The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A332599 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A332599

Triangle read by rows: T(n,k) = number of vertices in a "frame" of size n X k (see Comments in A331457 for definition).
(history; published version)

#25 by N. J. A. Sloane at Fri Mar 13 12:40:46 EDT 2020

STATUS

editing

approved

#24 by N. J. A. Sloane at Fri Mar 13 12:40:43 EDT 2020

FORMULA

Column 1 is A331755, for which there is an explicit formula.

Column 2 is A331763, for which no formula is known.

For m >= n >= 3, T(m,n) = A332600(m,n) - A331457(m,n) (Euler for genus 1 graph), and both A332600 and A331457 have explicit formulas.

EXAMPLE

5;

[5],

[13, 37;],

[35, 99, 152;],

[75, 213, 256, 364;],

[159, 401, ...448, 568, 776],

...

[275, 657, 704, 836, 1056, 1340],

[477, 1085, 1132, 1276, 1508, 1804, 2272],

[755, 1619, 1712, 1868, 2112, 2420, 2900, 3532],

[1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336],

[1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516],

...

CROSSREFS

Cf. A331457, A331755, A331763, A331776, A332599, A332600.

STATUS

approved

editing

#23 by N. J. A. Sloane at Fri Mar 13 12:36:45 EDT 2020

STATUS

editing

approved

#22 by N. J. A. Sloane at Fri Mar 13 12:36:43 EDT 2020

COMMENTS

There is a crucial difference between frames of size nX2 and size nXk with k = 1 or k >= 3. If k != 2, all regions are either triangles or quadrilaterals, but for k=2 regions with larger numbers of sides can appear. Remember also that for k <= 2, the "frame" has no hole, and the graph has genus 0, whereas for k >= 3 there is a nontrivial hole and the graph has genus 1.

STATUS

approved

editing

#21 by N. J. A. Sloane at Fri Mar 13 12:35:36 EDT 2020

STATUS

editing

approved

#20 by N. J. A. Sloane at Fri Mar 13 12:35:34 EDT 2020

DATA

5, 13, 37, 35, 99, 152, 75, 213, 256, 364, 159, 401, 448, 568, 776, 275, 657, 704, 836, 1056, 1340, 477, 1085, 1132, 1276, 1508, 1804, 2272, 755, 1619, 1712, 1868, 2112, 2420, 2900, 3532, 1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336, 1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516

KEYWORD

nonn,tabl,more,new

EXTENSIONS

More terms from N. J. A. Sloane, Mar 13 2020

STATUS

approved

editing

#19 by Bruno Berselli at Wed Mar 04 03:23:40 EST 2020

STATUS

reviewed

approved

#18 by Joerg Arndt at Wed Mar 04 02:20:40 EST 2020

STATUS

proposed

reviewed

#17 by Jon E. Schoenfield at Wed Mar 04 00:19:17 EST 2020

STATUS

editing

proposed

#16 by Jon E. Schoenfield at Wed Mar 04 00:19:15 EST 2020

EXAMPLE

5,;

13, 37,;

35, 99, 152,;

75, 213, 256, 364,;

159, 401, ...

STATUS

proposed

editing