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.

Previous Showing 11-17 of 17 results.

A256261 First differences of A256260.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 20, 28, 4, 12, 20, 28, 20, 44, 68, 60, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 92, 116, 140, 164, 124, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 92, 116, 140, 164, 124, 20, 44, 68, 92, 116, 140, 164, 188, 212, 236, 260, 284, 308, 332, 356, 252, 4, 12, 20, 28, 20, 44, 68, 60, 20, 44, 68, 92, 116, 140
Offset: 0

Views

Author

Omar E. Pol, Mar 30 2015

Keywords

Comments

First 27 terms agree with A169708. Both sequences share infinitely many terms.

Examples

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
4;
4,12;
4,12,20,28;
4,12,20,28,20,44,68,60;
4,12,20,28,20,44,68,60,20,44,68,92,116,140,164,124;
4,12,20,28,20,44,68,60,20,44,68,92,116,140,164,124,20,44,68,92,116,140,164,188,212,236,260,284,308,332,356,252;
...
It appears that the row sums give A000302.
It appears that the right border gives A173033.

Links

Crossrefs

Cf. A000302, A011782, A139251, A147582, A169708, A173033, A256258, A256251, A256260, A256263, A256264, A256265.

Formula

a(n) = 4*A256263(n), n >= 1.

A256251 First differences of A256250.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 20, 28, 4, 12, 20, 28, 36, 44, 52, 60, 4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100, 108, 116, 124, 132, 140, 148, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 4, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92, 100
Offset: 0

Views

Author

Omar E. Pol, Mar 20 2015

Keywords

Comments

Number of cells turned ON at n-th stage in the structure of A256250.
Apart from the initial 1, four times A006257 (Josephus problem).

Examples

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
1;
4;
4,12;
4,12,20,28;
4,12,20,28,36,44,52,60;
4,12,20,28,36,44,52,60,68,76,84,92,100,108,116,124;
4,12,20,28,36,44,52,60,68,76,84,92,100,108,116,124,132,140,148,156,164,172,180,188,196,204,212,220,228,236,244,252;
...
Row sums give A000302.
Right border gives A173033.

Links

Crossrefs

Cf. A000302, A006257, A011782, A028399, A139251, A147582, A162793, A169708, A256239.

Programs

  • PARI
    a(n) = if(n, 8*(n - 2^logint(n,2)) + 4, 1)
  • Sage
    [1] + [8*(n - 2^floor(log(n,base=2))) + 4 for n in range(1,77)] # Danny Rorabaugh, Apr 20 2015

Formula

a(0) = 1. For n >= 1; a(n) = 4*A006257(n).
For n>0, a(n) = 8*(n - 2^floor(log_2(n))) + 4 (by the formula of Gregory Pat Scandalis in A006257). - Danny Rorabaugh, Apr 20 2015

A262621 First differences of A262620.

Original entry on oeis.org

1, 4, 12, 4, 28, 20, 12, 4, 60, 52, 44, 36, 28, 20, 12, 4, 124, 116, 108, 100, 92, 84, 76, 68, 60, 52, 44, 36, 28, 20, 12, 4, 252, 244, 236, 228, 220, 212, 204, 196, 188, 180, 172, 164, 156, 148, 140, 132, 124, 116, 108, 100, 92, 84, 76, 68, 60, 52, 44, 36, 28, 20, 12, 4, 508, 500, 492, 484, 476, 468, 460, 452, 444, 436
Offset: 0

Views

Author

Omar E. Pol, Nov 03 2015

Keywords

Comments

Number of cells turned "ON" at n-th stage of cellular automaton of A262620.

Examples

			With the terms written as an irregular triangle in which row lengths are the terms of A011782 the sequence begins:
1;
4;
12, 4;
28, 20, 12, 4;
60, 52, 44, 36, 28, 20, 12, 4;
124, 116, 108, 100, 92, 84, 76, 68, 60, 52, 44, 36, 28, 20, 12, 4;
...

Links

Crossrefs

Cf. A147582, A160721, A261693, A262620.
Row sums give A000302. Row lengths give A011782. Right border gives A123932. Column 1 is A173033.

Formula

a(n) = 4 * A261693(n), n >= 1.

A256139 First differences of A256138.

Original entry on oeis.org

1, 4, 4, 12, 4, 12, 20, 28, 4, 12, 20, 36, 36, 28, 52, 60, 4, 12, 20, 36, 36, 36, 68, 100, 68, 28, 52, 92, 108, 76, 124, 124, 4, 12, 20, 36, 36, 36, 68, 100, 68, 36, 68, 116, 148, 132, 164, 228, 132, 28, 52, 92, 108, 108, 172, 268, 236, 108, 124, 220, 276, 196, 276, 252
Offset: 0

Views

Author

Omar E. Pol, Mar 20 2015

Keywords

Comments

Number of cells turned ON at n-th stage in the structure of A256138.
First differs from A169708 at a(11).

Examples

			Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
4;
4,12;
4,12,20,28;
4,12,20,36,36,28,52,60;
4,12,20,36,36,36,68,100,68,28,52,92,108,76,124,124;
4,12,20,36,36,36,68,100,68,36,68,116,148,132,164,228,132,28,52,92,108,108,172,268,236,108,124,220,276,196,276,252;
...
It appears that the right border gives A173033.

Links

Crossrefs

Cf. A028399, A011782, A151723, A151724, A160721, A169708, A170898, A170905, A173033, A256138.

Formula

a(n) = 2*A151724(n+1)/3, n >= 1.

A366058 Number of n-step self-avoiding walks on a 3D cubic lattice where no step is to a lattice point closer to the origin than the current point.

Original entry on oeis.org

1, 6, 30, 126, 462, 1566, 5070, 15966, 49422, 151326, 460110, 1392606, 4202382, 12656286, 38067150, 114398046, 343587342, 1031548446, 3096218190, 9291800286, 27881692302, 83657659806, 250998145230, 753044767326, 2259234965262, 6777906222366, 20334121320270, 61003169267166, 183011118414222
Offset: 0

Views

Author

Scott R. Shannon, Dec 15 2023

Keywords

Comments

Consider the n-step self-avoiding walks from the origin in the first octant that increase in L1 distance from the origin on each step. There are 3^n such walks since each of the n steps may occur in any of 3 ways. To account for all combinations of signs of coordinates, there are binomial(3,3)*2^3 = 8 octants so there would be 8*3^n n-step paths total, but they overlap where one or more coordinates of the endpoint are 0. They overlap pairwise on the binomial(3,2)*2^2 = 12 edges of the octahedron at distance n from the origin. Each edge represents 2^n paths, since holding one coordinate 0, either of the other two coordinates may be chosen for each step. So now we have 8*3^n - 12*2^n to avoid double counting the edges. However, since the edges overlap at each of the binomial(3,1)*2^1 = 6 octahedral vertices, we have now eliminated the vertices, so they must be added back in. There is only one n-step path from the origin to each octahedral vertex. Thus, there are 8*3^n - 12*2^n + 6 paths of length n that increase in distance from the origin at each step. - Shel Kaphan, Mar 10 2024

Examples

			a(2) = 30 as after two steps no walk can step closer to the origin than its current point, so a(2) = A001412(2) = 30.
a(3) = 126. Given the first two steps of the 3-step walk are to points (1,0,0) and (1,0,1) then a step to (0,0,1) is forbidden. This walk can be taken in 4*6 = 24 ways on the cubic lattice, so the total number of permitted walks is a(3) = A001412(3) - 24 = 150 - 24 = 126.

Crossrefs

Cf. A173033 (2D square lattice), A001412.
Cf. A371064.

Formula

Conjectured: a(n) = 6*(4*3^(n-1) - 4*2^(n-1) + 1), for n > 0.
a(n) = Sum_{i=1..d} (-1)^(d-i) * binomial(d,i) * 2^i * i^n, where d=3, n>=1, which simplifies to 8*3^n - 12*2^n + 6, equivalent to conjectured formula (and row 3 of A371064). - Shel Kaphan, Mar 09 2024

A173034 Sequence whose G.f is f such that: f(z)=8/(1-2*z)-12/(1-z)+z+5.

Original entry on oeis.org

1, 5, 20, 52, 116, 244, 500, 1012, 2036, 4084, 8180, 16372, 32756, 65524, 131060, 262132, 524276, 1048564, 2097140, 4194292, 8388596, 16777204, 33554420, 67108852, 134217716, 268435444, 536870900, 1073741812, 2147483636
Offset: 0

Views

Author

Richard Choulet, Feb 07 2010

Keywords

Comments

The Granvik array of A172119 is here written in "square": 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 :: 1 // 1 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 :: 2 // 1 :: 3 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 :: 4 // 1 :: 4 :: 7 :: 8 :: 8 :: 8 :: 8 :: 8 :: 8 :: 8 // This sequence gives the third diagonal under the main diagonal.

Examples

			From 1 + 5*z + 20*z^2 + 52*z^3 + 116*z^4 + 244*z^5 + 500*z^6 + O(z^7) we get a(6)=500.

Links

Crossrefs

Cf. A172119, A173033.

Formula

a(n) = 8*2^n-12, n>=2. - R. J. Mathar, Apr 20 2011

A269712 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 20", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 4, 12, 28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, 65532, 131068
Offset: 0

Views

Author

Robert Price, Mar 04 2016

Keywords

Comments

Initialized with a single black (ON) cell at stage zero.
Rules 28, 52, 60, 148, 156, 180, 188, 532, 540, 564, 572, 660, 668, 692 and 700 also generate this sequence.
Apparently a duplicate of A173033. - R. J. Mathar, Mar 09 2016

References

  • S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Crossrefs

Cf. A269711.

Programs

  • Mathematica
    rule=20; stages=300;
ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *)
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)

Formula

Conjectures from Colin Barker, Mar 08 2016: (Start)
a(n) = 4*(2^n-1) =A028399(n+2) for n>0.
a(n) = 3*a(n-1)-2*a(n-2) for n>2.
G.f.: (1+x+2*x^2) / ((1-x)*(1-2*x)).
(End)

Extensions

a(9)-a(15) from Lars Blomberg, Apr 15 2016
Previous Showing 11-17 of 17 results.

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