cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A332373 Main diagonal of A332372.

Original entry on oeis.org

9, 43, 139, 327, 703, 1259, 2183, 3487, 5355, 7723, 11031, 15111, 20535, 26995, 34875, 44195, 55819, 69063, 85203, 103587, 124859, 148599, 176579, 207627, 243323, 282539, 326759, 375287, 430615, 490303, 557779, 631267, 712091, 798939, 894315, 996459, 1109691
Offset: 2

Views

Author

N. J. A. Sloane, Feb 12 2020

Keywords

Crossrefs

Cf. A332372.

A332367 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of triangular cells in the partition for m >= n >= 2.

Original entry on oeis.org

4, 8, 20, 12, 32, 52, 16, 48, 80, 124, 20, 64, 108, 168, 228, 24, 84, 144, 228, 312, 428, 28, 104, 180, 288, 396, 544, 692, 32, 128, 224, 360, 496, 684, 872, 1100, 36, 152, 268, 432, 596, 824, 1052, 1328, 1604, 40, 180, 320, 520, 720, 1000, 1280, 1620, 1960, 2396
Offset: 2

Views

Author

N. J. A. Sloane, Feb 12 2020

Keywords

Examples

			Triangle begins:
4,
8, 20,
12, 32, 52,
16, 48, 80, 124,
20, 64, 108, 168, 228,
24, 84, 144, 228, 312, 428,
28, 104, 180, 288, 396, 544, 692,
32, 128, 224, 360, 496, 684, 872, 1100,
36, 152, 268, 432, 596, 824, 1052, 1328, 1604,
...

Links

Crossrefs

Cf. A332350, A332352, A331781, A332371, A332372, A332374.
For main diagonal see A332368.

Programs

  • Maple
    # Maple code for sequences mentioned in Theorem 12 of Alekseyev et al. (2015).
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;
VS := proc(m,n) local a,i,j; a:=0; # A331781
for i from 1 to m-1 do for j from 1 to n-1 do
if gcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
c3 := (m,n) -> VR(m,n,2)+4; # A332367
for m from 2 to 12 do lprint([seq(c3(m,n),n=2..m)]); od:
[seq(c3(n,n)/4,n=2..40)]; # A332368
c4 := (m,n) -> VR(m,n,1)/2 - VR(m,n,2) - 3; # A332369
for m from 2 to 12 do lprint([seq(c4(m,n),n=2..m)]); od:
[seq(c4(n,n),n=2..40)]; # A332370
ct := (m,n) -> c3(m,n)+c4(m,n); # A332371
for m from 2 to 12 do lprint([seq(ct(m,n),n=2..m)]); od:
[seq(ct(n,n),n=2..40)]; # A114043
et := (m,n) -> VR(m,n,1) - VR(m,n,2)/2 - VS(m,n) - 2; # A332372
for m from 2 to 12 do lprint([seq(et(m,n),n=2..m)]); od:
[seq(et(n,n),n=2..40)]; # A332373
vt := (m,n) ->  et(m,n) - ct(m,n) +1; # A332374
for m from 2 to 12 do lprint([seq(vt(m,n),n=2..m)]); od:
[seq(vt(n,n),n=2..40)]; # A332375

A332371 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of cells in the partition for m >= n >= 2.

Original entry on oeis.org

7, 14, 29, 23, 50, 87, 34, 75, 132, 201, 47, 106, 189, 290, 419, 62, 141, 252, 387, 560, 749, 79, 182, 327, 504, 731, 980, 1283, 98, 227, 410, 633, 920, 1235, 1618, 2041, 119, 278, 503, 778, 1133, 1522, 1995, 2518, 3107, 142, 333, 604, 935, 1362, 1829, 2398, 3027, 3736, 4493
Offset: 2

Views

Author

N. J. A. Sloane, Feb 12 2020

Keywords

Comments

Equals sum of triangles A332367 and A332369.

Examples

			Triangle begins:
7,
14, 29,
23, 50, 87,
34, 75, 132, 201,
47, 106, 189, 290, 419,
62, 141, 252, 387, 560, 749,
79, 182, 327, 504, 731, 980, 1283,
98, 227, 410, 633, 920, 1235, 1618, 2041,
119, 278, 503, 778, 1133, 1522, 1995, 2518, 3107,
...

Links

Crossrefs

Cf. A332350, A332352, A331781, A332367, A332372, A332374.
For main diagonal see A114043.

Programs

A332374 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of vertices in the partition for m >= n >= 2.

Original entry on oeis.org

3, 7, 15, 13, 28, 53, 21, 44, 82, 127, 31, 65, 122, 190, 285, 43, 89, 166, 256, 382, 511, 57, 118, 220, 339, 506, 678, 901, 73, 150, 279, 430, 642, 860, 1142, 1447, 91, 187, 348, 536, 801, 1073, 1424, 1804, 2249, 111, 227, 421, 647, 966, 1290, 1710, 2164, 2696, 3231
Offset: 2

Views

Author

N. J. A. Sloane, Feb 12 2020

Keywords

Comments

T(m,n) = A332372(m,n) - A332371(m,n) + 1 (this is Euler's formula).

Examples

			Triangle begins:
3,
7, 15,
13, 28, 53,
21, 44, 82, 127,
31, 65, 122, 190, 285,
43, 89, 166, 256, 382, 511,
57, 118, 220, 339, 506, 678, 901,
73, 150, 279, 430, 642, 860, 1142, 1447,
91, 187, 348, 536, 801, 1073, 1424, 1804, 2249,
...

Links

Crossrefs

Cf. A332350, A332352, A331781, A332367, A332371, A332372.
For main diagonal see A332375.

Programs

A332369 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of quadrilateral cells in the partition for m >= n >= 2.

Original entry on oeis.org

3, 6, 9, 11, 18, 35, 18, 27, 52, 77, 27, 42, 81, 122, 191, 38, 57, 108, 159, 248, 321, 51, 78, 147, 216, 335, 436, 591, 66, 99, 186, 273, 424, 551, 746, 941, 83, 126, 235, 346, 537, 698, 943, 1190, 1503, 102, 153, 284, 415, 642, 829, 1118, 1407, 1776, 2097, 123, 186, 345, 504, 777, 1002, 1349, 1696, 2139, 2528, 3047
Offset: 2

Views

Author

N. J. A. Sloane, Feb 12 2020

Keywords

Examples

			Triangle begins:
3,
6, 9,
11, 18, 35,
18, 27, 52, 77,
27, 42, 81, 122, 191,
38, 57, 108, 159, 248, 321,
51, 78, 147, 216, 335, 436, 591,
66, 99, 186, 273, 424, 551, 746, 941,
83, 126, 235, 346, 537, 698, 943, 1190, 1503,...

Links

Crossrefs

Cf. A332350, A332352, A331781, A332367, A332371, A332372, A332374.
For main diagonal see A332370.

Programs

Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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