A160739 -id:A160739 - cp's OEIS frontend

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

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

A160741 Numerator of P_6(2n), the Legendre polynomial of order 6 at 2n.

Original entry on oeis.org

-5, 10159, 867211, 10373071, 59271739, 227860495, 683245579, 1727242351, 3854919931, 7823790319, 14733641995, 26117017999, 44040338491, 71215667791, 111123125899, 168143944495, 247704167419, 356428995631, 502307776651, 694869638479, 945369767995

Offset: 0

N. J. A. Sloane, Nov 17 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A160739, A144126.

A160741 := proc(n)
        orthopoly[P](6,2*n) ;
        numer(%) ;
end proc: # R. J. Mathar, Oct 24 2011

Table[Numerator[LegendreP[6,2n]],{n,0,40}]

a(n)=numerator(pollegendre(6,n+n)) \\ Charles R Greathouse IV, Oct 24 2011

Vec(-(5 - 10194*x - 795993*x^2 - 4516108*x^3 - 4515933*x^4 - 796098*x^5 - 10159*x^6) / (1 - x)^7 + O(x^30)) \\ Colin Barker, Jul 23 2019

From Colin Barker, Jul 23 2019: (Start)
G.f.: -(5 - 10194*x - 795993*x^2 - 4516108*x^3 - 4515933*x^4 - 796098*x^5 - 10159*x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.
a(n) = -5 + 420*n^2 - 5040*n^4 + 14784*n^6.
(End)

A144126 P_6(2n+1), the Legendre polynomial of order 6 at 2n+1.

Original entry on oeis.org

1, 8989, 213445, 1651609, 7544041, 25289461, 69125869, 163456945, 346843729, 676661581, 1234422421, 2131762249, 3517093945, 5582925349, 8573842621, 12795158881, 18622228129, 26510424445, 37005786469, 50756327161

Offset: 0

Vladimir Joseph Stephan Orlovsky, Sep 11 2008

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A160739.

Table[LegendreP[6,2n+1],{n,0,50}] (* N. J. A. Sloane, Nov 17 2009 *)

a(n)=pollegendre(6,n+n+1) \\ Charles R Greathouse IV, Oct 25 2011

Vec((1 + 8982*x + 150543*x^2 + 346228*x^3 + 150543*x^4 + 8982*x^5 + x^6) / (1 - x)^7 + O(x^30)) \\ Colin Barker, Jul 23 2019

a(n) = 924*n^6 + 2772*n^5 + 3150*n^4 + 1680*n^3 + 420*n^2 + 42*n+1. - Vaclav Kotesovec, Jul 31 2013
From Colin Barker, Jul 23 2019: (Start)
G.f.: (1 + 8982*x + 150543*x^2 + 346228*x^3 + 150543*x^4 + 8982*x^5 + x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.
(End)

Definition corrected by N. J. A. Sloane, Nov 17 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

cp's OEIS Frontend

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

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A160741 Numerator of P_6(2n), the Legendre polynomial of order 6 at 2n.

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A144126 P_6(2n+1), the Legendre polynomial of order 6 at 2n+1.

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A160731 First differences of A160730.

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A160733 First differences of A160732.

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