A233781 -id:A233781 - cp's OEIS frontend

A233782 a(n) = A233781(n+1)/2.

1, 1, 2, 1, 2, 3, 4, 1, 2, 3, 5, 5, 4, 7, 8, 1, 2, 3, 5, 5, 5, 9, 12, 9, 4, 7, 11, 13, 8, 15, 16, 1, 2, 3, 5, 5, 5, 9, 12, 9, 5, 9, 15, 19, 13, 21, 26, 17, 4, 7, 11, 13, 11, 21, 28, 25, 8, 15, 23, 29, 16, 31, 32, 1, 2, 3, 5, 5, 5, 9, 12, 9, 5, 9, 15, 19

Offset: 1

Omar E. Pol, Dec 17 2013

First 22 terms coincide with a(2)-a(23) of A169778.

			Written as an irregular triangle in which the row lengths is A000079 the sequence begins:
1;
1,2;
1,2,3,4;
1,2,3,5,5,4,7,8;
1,2,3,5,5,5,9,12,9,4,7,11,13,8,15,16;
1,2,3,5,5,5,9,12,9,5,9,15,19,13,21,26,17,4,7,11,13,11,21,28,25,8,15,23,29,16,31,32;

Right border gives A000079.

Cf. A139251, A169778, A220501, A220521, A231351, A233781, A233780.

A233780 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 21, 29, 31, 35, 41, 51, 61, 69, 83, 99, 101, 105, 111, 121, 131, 141, 159, 183, 201, 209, 223, 245, 271, 287, 317, 349, 351, 355, 361, 371, 381, 391, 409, 433, 451, 461, 479, 509, 547, 573, 615, 667, 701, 709, 723, 745, 771

Offset: 0

Omar E. Pol, Dec 15 2013

nonn

The D-toothpicks placed in northwest or northeast direction both are prohibited, except in the substructures in which the symmetry could be broken, so a(44) = 509, not 507. For another version with broken symmetry in some substructures see A233764. See also A231348, a simpler cellular automaton based in triangles which has essentially a similar structure.
A233781 (the first differences) gives the number of toothpicks or D-toothpicks added at n-th stage.
First differs from A169780 at a(24).
First differs from A233970 at a(25).
First differs from A233764 at a(44).

Cf. A139250, A169780, A220500, A231348, A233764, A233781, A233970.

A233971 Number of toothpicks added at n-th stage to the structure of A233970.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 10, 10, 8, 14, 16, 2, 4, 6, 10, 10, 10, 18, 24, 22, 8, 14, 22, 26, 16, 30, 32, 2, 4, 6, 10, 10, 10, 18, 24, 22, 10, 18, 28, 38, 28, 46, 56, 54, 8, 14, 22, 26, 22, 42, 56, 62, 16, 30, 46, 58, 32, 62, 64, 2, 4, 6, 10, 10

Offset: 0

Omar E. Pol, Dec 18 2013

Essentially the first differences of A233970.
First differs from A170905 at a(24).
First differs from both A233765 and A233781 at a(25).

			Written as an irregular triangle in which the row lengths is A011782 the sequence (starting from 1) begins:
1;
2;
2,4;
2,4,6,8;
2,4,6,10,10,8,14,16;
2,4,6,10,10,10,18,24,22,8,14,22,26,16,30,32;
2,4,6,10,10,10,18,24,22,10,18,28,38,28,46,56,54,8,14,22,26,22,42,56,62,16,30,46,58,32,62,64;
Right border gives A000079.

Cf. A000079, A011782, A139251, A161207, A161831, A170905, A182633, A182635, A182841, A233761, A233765, A233781, A233970, A233972.

A233761 Number of toothpicks or D-toothpicks added at n-th stage in the structure of the D-toothpick "wide" triangle of A233760.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 12, 14, 12, 14, 16, 2, 4, 6, 12, 14, 20, 22, 26, 22, 12, 18, 32, 38, 28, 30, 32, 2, 4, 6, 12, 14, 20, 22, 26, 22, 20, 30, 44, 62, 52, 46, 54, 38, 12, 18, 36, 46, 56, 62, 74, 74, 30, 42, 72, 86, 60, 62, 64, 2, 4, 6, 12

Offset: 0

Omar E. Pol, Dec 16 2013

Essentially the first differences of A233760.

			Written as an irregular triangle in which the row lengths is A011782 the sequence (starting from 1) begins:
1;
2;
2,4;
2,4,6,8;
2,4,6,12,14,12,14,16;
2,4,6,12,14,20,22,26,22,12,18,32,38,28,30,32;
2,4,6,12,14,20,22,26,22,20,30,44,62,52,46,54,38,12,18,36,46,56,62,74,74,30,42,72,86,60,62,64;
Right border gives A000079.

Cf. A000079, A011782, A139251, A220501, A220521, A233760, A233763, A233781.

a(2n) = A220521(2n); a(2n+1) = A220521(2n+1) - 2.

A233765 Number of toothpicks or D-toothpicks added at n-th stage to the structure of the toothpick "wide" triangle of A233764.

Original entry on oeis.org

0, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 6, 10, 10, 8, 14, 16, 2, 4, 6, 10, 10, 10, 18, 24, 18, 8, 14, 22, 26, 16, 30, 32, 2, 4, 6, 10, 10, 10, 18, 24, 18, 10, 18, 28, 38, 30, 50, 54, 34, 8, 14, 22, 26, 22, 42, 56, 50, 16, 30, 46, 58, 32, 62, 64, 2, 4, 6, 10, 10

Offset: 0

Omar E. Pol, Dec 16 2013

Essentially the first differences of A233764.
First differs from A170905 at a(24).
First differs from A233971 at a(25).
First differs from A233781 at a(44).

			Written as an irregular triangle in which the row lengths is A011782 the sequence (starting from 1) begins:
1;
2;
2,4;
2,4,6,8;
2,4,6,10,10,8,14,16;
2,4,6,10,10,10,18,24,18,8,14,22,26,16,30,32;
2,4,6,10,10,10,18,24,18,10,18,28,38,30,50,54,34,8,14,22,26,22,42,56,50,16,30,46,58,32,62,64;
Right border gives A000079.

Cf. A000079, A011782, A139251, A220501, A220521, A231349, A233764, A233781, A233971.

cp's OEIS Frontend

A233782 a(n) = A233781(n+1)/2.

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A233780 Number of toothpicks and D-toothpicks after n-th stage in a D-toothpick "wide" triangle (see Comments lines for definition).

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A233971 Number of toothpicks added at n-th stage to the structure of A233970.

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A233761 Number of toothpicks or D-toothpicks added at n-th stage in the structure of the D-toothpick "wide" triangle of A233760.

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A233765 Number of toothpicks or D-toothpicks added at n-th stage to the structure of the toothpick "wide" triangle of A233764.

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