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 1-6 of 6 results.

A335179 a(n) = A115004(n) - A334701(n).

Original entry on oeis.org

0, 2, 7, 26, 55, 118, 203, 350, 513, 802, 1153, 1594, 2105, 2878, 3701, 4770, 5973, 7510, 9221, 11282, 13411, 16202, 19255, 22674, 26371, 31018, 35865, 41422, 47377, 54114, 61297, 69470, 77803, 87758, 98311, 109766, 121851, 136006, 150443, 166278, 182903, 201110, 220179, 241030, 262447, 286886, 312411, 339422, 367551
Offset: 1

Views

Author

N. J. A. Sloane, Jun 22 2020

Keywords

Comments

It would be nice to have a formula or recurrence.

Crossrefs

Cf. A115004, A334701.

A335677 a(n) = A159065(n+1) - A334701(n).

Original entry on oeis.org

0, 1, 3, 11, 23, 47, 79, 139, 193, 303, 437, 595, 777, 1079, 1361, 1747, 2181, 2743, 3369, 4147, 4867, 5883, 6995, 8203, 9507, 11243, 12949, 14955, 17105, 19527, 22109, 25123, 27979, 31563, 35363, 39455, 43771, 49031, 54095, 59743, 65671, 72143, 78919, 86459, 93967, 102843, 112119, 121795, 131871
Offset: 1

Views

Author

N. J. A. Sloane, Jun 22 2020

Keywords

Comments

It would be nice to have a formula or recurrence.

Crossrefs

Cf. A159065, A334701.

A335682 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = number of simple interior vertices in figure formed by taking m equally spaced points on a line and n equally spaced points on a parallel line, and joining each of the m points to each of the n points by a line segment.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 3, 3, 0, 0, 6, 6, 6, 0, 0, 10, 12, 12, 10, 0, 0, 15, 18, 24, 18, 15, 0, 0, 21, 27, 36, 36, 27, 21, 0, 0, 28, 36, 54, 54, 54, 36, 28, 0, 0, 36, 48, 72, 82, 82, 72, 48, 36, 0, 0, 45, 60, 96, 108, 124, 108, 96, 60, 45, 0, 0, 55, 75, 120, 144, 163, 163, 144, 120, 75, 55, 0
Offset: 1

Views

Author

Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Jun 28 2020

Keywords

Comments

A simple interior vertex is a vertex where exactly two lines cross. In graph theory terms, this is an interior vertex of degree 4.
The case m=n (the main diagonal) is dealt with in A334701. A306302 has illustrations for the diagonal case for m = 1 to 15.
Also A335678 has colored illustrations for many values of m and n.
This is the only one of the five arrays (A335678-A335682) that does not have an explicit formula.
Let G_m(x) = g.f. for row m. For m <= 9, G_m appears to be a rational function of x with denominator D_m(x), where (writing C_k for the k-th cyclotomic polynomial):
D_3 = D_4 = C_1^3*C_2
D_5 = C_1^3*C_2*C_4
D_6 = C_1^3*C_2*C_4*C_5
D_7 = C_1^3*C_2*C_3*C_4*C_5*C_6
D_8 = D_9 = C_1^3*C_2*C_3*C_4*C_5*C_6*C_7

Examples

			The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
0, 3, 6, 12, 18, 27, 36, 48, 60, 75, 90, 108, ...
0, 6, 12, 24, 36, 54, 72, 96, 120, 150, 180, 216, ...
0, 10, 18, 36, 54, 82, 108, 144, 180, 226, 270, 324, ...
0, 15, 27, 54, 82, 124, 163, 217, 272, 342, 408, 489, ...
0, 21, 36, 72, 108, 163, 214, 286, 358, 451, 536, 642, ...
0, 28, 48, 96, 144, 217, 286, 382, 478, 602, 715, 856, ...
0, 36, 60, 120, 180, 272, 358, 478, 598, 754, 894, 1070, ...
0, 45, 75, 150, 226, 342, 451, 602, 754, 950, 1126, 1347, ...
0, 55, 90, 180, 270, 408, 536, 715, 894, 1126, 1334, 1597, ...
0, 66, 108, 216, 324, 489, 642, 856, 1070, 1347, 1597, 1912, ...
...
The initial antidiagonals are:
0
0, 0
0, 1, 0
0, 3, 3, 0
0, 6, 6, 6, 0
0, 10, 12, 12, 10, 0
0, 15, 18, 24, 18, 15, 0
0, 21, 27, 36, 36, 27, 21, 0
0, 28, 36, 54, 54, 54, 36, 28, 0
0, 36, 48, 72, 82, 82, 72, 48, 36, 0
0, 45, 60, 96, 108, 124, 108, 96, 60, 45, 0
0, 55, 75, 120, 144, 163, 163, 144, 120, 75, 55, 0
...

Links

Crossrefs

This is one of a set of five arrays: A335678, A335679, A335680, A335681, A335682.
For the diagonal case see A306302, A331755, A334701.

A336489 Consider the figure made up of a row of n adjacent congruent rectangles, with diagonals of all possible rectangles drawn; a(n) = number of interior vertices where exactly three lines cross.

Original entry on oeis.org

0, 1, 2, 8, 18, 32, 50, 102, 126, 198, 286, 390, 510, 742, 878, 1130, 1414, 1770, 2166, 2778, 3186, 3858, 4594, 5342, 6146, 7342, 8294, 9662, 11134, 12710, 14390, 16582, 18226, 20518, 22946, 25662, 28530, 32270, 35250, 38862, 42650, 46614, 50754, 55990, 60730
Offset: 1

Views

Author

Lars Blomberg, Jul 25 2020

Keywords

Links

Crossrefs

Cf. A334701, A336490, A336491.

A336490 Consider the figure made up of a row of n adjacent congruent rectangles, with diagonals of all possible rectangles drawn; a(n) is the number of interior vertices where exactly four lines cross.

Original entry on oeis.org

0, 0, 1, 2, 2, 10, 22, 18, 32, 62, 100, 118, 136, 154, 274, 346, 426, 554, 698, 726, 884, 1058, 1248, 1454, 1784, 2146, 2714, 2978, 3250, 3722, 4226, 4490, 5104, 5758, 6674, 7350, 8058, 8798, 10062, 11238, 12478, 14046, 15702, 16526, 18016, 19570, 21188, 22870
Offset: 1

Views

Author

Lars Blomberg, Jul 25 2020

Keywords

Links

Crossrefs

Cf. A334701, A336489, A336491.

A336491 Consider the figure made up of a row of n adjacent congruent rectangles, with diagonals of all possible rectangles drawn; a(n) = number of interior vertices where exactly five lines cross.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 2, 12, 26, 20, 10, 38, 74, 118, 102, 114, 126, 178, 238, 306, 382, 466, 558, 762, 774, 778, 774, 942, 1126, 1326, 1542, 1910, 2310, 2742, 2762, 3070, 3394, 3734, 4090, 4294, 4498, 4966, 5458, 6434, 6910, 7402, 7910, 8834, 9806, 10410, 11030
Offset: 1

Views

Author

Lars Blomberg, Jul 25 2020

Keywords

Links

Crossrefs

Cf. A334701, A336489, A336490.
Showing 1-6 of 6 results.

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

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