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 1-4 of 4 results.

A194442 Number of toothpicks and D-toothpicks after n-th stage in the D-toothpick "narrow" triangle of the second kind.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 19, 26, 34, 38, 42, 50, 62, 70, 78, 91, 107, 111, 115, 123, 135, 151, 167, 187, 211, 223, 231, 247, 275, 291, 307, 332, 364, 368, 372, 380, 392, 408, 424, 446, 478, 504, 524, 548, 588, 620, 660, 693, 741, 761, 769, 785, 813, 853, 897, 947
Offset: 0

Views

Author

Omar E. Pol, Aug 29 2011

Keywords

Comments

If n = 2^k, k >= 1, then the structure looks like an isosceles triangle. For the D-toothpick "wide" triangle of the second kind see A194440.
The structure is essentially one of the wedges of several D-toothpick structures. For more information see A194270. The first differences (A194443) give the number of toothpicks or D-toothpicks added at n-th stage. - Omar E. Pol, Mar 28 2013

Links

Crossrefs

Cf. A047999, A139250, A160120, A160406, A161830, A194270, A194277, A194440, A194443, A194444, A194694, A194695, A194697, A220496, A220522.

A194443 Number of toothpicks or D-toothpicks added at n-th stage to the structure of A194442.

Original entry on oeis.org

0, 1, 2, 4, 4, 4, 4, 7, 8, 4, 4, 8, 12, 8, 8, 13, 16, 4, 4, 8, 12, 16, 16, 20, 24, 12, 8, 16, 28, 16, 16, 25, 32, 4, 4, 8, 12, 16, 16, 22, 32, 26, 20, 24, 40, 32, 40, 33, 48, 20, 8, 16, 28, 40, 44, 50, 60, 28, 16, 32, 60, 32, 32, 49, 64, 4, 4, 8
Offset: 0

Views

Author

Omar E. Pol, Aug 29 2011

Keywords

Comments

Essentially the first differences of A194442. It appears that the structure of the "narrow" triangle is much more regular about n=2^k, see formula section.

Examples

			If written as a triangle:
0,
1,
2,
4,4,
4,4,7,8,
4,4,8,12,8,8,13,16,
4,4,8,12,16,16,20,24,12,8,16,28,16,16,25,32,
4,4,8,12,16,16,22,32,26,20,24,40,32,40,33,48,20,8,16,28...
.
It appears that rows converge to A194697.

Links

Crossrefs

Cf. A139251, A160121, A160407, A161831, A194271, A194441, A194442, A194445, A194694, A194695, A194697.

Formula

Conjectures for n = 2^k+j, if -6<=j<=6:
a(2^k-6) = 2^(k-2), if k >= 3.
a(2^k-5) = 2^(k-1), if k >= 3.
a(2^k-4) = 2^k-4, if k >= 2.
a(2^k-3) = 2^(k-1), if k >= 3.
a(2^k-2) = 2^(k-1), if k >= 2.
a(2^k-1) = 3*2^(k-2)+1, if k >= 2.
a(2^k+0) = 2^k, if k >= 0.
a(2^k+1) = 4, if k >= 1.
a(2^k+2) = 4, if k >= 1.
a(2^k+3) = 8, if k >= 3.
a(2^k+4) = 12, if k >= 3.
a(2^k+5) = 16, if k >= 4.
a(2^k+6) = 16, if k >= 4.
End of conjectures.

A194695 Number of toothpicks or D-toothpicks added at n-th stage to the D-toothpick "corner" structure of A194694.

Original entry on oeis.org

2, 2, 4, 6, 8, 8, 11, 16, 13, 10, 12, 20, 22, 26, 24, 36, 21, 10, 12
Offset: 1

Views

Author

Omar E. Pol, Sep 03 2011

Keywords

Comments

Sequence related to the D-toothpick "narrow" triangle (See A194442 and A194443). First differences of A194694.

Examples

			Written as a triangle:
2,
2,
4,
6,8,
8,11,16,13,
10,12,20,22,26,24,36,21,
10,12

Links

Crossrefs

Cf. A152980, A194271, A194442, A194443, A194693, A194694, A194697

Formula

a(n) = A194697(n)/2.

A194696 a(n) = 2*A194693(n).

Original entry on oeis.org

4, 4, 8, 12, 16, 16, 24, 26, 24, 20, 32, 40, 64, 40, 48, 54, 40, 20, 32, 48, 72, 82, 96, 96, 108, 68, 88, 100, 160, 96, 96, 110, 72, 20, 32, 48, 72, 82, 96, 108, 136, 124, 160, 160, 248, 190, 208, 178, 188, 88, 96, 136
Offset: 1

Views

Author

Omar E. Pol, Sep 03 2011

Keywords

Comments

Conjecture: number of toothpicks or D-toothpicks added to the structure of A194440 at stage 2^k+n, if k tends to infinity. It appears that rows of A194441 when written as a triangle converge to this sequence.

Examples

			Written as a triangle:
4,
4,
8,
12,16,
16,24,26,24,
20,32,40,64,40,48,54,40,
20,32,48,72,82,96,96,108,68,88,100,160,96,96,110,72,
20,32,48,72,82,96,108,136,124,160,160,248,190,208,178,...

Links

Crossrefs

Cf. A152980, A194270, A194440, A194692, A194693, A194697.
Showing 1-4 of 4 results.

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

Content is available under The OEIS End-User License Agreement.