cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A160740 Toothpick sequence starting from a cross formed by 4 toothpicks.

Original entry on oeis.org

0, 4, 8, 16, 24, 32, 40, 56, 72, 80, 88, 104, 120, 136, 160, 200, 232, 240, 248, 264, 280, 296, 320, 360, 392, 408, 432, 472, 512, 560, 640, 744, 808, 816, 824, 840, 856, 872, 896, 936, 968, 984, 1008, 1048, 1088, 1136, 1216, 1320, 1384, 1400, 1424, 1464, 1504, 1552
Offset: 0

Views

Author

Omar E. Pol, May 25 2009

Keywords

Comments

On the infinite square grid we start at stage 0 with no toothpicks. Toothpicks have length 2. At stage 1 we place two consecutive toothpicks in the vertical direction and two consecutive toothpicks in the horizontal direction forming a cross centered at the origin. At stage 2 we place four toothpicks. At stage 3 we place eight toothpicks. For more information about the toothpick sequences see A139250. - Omar E. Pol, Nov 24 2011

Links

Crossrefs

Cf. A139250, A139251, A160736, A160738, A160741.
Cf. A160170, A160426, A194432, A194434.

Formula

a(n) = 4*A160406(n).

Extensions

More terms from N. J. A. Sloane, May 25 2009

A160739 16*P_6(n), 16 times the Legendre Polynomial of order 6 at n.

Original entry on oeis.org

-5, 16, 10159, 143824, 867211, 3415120, 10373071, 26425744, 59271739, 120704656, 227860495, 404631376, 683245579, 1106013904, 1727242351, 2615311120, 3854919931, 5549499664, 7823790319, 10826585296, 14733641995, 19750758736, 26117017999
Offset: 0

Views

Author

N. J. A. Sloane, Nov 17 2009

Keywords

Links

Crossrefs

Cf. A144126, A160741.

Programs

  • Magma
    [231*n^6 -315*n^4 +105*n^2 -5: n in [0..30]]; // G. C. Greubel, May 02 2018
  • Maple
    A160739 := proc(n)
        16*orthopoly[P](6,n) ;
end proc: # R. J. Mathar, Oct 24 2011
  • Mathematica
    Table[16 LegendreP[6,n],{n,0,40}]
  • PARI
    a(n)=16*pollegendre(6,n) \\ Charles R Greathouse IV, Mar 18 2017
  • PARI
    Vec(-(5 - 51*x - 9942*x^2 - 73222*x^3 - 73047*x^4 - 10047*x^5 - 16*x^6) / (1 - x)^7 + O(x^30)) \\ Colin Barker, Jul 23 2019

Formula

a(n) = 231*n^6 - 315*n^4 + 105*n^2 - 5. - Vaclav Kotesovec, Jul 31 2013
From Colin Barker, Jul 23 2019: (Start)
G.f.: -(5 - 51*x - 9942*x^2 - 73222*x^3 - 73047*x^4 - 10047*x^5 - 16*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)

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

Views

Author

Omar E. Pol, May 25 2009

Keywords

Examples

			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.

Links

Crossrefs

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

Formula

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

Extensions

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

Views

Author

Omar E. Pol, May 25 2009

Keywords

Links

Crossrefs

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

Extensions

Terms after a(9) from Nathaniel Johnston, Mar 30 2011
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.