A141727 -id:A141727 - cp's OEIS frontend

A141729 Numbers of 1's in the rows of triangle A141727.

1, 2, 3, 3, 6, 4, 5, 8, 8, 8, 8, 10, 9, 12, 12, 15, 18, 14, 16, 16, 16, 16, 18, 16, 17, 26, 20, 27, 24, 24, 25, 32, 32, 30, 30, 36, 28, 32, 36, 34, 26, 32, 28, 34, 30, 36, 36, 42, 33, 46, 42, 41, 38, 44, 41, 50, 42, 56, 44, 58, 45, 60, 52, 63, 66, 58, 62, 62, 52, 56, 64, 64, 46

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 02 2008

Cf. A141727, A141728, A141730 - A141746.

A141730 Numbers of 0's in the rows of triangle A141727.

Original entry on oeis.org

0, 1, 3, 4, 3, 7, 8, 7, 9, 11, 13, 13, 16, 15, 17, 16, 15, 21, 21, 23, 25, 27, 27, 31, 32, 25, 33, 28, 33, 35, 36, 31, 33, 37, 39, 35, 45, 43, 41, 45, 55, 51, 57, 53, 59, 55, 57, 53, 64, 53, 59, 62, 67, 63, 68, 61, 71, 59, 73, 61, 76, 63, 73, 64, 63, 73, 71, 73, 85, 83, 77, 79

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 02 2008

Cf. A141727 - A141729, A141731 - A141746.

A141733 Binary digits, representing the rows of triangle A141727, written in base 10.

Original entry on oeis.org

1, 5, 18, 84, 311, 1286, 4636, 21607, 79398, 328540, 1183512, 5518960, 20382304, 84281919, 303834326, 1416057916, 5203506983, 21531002534, 77561732700, 361685609752, 1335790797424, 5523583535712, 19912388519360, 92801359368576

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 02 2008

Cf. A141727 - A141732, A141734 - A141746.

A141735 List of the 0's and 1's digits of triangle A141727 along a boustrophedon path. First case: see example below.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 02 2008

			First boustrophedon path:
................................/1.
.............................../_____
...............................1.0.1.\
............................._________\
.........................../.1.0.0.1.0.
.........................../______________
...........................1.0.1.0.1.0.0.\
.........................__________________\
......................./.1.0.0.1.1.0.1.1.1
......................./_____________________
.......................1.0.1.0.0.0.0.0.1.1.0

Cf. A141727 - A141734, A141736 - A141746.

A141736 List of the 0's and 1's digits of triangle A141727 along a boustrophedon path. Second case: see example below.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 02 2008

			Second boustrophedon path:
.................................1\
...............................____\
............................../1.0.1.
............................./________
.............................1.0.0.1.0\
............................___________\
........................../1.0.1.0.1.0.0
........................./________________
.........................1.0.0.1.1.0.1.1.1\
........................___________________\
.......................1.0.1.0.0.0.0.0.1.1.0

Cf. A141727 - A141735, A141737 - A141746.

A141739 Cumulative sums of the rows of triangle A141727.

Original entry on oeis.org

1, 3, 5, 8, 14, 18, 23, 31, 39, 47, 55, 65, 74, 86, 98, 113, 131, 145, 161, 177, 193, 209, 227, 243, 260

Offset: 0

Paolo P. Lava, Jul 03 2008

Cf. A141727 - A141738, A141740 - A141746.

A141741 Starting from the 1 in the first line of triangle A141727 choose one of the three digits below it. Repeat down to the other rows. Sequence gives the numbers in base 10 expressed by the collected digits that cannot be reached following any possible path.

Original entry on oeis.org

35, 41, 45, 70, 71, 78, 82, 83, 90, 91, 94, 110

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jul 07 2008

			35 in binary is 100011. We do not have any path starting from the 1 in the first row that leads to this sequence of digits.

Cf. A141727 - A141740, A141742 - A141746.

A141746 Binary digits, representing the rows of triangle A141743, written in base 10.

Original entry on oeis.org

0, 5, 7, 76, 18, 1281, 1872, 19850, 5518, 334438, 470902, 4989654, 1187177, 83965631, 122705027, 1300996004, 361349378, 21919223969, 30854252783, 327011301795, 77754211611

Offset: 0

Paolo P. Lava & Giorgio Balzarotti, Jul 07 2008

Cf. A141727 - A141745.

A141728 Triangle read by rows T(n,k). Triangle elements are 0 and 1. Starting with 1 in the top add below a second row of (2n-1) elements (with n=2 -> 3). Moving from left to right add 1 if the number of adjacent 1's is even or add 0 if it is odd. See example below.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jul 02 2008

Any diagonal, read top down from right to left, expresses a periodic sequence of 0'0's and 1's Lengths of the periods are alway powers of 2. Here below the periods for the first 20 diagonals:
10
0
0110
0110
1000
0
01011010
00011110
11011000
11110000
11001010
01100000
01000110
0110
1011011101001000
0111111110000000
0000111101011010
1110000100011110
0100000111011000
1001011100001110

			.....................................1 First Row
..................................0 ... Add 0 to have an odd number of adjacent 1's
.....................................1 First Row
...................................0.0 ... Add again 0 to have an odd number of adjacent 1's
......................................1 First Row
...................................0.0.0 ... Again add 0 to have an odd number of adjacent 1's
The second row is now complete.
.....................................1 First Row
...................................0.0.0 Second Row
.................................1 ... Add 1 because there are no adjacent 1's
.....................................1 First Row
...................................0.0.0 Second Row
.................................1.0 ... Add 0 because there is one adjacent 1 (third row)
.....................................1 First Row
...................................0.0.0 Second Row
.................................1.0.1 ... Add 1 because there is no adjacent 1
.....................................1 First Row
...................................0.0.0 Second Row
.................................1.0.1.0 ... Add 0 because there is only an 1 adjacent (third row)
.....................................1 First Row
...................................0.0.0 Second Row
.................................1.0.1.0.1 ... Add 1 because there is no adjacent 1
The third row is now complete. Then repeat the process for the other rows.

Paolo P. Lava, Picture of Triangle A141728

Cf. A141727, A141729-A141746.

A141731 Numbers of 1's in the rows of triangle A141728.

Original entry on oeis.org

1, 0, 3, 2, 4, 3, 6, 9, 7, 7, 9, 8, 11, 8, 13, 16, 18, 14, 14, 14, 22, 20, 20, 25, 22, 23, 24, 17, 29, 22, 29, 36, 38, 34, 32, 32, 38, 36, 34, 36, 36, 40, 44, 44, 34, 30, 42, 45, 50, 45, 38, 43, 49, 50, 55, 56, 58, 62, 56, 49, 58, 51, 62, 71, 73, 65, 65, 63, 69, 69, 69, 67, 67, 75

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 02 2008

Cf. A141727 - A141730, A141732 - A141746.

cp's OEIS Frontend

A141729 Numbers of 1's in the rows of triangle A141727.

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A141730 Numbers of 0's in the rows of triangle A141727.

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A141733 Binary digits, representing the rows of triangle A141727, written in base 10.

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A141735 List of the 0's and 1's digits of triangle A141727 along a boustrophedon path. First case: see example below.

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A141736 List of the 0's and 1's digits of triangle A141727 along a boustrophedon path. Second case: see example below.

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A141739 Cumulative sums of the rows of triangle A141727.

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A141741 Starting from the 1 in the first line of triangle A141727 choose one of the three digits below it. Repeat down to the other rows. Sequence gives the numbers in base 10 expressed by the collected digits that cannot be reached following any possible path.

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A141746 Binary digits, representing the rows of triangle A141743, written in base 10.

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A141728 Triangle read by rows T(n,k). Triangle elements are 0 and 1. Starting with 1 in the top add below a second row of (2n-1) elements (with n=2 -> 3). Moving from left to right add 1 if the number of adjacent 1's is even or add 0 if it is odd. See example below.

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A141731 Numbers of 1's in the rows of triangle A141728.

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