A171175 -id:A171175 - cp's OEIS frontend

A171178 Permutation of the natural numbers: 0 together with the partial sums of A171177.

0, 2, 1, 6, 3, 5, 4, 12, 7, 11, 8, 10, 9, 20, 13, 19, 14, 18, 15, 17, 16, 30, 21, 29, 22, 28, 23, 27, 24, 26, 25, 42, 31, 41, 32, 40, 33, 39, 34, 38, 35, 37, 36, 56, 43, 55, 44, 54, 45, 53, 46, 52, 47, 51, 48, 50, 49, 72, 57, 71, 58, 70, 59, 69, 60, 68

Offset: 0

Omar E. Pol, Feb 23 2010

a(n) is also the value of "x" and "y" of the n-th point (x,y), located on the infinite straight line (0,0),(1,1)..., that is intercepted by the path in structure of A171166.

For another version see A171175.

Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A005132, A171164, A161165, A161166, A171172, A171173, A171174, A171175, A171176, A171178, A172310.

A210606 Length of the n-th edge of an L-toothpick structure which gives Recamán's sequence A005132.

Original entry on oeis.org

1, 3, 5, 3, 4, 4, 5, 11, 13, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17

Offset: 1

Omar E. Pol, Mar 24 2012

Consider a toothpick structure formed by L-toothpicks connected by their endpoints. The endpoints of the L-toothpicks are placed on the main diagonal of the first quadrant. At stage 1 we place an L-toothpick with one of its endpoints on the origin. At stage n we place an L-toothpick of size n. The L-toothpicks are placed alternately, on one or another sector of the first quadrant, trying to make the structure have an exposed endpoint closest to the origin. The total length of all L-toothpicks after the n-th stage is A002378(n). The value of x and y of the endpoint of the structure after the n-th stage is equal to the n-th term of Recamán's sequence A005132(n). Note that we can get other illustrations of initial terms of Recamán's sequence by replacing each L-toothpick by a Q-toothpick or by a semicircumference. This structure is also one of the three views of the three-dimensional model for Recamán's sequence. For more information about L-toothpicks and Q-toothpicks, see A172310 and A187210.

			The summands are the size of the L-toothpicks:
a(1) = 1.
a(2) = 1 + 2 = 3.
a(3) = 2 + 3 = 5.
a(4) = 3.
a(5) = 4.
a(6) = 4.
a(7) = 5.
a(8) = 5 + 6 = 11.
a(9) = 6 + 7 = 13.
a(10) = 7.

First differences of A210607.

Cf. A005132, A064289, A119632, A139250, A160356, A160357, A171175, A171178, A172310, A187210, A210604, A210605, A210608-A210613.

A171173 Triangle read by rows in which row n lists A033627(n) together with the first 2n-1 positive integers.

Original entry on oeis.org

2, 1, 4, 1, 2, 3, 7, 1, 2, 3, 4, 5, 10, 1, 2, 3, 4, 5, 6, 7, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 25, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 28, 1

Offset: 1

Omar E. Pol, Feb 23 2010

The same as A171172 except the initial term.
Also, a(n) is the length of each component of the n-th L-toothpick added to the structure of A171165.
See also A171175, a permutation of the natural numbers.

			Triangle begins:
2,1,
4,1,2,3,
7,1,2,3,4,5,
10,1,2,3,4,5,6,7,
13,1,2,3,4,5,6,7,8,9,
16,1,2,3,4,5,6,7,8,9,10,11,
19,1,2,3,4,5,6,7,8,9,10,11,12,13,
22,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
25,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,

Harvey P. Dale, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000027, A033627, A172310, A171164, A161165, A161166, A171172, A171174, A171175, A171176, A171177, A171178.

Join[{2,1},Flatten[Table[Flatten[{3n+1,Range[2n+1]}],{n,10}]]] (* Harvey P. Dale, Nov 24 2011 *)

A171176 Triangle read by rows in which row n lists 3n-1 together with the first 2n-1 positive integers, in reverse order.

Original entry on oeis.org

2, 1, 5, 3, 2, 1, 8, 5, 4, 3, 2, 1, 11, 7, 6, 5, 4, 3, 2, 1, 14, 9, 8, 7, 6, 5, 4, 3, 2, 1, 17, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 23, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 26, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Omar E. Pol, Feb 23 2010

a(n) is also the length of the n-th L-toothpick added to the structure of A171166.

			Triangle begins:
   2,  1;
   5,  3,  2,  1;
   8,  5,  4,  3,  2,  1;
  11,  7,  6,  5,  4,  3,  2,  1;
  14,  9,  8,  7,  6,  5,  4,  3,  2, 1;
  17, 11, 10,  9,  8,  7,  6,  5,  4, 3, 2, 1;
  20, 13, 12, 11, 10,  9,  8,  7,  6, 5, 4, 3, 2, 1;
  23, 15, 14, 13, 12, 11, 10,  9,  8, 7, 6, 5, 4, 3, 2, 1;
  26, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1;

Harvey P. Dale, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A000027, A016789, A172310, A171164, A161165, A161166, A171172, A171173, A171174, A171175, A171177, A171178.

Cf. A132209 (row sums). - R. J. Mathar, Mar 01 2010

Table[{3n-1,Reverse[Range[2n-1]]},{n,10}]//Flatten (* Harvey P. Dale, Jun 26 2022 *)

A171174 Triangle read by rows in which row n lists A033627(n) together with the first 2n-1 numbers <> 0 of A038608.

Original entry on oeis.org

2, -1, 4, -1, 2, -3, 7, -1, 2, -3, 4, -5, 10, -1, 2, -3, 4, -5, 6, -7, 13, -1, 2, -3, 4, -5, 6, -7, 8, -9, 16, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 19, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 22, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 25, -1, 2, -3, 4

Offset: 1

Omar E. Pol, Feb 23 2010

Absolute values give A171173.

Note that the partial sums of this sequence gives A171175, a permutation of the natural numbers.

			Triangle begins:
2, -1,
4, -1,2,-3,
7, -1,2,-3,4,-5,
10,-1,2,-3,4,-5,6,-7,
13,-1,2,-3,4,-5,6,-7,8,-9,
16,-1,2,-3,4,-5,6,-7,8,-9,10,-11,
19,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,
22,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,
25,-1,2,-3,4,-5,6,-7,8,-9,10,-11,12,-13,14,-15,16,-17,

Cf. A033627, A038608, A172310, A171164, A161165, A161166, A171172, A171173, A171175, A171176, A171177, A171178.

A171177 Triangle read by rows in which row n lists 3n-1 together with the first 2n-1 numbers <> 0 of A038608, in reverse order.

Original entry on oeis.org

2, -1, 5, -3, 2, -1, 8, -5, 4, -3, 2, -1, 11, -7, 6, -5, 4, -3, 2, -1, 14, -9, 8, -7, 6, -5, 4, -3, 2, -1, 17, -11, 10, -9, 8, -7, 6, -5, 4, -3, 2, -1, 20, -13, 12, -11, 10, -9, 8, -7, 6, -5, 4, -3, 2, -1, 23, -15, 14, -13, 12, -11, 10, -9, 8, -7, 6, -5, 4, -3, 2, -1, 26, -17, 16, -15

Offset: 1

Omar E. Pol, Feb 23 2010

Absolute values give A171176.

Note that the partial sums of this sequence give A171178, a permutation of the natural numbers.

			Triangle begins:
2, -1;
5, -3, 2, -1;
8, -5, 4, -3, 2, -1;
11,-7, 6, -5, 4, -3, 2, -1;
14,-9, 8, -7, 6, -5, 4, -3, 2, -1;
17,-11,10,-9, 8, -7, 6, -5, 4, -3, 2, -1;
20,-13,12,-11,10,-9, 8, -7, 6, -5, 4, -3, 2, -1;
23,-15,14,-13,12,-11,10,-9, 8, -7, 6, -5, 4, -3, 2, -1;
26,-17,16,-15,14,-13,12,-11,10,-9, 8, -7, 6, -5, 4, -3, 2, -1;

Cf. A016789, A038608, A172310, A171164, A161165, A161166, A171172, A171173, A171174, A171175, A171176, A171178.

A171172 Triangle read by rows in which row n lists 3n-2 together with the first 2n-1 positive integers.

Original entry on oeis.org

1, 1, 4, 1, 2, 3, 7, 1, 2, 3, 4, 5, 10, 1, 2, 3, 4, 5, 6, 7, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 22, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 25, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 28, 1

Offset: 1

Omar E. Pol, Feb 23 2010

			Triangle begins:
1,1,
4,1,2,3,
7,1,2,3,4,5,
10,1,2,3,4,5,6,7,
13,1,2,3,4,5,6,7,8,9,
16,1,2,3,4,5,6,7,8,9,10,11,
19,1,2,3,4,5,6,7,8,9,10,11,12,13,
22,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,
25,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,

Cf. A016777, A172310, A171173, A171174, A171175, A171176, A171177, A171178.

A171164 A polyspiral path: a(n) represents the n-th vertex of a lattice path with an infinite number of finite square spirals.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 12, 13, 14, 16, 18, 21, 24, 31, 38, 39, 40, 42, 44, 47, 50, 54, 58, 63, 68, 78, 88, 89, 90, 92, 94, 97, 100, 104, 108, 113, 118, 124, 130, 137, 144, 157, 170, 171, 172, 174, 176, 179, 182, 186, 190, 195, 200, 206, 212

Offset: 0

Omar E. Pol, Mar 14 2010

Note that the vertex 0 and the vertex 4 both are overlapping.
For other versions see A171165 and A171166.
Also, partial sums of the sequence formed by 0 together with the numbers of A171172 repeated.

Cf. A171165, A171166, A171172, A171173, A171174, A171175, A171176, A171177, A171178.

A210607 Vertex number of an L-toothpick structure which give Recamán's sequence A005132.

Original entry on oeis.org

0, 1, 4, 9, 12, 16, 20, 25, 36

Offset: 1

Omar E. Pol, Mar 24 2012

For more information see A210606.

First differences give A210606.

A002378, A005132, A119632, A139250, A160356, A171175, A171178, A172310, A187210, A210607-A210613.

A171165 A polyspiral path: a(n) represents the n-th vertex of a lattice path with an infinite number of finite square spirals.

Original entry on oeis.org

0, 2, 4, 5, 6, 10, 14, 15, 16, 18, 20, 23, 26, 33, 40, 41, 42, 44, 46, 49, 52, 56, 60, 65, 70, 80, 90, 91, 92, 94, 96, 99, 102, 106, 110, 115, 120, 126, 132, 139, 146, 159, 172, 173, 174, 176, 178, 181, 184, 188, 192, 197, 202, 208, 214

Offset: 0

Omar E. Pol, Mar 14 2010

Also, partial sums of the sequence formed by 0 together the numbers of A171173 repeated.

For another version see A171166.

Cf. A171166, A171173, A171174, A171175, A171176, A171177, A171178.

cp's OEIS Frontend

A171178 Permutation of the natural numbers: 0 together with the partial sums of A171177.

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A210606 Length of the n-th edge of an L-toothpick structure which gives Recamán's sequence A005132.

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A171173 Triangle read by rows in which row n lists A033627(n) together with the first 2n-1 positive integers.

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A171176 Triangle read by rows in which row n lists 3n-1 together with the first 2n-1 positive integers, in reverse order.

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A171174 Triangle read by rows in which row n lists A033627(n) together with the first 2n-1 numbers <> 0 of A038608.

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A171177 Triangle read by rows in which row n lists 3n-1 together with the first 2n-1 numbers <> 0 of A038608, in reverse order.

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A171172 Triangle read by rows in which row n lists 3n-2 together with the first 2n-1 positive integers.

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A171164 A polyspiral path: a(n) represents the n-th vertex of a lattice path with an infinite number of finite square spirals.

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A210607 Vertex number of an L-toothpick structure which give Recamán's sequence A005132.

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A171165 A polyspiral path: a(n) represents the n-th vertex of a lattice path with an infinite number of finite square spirals.

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