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.

A170885 First differences of A170884.

Original entry on oeis.org

0, 3, 4, 8, 12, 12, 12, 16, 32, 56, 68, 40, 20, 16, 32, 56, 80, 88, 104, 152, 256, 376, 388, 164, 64, 24, 40, 56, 80, 88, 104, 152, 256, 376, 424, 296, 256, 272, 416, 552
Offset: 0

Views

Author

David Applegate, Omar E. Pol and N. J. A. Sloane, Jan 09 2010

Keywords

Links

Extensions

More terms from Nathaniel Johnston, Nov 13 2010

A160160 Toothpick sequence in the three-dimensional grid.

Original entry on oeis.org

0, 1, 3, 7, 15, 23, 31, 39, 55, 87, 143, 175, 191, 199, 215, 247, 303, 359, 423, 503, 655, 887, 1239, 1383, 1431, 1463, 1487, 1527, 1583, 1639, 1703, 1783, 1935, 2167, 2519, 2735, 2903, 3079, 3351, 3711, 4207, 4655, 5191, 5855, 7023, 8511, 10511, 11279, 11583, 11919, 12183, 12375, 12487, 12607
Offset: 0

Views

Author

Omar E. Pol, May 03 2009, May 06 2009

Keywords

Comments

Similar to A139250, except the toothpicks are placed in three dimensions, not two. The first toothpick is in the z direction. Thereafter, new toothpicks are placed at free ends, as in A139250, perpendicular to the existing toothpick, but choosing in rotation the x-direction, y-direction, z-direction, x-direction, etc.
The graph of this sequence has a nice self-similar shape: it looks the when the x-range is multiplied by 2, e.g. a(0..125) vs a(0..250) or a(0..500). - M. F. Hasler, Dec 12 2018

Links

Crossrefs

Cf. A139250, A160120, A160161, A160170, A170884, A170885, A170876.

Programs

  • PARI
    A160160_vec(n,o=1)={local(s(U)=[Vecsmall(Vec(V)+U)|V<-E], E=[Vecsmall([1,1,1])], J=[], M,A,B,U); [if(i>4, M+=8*#E=setminus(setunion(A=s(U=matid(3)[i%3+1,]), B=select(vecmin,s(-U))), J=setunion(setunion(setintersect(A,B),E),J)),M=1<M. F. Hasler, Dec 11 2018
  • PARI
    A160160(n)=sum(k=1,n,A160161[k]) \\ if A160161=A160161_vec(n) has already been computed. - M. F. Hasler, Dec 12 2018

Formula

Partial sums of A160161: a(n) = Sum_{1 <= k <= n} A160161(k) for all n >= 0. - M. F. Hasler, Dec 12 2018

Extensions

Edited by N. J. A. Sloane, Jan 02 2009
Extended to a(76) with C++ program and illustrations by R. J. Mathar, Jan 09 2010
Extended to 500 terms by M. F. Hasler, Dec 12 2018

A147614 a(n) = number of grid points that are covered after n-th stage of A139250, assuming the toothpicks have length 2.

Original entry on oeis.org

0, 3, 7, 13, 19, 27, 39, 53, 63, 71, 83, 99, 119, 147, 183, 217, 235, 243, 255, 271, 291, 319, 355, 391, 419, 447, 487, 539, 607, 699, 803, 885, 919, 927, 939, 955, 975, 1003, 1039, 1075, 1103, 1131, 1171, 1223, 1291, 1383, 1487, 1571, 1615
Offset: 0

Views

Author

David Applegate, Apr 29 2009

Keywords

Comments

a(n) is also the number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton whose virtual skeleton is a polyedge as the toothpick structure of A139250. [From Omar E. Pol, May 18 2009]
It appears that the number of grid points that are covered after n-th stage of A139250, assuming the toothpicks have length 2*k, is equal to (2*k-2) * A139250(n) + a(n), k>0. See formulas in A160420 and A160422. [From Omar E. Pol, Nov 15 2010]
More generally, it appears that a(n) is also the number of grid points that are covered by the "special points" of the toothpicks of A139250, after n-th stage, assuming the toothpicks have length 2*k, k>0 and that each toothpick has three special points: the midpoint and two endpoints.
Note that if k>1 then also there are 2*k-2 grid points that are covered by each toothpick, but these points are not considered for this sequence. [From Omar E. Pol, Nov 15 2010]
Contribution from Omar E. Pol, Sep 16 2012 (Start):
It appears that a(n)/A139250(n) converge to 4/3.
It appears that a(n)/A160124(n) converge to 2.
It appears that a(n)/A139252(n) converge to 4.
(End)

Links

Crossrefs

Cf. A139250, A139251, A139252, A160124, A160420, A160422, A170884.

Formula

Since A160124(n) = 1+2*A139250(n)-A147614(n), n>0 (see A160124), and we have recurrences for A160125 (hence A160124) and A139250, we have a recurrence for this sequence. - N. J. A. Sloane, Feb 02 2010
a(n) = A187220(n+1)-A160124(n), n>0. - Omar E. Pol, Feb 15 2013

A160430 The 3-D toothpick sequence A160160, but using toothpicks of length 4; a(n) is the number of nodes occupied after n steps.

Original entry on oeis.org

0, 5, 13, 29, 57, 85, 113, 145, 209, 329, 509, 613, 665, 697, 761, 881, 1073, 1273, 1505, 1817, 2377, 3217, 4309, 4761, 4921, 5009, 5097, 5233, 5425, 5625, 5857, 6169, 6729, 7569, 8697, 9425, 10017, 10641, 11601, 12873
Offset: 0

Views

Author

Omar E. Pol, May 13 2009

Keywords

Examples

			Each toothpick looks like this: o-o-o-o-o.
The initial (z-axis) toothpick occupies 5 nodes. The next two, in the x-direction, add 8 further nodes, and so on.

Links

Crossrefs

Cf. A139250, A139251, A147562, A160118, A160120, A160160, A160170, A160379, A160420.

Formula

a(n) = A170884(n) + 2*A160160(n)

Extensions

Edited by N. J. A. Sloane, Jan 02 2010
Formula and more terms from Nathaniel Johnston, Nov 14 2010
Showing 1-4 of 4 results.

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

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