagonals; T(i,j)=(i+1)*(j+1)-H(i,j), where H is the array in A049615; thus T(i,j) is the number of lattice points in rectangle having diagonal (0,0)-to-(i,j) that are visible from (i,j).

1, 2, 2, 2, 4, 2, 2, 5, 5, 2, 2, 6, 6, 6, 2, 2, 7, 8, 8, 7, 2, 2, 8, 9, 10, 9, 8, 2, 2, 9, 11, 12, 12, 11, 9, 2, 2, 10, 12, 15, 14, 15, 12, 10, 2, 2, 11, 14, 16, 18, 18, 16, 14, 11, 2, 2, 12, 15, 19, 19, 22, 19, 19, 15, 12, 2, 2, 13, 17, 21, 23, 24, 24, 23, 21

Offset: 0

Clark Kimberling

			Diagonals (each starting on row 0):
  {1};
  {2,2};
  {2,4,2};
  ...
Array begins:
  1  2  2  2  2  2  2
  2  4  5  6  7  8  9
  2  5  6  8  9 11 12
  2  6  8 10 12 15 16
  2  7  9 12 14 18 19
  2  8 11 15 18 22 24
  2  9 12 16 19 24 26

Cf. A049615.

T(n,k) = (n+1)*(k+1) - sum(i=0, n, sum(j=0, k, gcd(i,j)>1));
matrix(7,7,n,k, T(n-1,k-1)) \\ Michel Marcus, Aug 06 2021

A049617 a(n) = Sum_{i=0..2n} (-1)^i * T(i,2n-i) where T is A049615.

Original entry on oeis.org

0, 2, 5, 8, 13, 18, 23, 30, 39, 46, 55, 66, 75, 88, 101, 110, 127, 144, 157, 176, 193, 206, 227, 250, 267, 288, 313, 332, 357, 386, 403, 434, 467, 488, 521, 546, 571, 608, 645, 670, 703, 744, 769, 812, 853, 878, 923, 970, 1003, 1046, 1087

Offset: 0

Clark Kimberling

nonn

A049617 := proc(n)
    add((-1)^i*A049615(i,2*n-i),i=0..2*n) ;
end proc: # R. J. Mathar, Oct 26 2015

A049616 a(n) = Sum_{i=1..n} T(i,n-i), where T is A049615.

Original entry on oeis.org

0, 0, 2, 6, 13, 22, 36, 52, 73, 98, 130, 164, 207, 252, 306, 368, 439, 512, 598, 686, 787, 898, 1022, 1148, 1291, 1440, 1604, 1778, 1969, 2162, 2378, 2596, 2831, 3080, 3348, 3628, 3933, 4240, 4568, 4912, 5281, 5652, 6054, 6458, 6887, 7338, 7814, 8292, 8803

Offset: 0

Clark Kimberling

nonn

More terms from Sean A. Irvine, Aug 05 2021

A049618 a(n) = Sum_{i=0..floor(n/2)} T(2i,n-2i) where T is A049615.

Original entry on oeis.org

0, 0, 2, 3, 9, 11, 22, 26, 43, 49, 74, 82, 115, 126, 168, 184, 239, 256, 322, 343, 421, 449, 544, 574, 683, 720, 846, 889, 1035, 1081, 1244, 1298, 1479, 1540, 1746, 1814, 2045, 2120, 2372, 2456, 2737, 2826, 3130, 3229, 3557, 3669, 4032, 4146, 4535, 4661, 5080

Offset: 0

Clark Kimberling

nonn

More terms from Sean A. Irvine, Aug 05 2021

A049619 a(n) = Sum_{i=0..floor((n+1)/2)} T(2i+1,n-2i-1) where T is A049615.

Original entry on oeis.org

0, 0, 0, 3, 4, 11, 14, 26, 30, 49, 56, 82, 92, 126, 138, 184, 200, 256, 276, 343, 366, 449, 478, 574, 608, 720, 758, 889, 934, 1081, 1134, 1298, 1352, 1540, 1602, 1814, 1888, 2120, 2196, 2456, 2544, 2826, 2924, 3229, 3330, 3669, 3782, 4146, 4268, 4661, 4792

Offset: 0

Clark Kimberling

nonn

More terms from Sean A. Irvine, Aug 05 2021

A049620 a(n) = T(n,n), array T as in A049615.

Original entry on oeis.org

0, 0, 3, 6, 11, 14, 23, 26, 35, 42, 55, 58, 75, 78, 95, 110, 127, 130, 155, 158, 183, 202, 227, 230, 263, 274, 303, 322, 355, 358, 403, 406, 439, 466, 503, 526, 575, 578, 619, 650, 699, 702, 763, 766, 815, 858, 907, 910, 975, 990, 1051, 1090, 1147, 1150, 1223, 1254, 1319, 1362, 1423, 1426, 1515, 1518, 1583

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A049615, A049616, A049617, A049618, A049619, A049621, A049622, A049623.

T[n_, k_]:= Length[Select[Flatten[Table[{x, y}, {x,0,n}, {y,0,k}], 1], GCD @@ # > 1 &]]; Table[T[n, n], {n,0,65}] (* G. C. Greubel, Dec 16 2019 *)

Terms a(38) onward added by G. C. Greubel, Dec 16 2019

A049621 a(n) = T(n,n+1), array T as in A049615.

Original entry on oeis.org

0, 1, 4, 8, 12, 18, 24, 30, 38, 48, 56, 66, 76, 86, 102, 118, 128, 142, 156, 170, 192, 214, 228, 246, 268, 288, 312, 338, 356, 380, 404, 422, 452, 484, 514, 550, 576, 598, 634, 674, 700, 732, 764, 790, 836, 882, 908, 942, 982, 1020, 1070, 1118, 1148, 1186, 1238, 1286, 1340, 1392, 1424, 1470, 1516

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A049615, A049616, A049617, A049618, A049619, A049620, A049622, A049623.

T[n_, k_]:= Length[Select[Flatten[Table[{x, y}, {x,0,n}, {y,0,k}], 1], GCD @@ # > 1 &]]; Table[T[n, n+1], {n,0,65}] (* G. C. Greubel, Dec 16 2019 *)

Terms a(37) onward added by G. C. Greubel, Dec 16 2019

A049622 a(n) = T(n,n+2), array T as in A049615.

Original entry on oeis.org

1, 2, 6, 9, 16, 19, 28, 33, 44, 49, 64, 67, 84, 93, 110, 119, 140, 143, 168, 179, 204, 215, 244, 251, 282, 297, 328, 339, 378, 381, 420, 435, 470, 495, 538, 551, 596, 613, 658, 675, 730, 733, 788, 811, 860, 883, 940, 949, 1012, 1039, 1098, 1119, 1184, 1201, 1270, 1307, 1370, 1393, 1468, 1471, 1548, 1577

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A049615, A049616, A049617, A049618, A049619, A049620, A049621, A049623.

T[n_, k_]:= Length[Select[Flatten[Table[{x, y}, {x,0,n}, {y,0,k}], 1], GCD @@ # > 1 &]]; Table[T[n, n+2], {n,0,65}] (* G. C. Greubel, Dec 16 2019 *)

Terms a(37) onward added by G. C. Greubel, Dec 16 2019

A049623 a(n) = T(n,n+3), array T as in A049615.

Original entry on oeis.org

2, 3, 7, 12, 17, 22, 31, 38, 45, 56, 65, 74, 91, 100, 111, 130, 141, 154, 177, 190, 205, 230, 249, 264, 291, 312, 329, 360, 379, 396, 433, 452, 481, 518, 539, 570, 611, 636, 659, 704, 731, 756, 809, 834, 861, 914, 947, 978, 1031, 1066, 1099, 1154, 1199, 1232, 1291, 1336, 1371, 1436, 1469, 1502, 1575

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A049615, A049616, A049617, A049618, A049619, A049620, A049621, A049622.

T[n_, k_]:= Length[Select[Flatten[Table[{x, y}, {x,0,n}, {y,0,k}], 1], GCD @@ # > 1 &]]; Table[T[n, n+3], {n,0,65}] (* G. C. Greubel, Dec 16 2019 *)

Terms a(37) onward added by G. C. Greubel, Dec 16 2019

A049626 a(n) = T(n,4), array T as in A049615.

Original entry on oeis.org

3, 3, 6, 8, 11, 12, 16, 17, 20, 22, 25, 26, 30, 31, 34, 36, 39, 40, 44, 45, 48, 50, 53, 54, 58, 59, 62, 64, 67, 68, 72, 73, 76, 78, 81, 82, 86, 87, 90, 92, 95, 96, 100, 101, 104, 106, 109, 110, 114, 115, 118, 120, 123, 124, 128, 129, 132, 134, 137, 138, 142

Offset: 0

Clark Kimberling

nonn

More terms from Sean A. Irvine, Aug 05 2021

cp's OEIS Frontend

A049627 Array T read by diagonals; T(i,j)=(i+1)*(j+1)-H(i,j), where H is the array in A049615; thus T(i,j) is the number of lattice points in rectangle having diagonal (0,0)-to-(i,j) that are visible from (i,j).

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PARI

A049617 a(n) = Sum_{i=0..2n} (-1)^i * T(i,2n-i) where T is A049615.

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A049616 a(n) = Sum_{i=1..n} T(i,n-i), where T is A049615.

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A049618 a(n) = Sum_{i=0..floor(n/2)} T(2i,n-2i) where T is A049615.

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A049619 a(n) = Sum_{i=0..floor((n+1)/2)} T(2i+1,n-2i-1) where T is A049615.

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A049620 a(n) = T(n,n), array T as in A049615.

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Mathematica

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A049621 a(n) = T(n,n+1), array T as in A049615.

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Mathematica

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A049622 a(n) = T(n,n+2), array T as in A049615.

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Mathematica

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A049623 a(n) = T(n,n+3), array T as in A049615.

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Mathematica

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A049626 a(n) = T(n,4), array T as in A049615.

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