A141746 -id:A141746 - cp's OEIS frontend

A141727 Triangle T(n,k) read by rows. Entries are 0 and 1. Start with 1 in the top row, add a second row of 2n-1 elements (with n=2 -> 3). Moving from left to right add 0 if the number of adjacent 1's is even or add 1 if it is odd.

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

Offset: 0

Paolo P. Lava & Giorgio Balzarotti, Jul 02 2008

Any diagonal, read top down from right to left, is a periodic sequence of 0's and 1's. The lengths of the periods are always powers of 2. Here are the periods for the first 20 diagonals:
1
0
10
10
0110
0
0100
1000
11110000
1110
01001110
00101000
01011100
1000
11100000
11001110
0111000110001110
01101000
0011011010011100
0010001010001000
If we draw a large number of rows we obtain an interesting figure with several large islands of zeros.

.....................................1 First Row
...................................1 ... Add 1 to have an even number of adjacent 1's (2)
.....................................1 First Row
...................................1.0 ... Add 0 because there are two adjacent 1's (in the first and second rows)
......................................1 First Row
....................................1.0.1 ... Again add 1 to have an even number of adjacent 1's (2)
The second row is now complete.
.....................................1 First Row
...................................1.0.1 Second Row
.................................1 ... Add 1 because there is only an 1 adjacent (second row)
.....................................1 First Row
...................................1.0.1 Second Row
.................................1.0 ... Add 0 because there are two 1's adjacent (second and third row)
.....................................1 First Row
...................................1.0.1 Second Row
.................................1.0.0 ... Again add 0 because there are two 1's adjacent (second row)
.....................................1 First Row
...................................1.0.1 Second Row
.................................1.0.0.1 ... Add 1 because there is only an 1 adjacent (second row)
.....................................1 First Row
...................................1.0.1 Second Row
.................................1.0.0.1.0 ... Add 0 because there are two 1's adjacent (second and third row)
The third row is now complete. Then repeat the process for the other rows.
The triangle begins:
...........................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
.........1..0..0..1..0..0..0..0..1..1..1..0..0
......1..0..1..0..1..0..0..0..1..1..0..0..1..1..1
...1..0..0..1..1..0..1..1..0..0..0..1..0..0..1..1..0
1..0..1..0..0..0..0..0..0..1..1..0..1..0..1..1..1..0..0

Paolo P. Lava, Picture of Triangle A141727

Cf. A141728-A141746.

Minor edits by N. J. A. Sloane, Sep 10 2012

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.

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

Original entry on oeis.org

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.

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.

A141732 Numbers of 0's in the rows of triangle A141728.

Original entry on oeis.org

0, 3, 2, 5, 5, 8, 7, 6, 10, 12, 12, 15, 14, 19, 16, 15, 15, 21, 23, 25, 19, 23, 25, 22, 27, 28, 29, 38, 28, 37, 32, 27, 27, 33, 37, 39, 35, 39, 43, 43, 45, 43, 41, 43, 55, 61, 51, 50, 47, 54, 63, 60, 56, 57, 54, 55, 55, 53, 61, 70, 63, 72, 63, 56, 56, 66, 68, 72, 68, 70, 72, 76

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 02 2008

Cf. A141727 - A141731, A141733 - 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.

A141734 Binary digits, representing the rows of triangle A141728, written in base 10.

Original entry on oeis.org

1, 0, 21, 24, 293, 7, 5452, 6637, 74664, 6458, 1384558, 1578150, 19209993, 474240, 357331029, 434949016, 4893636133, 421910023, 90731484851, 103435736251, 1258858815339, 30658739531, 23416607110667, 28506443273867

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jul 02 2008

Cf. A141727 - A141733, A141735 - 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.

cp's OEIS Frontend

A141727 Triangle T(n,k) read by rows. Entries are 0 and 1. Start with 1 in the top row, add a second row of 2n-1 elements (with n=2 -> 3). Moving from left to right add 0 if the number of adjacent 1's is even or add 1 if it is odd.

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

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A141732 Numbers of 0's in the rows of triangle A141728.

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

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A141734 Binary digits, representing the rows of triangle A141728, 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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