A160737 -id:A160737 - cp's OEIS frontend

A160736 Toothpick sequence starting from a right angle formed by 2 toothpicks: a(n)=A160406(n)*2.

0, 2, 4, 8, 12, 16, 20, 28, 36, 40, 44, 52, 60, 68, 80, 100, 116, 120, 124, 132, 140, 148, 160, 180, 196, 204, 216, 236, 256, 280, 320, 372, 404, 408, 412, 420, 428, 436, 448, 468, 484, 492, 504, 524, 544, 568

Offset: 0

Omar E. Pol, May 25 2009

nonn

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A160737, A160738, A160740.

Terms after a(9) from Nathaniel Johnston, Mar 31 2011

A160738 Toothpick sequence starting from a T formed by 3 toothpicks: a(n)=A160406(n)*3.

Original entry on oeis.org

0, 3, 6, 12, 18, 24, 30, 42, 54, 60, 66, 78, 90, 102, 120, 150, 174, 180, 186, 198, 210, 222, 240, 270, 294, 306, 324, 354, 384, 420, 480, 558, 606, 612, 618, 630, 642, 654, 672, 702, 726, 738, 756, 786, 816, 852, 912, 990, 1038, 1050, 1068, 1098, 1128, 1164

Offset: 0

Omar E. Pol, May 25 2009, Jun 19 2009

nonn

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A160736, A160737, A160739, A160740.

More terms from R. J. Mathar, Jul 28 2009

A160731 First differences of A160730.

Original entry on oeis.org

2, 2, 4, 6, 6, 6, 12, 14, 12, 6, 12, 14, 16, 18, 32, 34, 20, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 36, 6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 40, 18, 32, 38, 44, 62

Offset: 1

Omar E. Pol, May 25 2009

nonn

			Placing the entries starting from a(4) in a triangle with rows that have length equal to powers of two gives:
6, 6
6, 12, 14, 12
6, 12, 14, 16, 18, 32, 34, 20
6, 12, 14, 16, 18, 32, 34, 24, 18, 32, 38, 44, 62, 92, 82, 36
...
The rows of this triangle tend to 2*A168114.

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A160730, A160733, A160737, A160739, A160741.

From Nathaniel Johnston, Mar 28 2011: (Start)
a(n) = 2*A168113(n)
a(2^(n+2) + 1) = 4(2^n + 1), n >= 1.
(End)

Terms after a(11) from Nathaniel Johnston, Mar 28 2011

A160733 First differences of A160732.

Original entry on oeis.org

3, 3, 6, 8, 8, 6, 12, 14, 8, 8, 16, 20, 18, 18, 30, 26, 8, 8, 16, 20, 18, 20, 34, 32, 20, 28, 48, 54, 50, 60, 78, 50, 8, 8, 16, 20, 18, 20, 34, 32, 20, 28, 48, 54, 50, 62, 82, 56, 20, 28, 48, 54, 52, 70, 96

Offset: 1

Omar E. Pol, May 25 2009

nonn

Nathaniel Johnston, Table of n, a(n) for n = 1..274
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A139251, A160731, A160732, A160737, A160739, A160741.

Terms after a(9) from Nathaniel Johnston, Mar 30 2011

A335338 P_5(2n+1), the Legendre polynomial of order 5 at 2n+1.

Original entry on oeis.org

1, 1683, 23525, 129367, 458649, 1256651, 2904733, 5950575, 11138417, 19439299, 32081301, 50579783, 76767625, 112825467, 161311949, 225193951, 307876833, 413234675, 545640517, 709996599, 911764601, 1156995883, 1452361725, 1805183567, 2223463249, 2715913251, 3291986933

Offset: 0

Seiichi Manyama, Jun 02 2020

Eric Weisstein's World of Mathematics, Legendre Polynomial.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Row 5 of A335333.

Cf. A160737.

a[n_] := LegendreP[5, 2*n + 1]; Array[a, 27, 0] (* Amiram Eldar, May 03 2021 *)

PARI
```
a(n) = pollegendre(5, 2*n+1)
```

a(n) = 252*n^5+630*n^4+560*n^3+210*n^2+30*n+1

N=40; x='x+O('x^N); Vec((1+x)*(1+1676*x+11766*x^2+1676*x^3+x^4)/(1-x)^6)

a(n) = A160737(2*n+1)/4.
a(n) = 252*n^5 + 630*n^4 + 560*n^3 + 210*n^2 + 30*n + 1 = (2*n + 1) * (126*n^4 + 252*n^3 + 154*n^2 + 28*n + 1).
G.f.: (1+x)*(1+1676*x+11766*x^2+1676*x^3+x^4)/(1-x)^6.

cp's OEIS Frontend

A160736 Toothpick sequence starting from a right angle formed by 2 toothpicks: a(n)=A160406(n)*2.

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A160738 Toothpick sequence starting from a T formed by 3 toothpicks: a(n)=A160406(n)*3.

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A160731 First differences of A160730.

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A160733 First differences of A160732.

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A335338 P_5(2n+1), the Legendre polynomial of order 5 at 2n+1.

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