Original entry on oeis.org
3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888
Offset: 0
- Jason I. Brown, Discrete Structures and Their Interactions, CRC Press, 2013, p. 71.
- T. Ito, Method, equipment, program and storage media for producing tables, Publication number JP2004-272104A, Japan Patent Office (written in Japanese, a(2)=12, a(3)=24, a(4)=48, a(5)=96, a(6)=192, a(7)=384 (a(7)=284 was corrected)).
- Kenji Ohkuma, Atsuhiro Yamagishi and Toru Ito, Cryptography Research Group Technical report, IT Security Center, Information-Technology Promotion Agency, JAPAN.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- K. Bezdek and Tudor Zamfirescu, A Characterization of 3-dimensional Convex Sets with an Infinite X-ray Number, in: Coll. Math. Soc. J. Bolyai 63., Intuitive Geometry, Szeged (Hungary), North-Holland, Amsterdam, 1991, pp. 33-38.
- Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
- Yuri Brudnyi and Pavel Shvartsman, Generalizations of Whitney's extension theorem, International Mathematics Research Notices 1994.3 (1994): 129-139.
- J. W. Cannon and P. Wagreich, Growth functions of surface groups, Mathematische Annalen, 1992, Volume 293, pp. 239-257. See Prop. 3.1.
- Tomislav Došlić, Kepler-Bouwkamp Radius of Combinatorial Sequences, Journal of Integer Sequences, Vol. 17, 2014, #14.11.3.
- John Elias, Illustration: 2^n+1 hexagram perimeters
- Lukas Fleischer and Jeffrey Shallit, Words With Few Palindromes, Revisited, arxiv preprint arXiv:1911.12464 [cs.FL], November 27 2019.
- A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
- Tanya Khovanova, Recursive Sequences
- Roberto Rinaldi and Marco Ripà, Optimal cycles enclosing all the nodes of a k-dimensional hypercube, arXiv:2212.11216 [math.CO], 2022.
- Edwin Soedarmadji, Latin Hypercubes and MDS Codes, Discrete Mathematics, Volume 306, Issue 12, Jun 28 2006, Pages 1232-1239
- D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739 - 741, 1968.
- Index entries for linear recurrences with constant coefficients, signature (2).
Coordination sequences for triangular tilings of hyperbolic space:
A001630,
A007283,
A054886,
A078042,
A096231,
A163876,
A179070,
A265057,
A265058,
A265059,
A265060,
A265061,
A265062,
A265063,
A265064,
A265065,
A265066,
A265067,
A265068,
A265069,
A265070,
A265071,
A265072,
A265073,
A265074,
A265075,
A265076,
A265077.
Subsequence of the following sequences:
A029744,
A029747,
A029748,
A029750,
A362804 (after 3),
A364494,
A364496,
A364289,
A364291,
A364292,
A364295,
A364497,
A364964,
A365422.
Row sums of (5, 1)-Pascal triangle
A093562 and of (1, 5) Pascal triangle
A096940.
-
a007283 = (* 3) . (2 ^)
a007283_list = iterate (* 2) 3
-- Reinhard Zumkeller, Mar 18 2012, Feb 20 2012
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[3*2^n: n in [0..30]]; // Vincenzo Librandi, May 18 2011
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A007283:=n->3*2^n; seq(A007283(n), n=0..50); # Wesley Ivan Hurt, Dec 03 2013
-
Table[3(2^n), {n, 0, 32}] (* Alonso del Arte, Mar 24 2011 *)
-
A007283(n):=3*2^n$
makelist(A007283(n),n,0,30); /* Martin Ettl, Nov 11 2012 */
-
a(n)=3*2^n
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a(n)=3<Charles R Greathouse IV, Oct 10 2012
-
def A007283(n): return 3<Chai Wah Wu, Feb 14 2023
-
(List.fill(40)(2: BigInt)).scanLeft(1: BigInt)( * ).map(3 * ) // _Alonso del Arte, Nov 28 2019
A182368
Triangle T(n,k), n>=1, 0<=k<=n^2, read by rows: row n gives the coefficients of the chromatic polynomial of the square grid graph G_(n,n), highest powers first.
Original entry on oeis.org
1, 0, 1, -4, 6, -3, 0, 1, -12, 66, -216, 459, -648, 594, -323, 79, 0, 1, -24, 276, -2015, 10437, -40614, 122662, -292883, 557782, -848056, 1022204, -960627, 682349, -346274, 112275, -17493, 0, 1, -40, 780, -9864, 90798, -647352, 3714180, -17590911, 69997383
Offset: 1
3 example graphs: o---o---o
. | | |
. o---o o---o---o
. | | | | |
. o o---o o---o---o
Graph: G_(1,1) G_(2,2) G_(3,3)
Vertices: 1 4 9
Edges: 0 4 12
The square grid graph G_(2,2) is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => row 2 = [1, -4, 6, -3, 0].
Triangle T(n,k) begins:
1, 0;
1, -4, 6, -3, 0;
1, -12, 66, -216, 459, -648, 594, ...
1, -24, 276, -2015, 10437, -40614, 122662, ...
1, -40, 780, -9864, 90798, -647352, 3714180, ...
1, -60, 1770, -34195, 486210, -5421612, 49332660, ...
1, -84, 3486, -95248, 1926585, -30755376, 403410654, ...
1, -112, 6216, -227871, 6205479, -133865298, 2382122274, ...
1, -144, 10296, -487280, 17169852, -480376848, 11114098408, ...
...
Sums of absolute values of row elements give:
A080690(n).
-
Reverse /@ CoefficientList[Table[ChromaticPolynomial[GridGraph[{n, n}], x], {n, 5}], x] // Flatten (* Eric W. Weisstein, May 01 2017 *)
A067771
Number of vertices in Sierpiński triangle of order n.
Original entry on oeis.org
3, 6, 15, 42, 123, 366, 1095, 3282, 9843, 29526, 88575, 265722, 797163, 2391486, 7174455, 21523362, 64570083, 193710246, 581130735, 1743392202, 5230176603, 15690529806, 47071589415, 141214768242, 423644304723, 1270932914166
Offset: 0
Martin Wessendorf (martinw(AT)mail.ahc.umn.edu), Feb 09 2002
Order 0 is a triangle, so a(0) = 3.
Order 1 has three corners (degree 2) and three other vertices, so a(1) = 6.
3 example graphs: o
/ \
o---o
/ \ / \
o o---o---o
/ \ / \ / \
o o---o o---o o---o
/ \ / \ / \ / \ / \ / \ / \
o---o o---o---o o---o---o---o---o
Graph: S_1 S_2 S_3
Vertices: 3 6 15
Edges: 3 9 27
- Peter Wessendorf and Kristina Downing, personal communication.
- Vincenzo Librandi, Table of n, a(n) for n = 0..600
- Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
- Paul Bradley and Peter Rowley, Orbits on k-subsets of 2-transitive Simple Lie-type Groups, 2014.
- A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
- András Kaszanyitzky, Triangular fractal approximating graphs and their covering paths and cycles, arXiv:1710.09475 [math.CO], 2017. See Table 2.
- C. Lanius, Fractals.
- Eric Weisstein's World of Mathematics, Dorogovtsev-Goltsev-Mendes Graph.
- Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph.
- Eric Weisstein's World of Mathematics, Total Domination Number.
- Index entries for linear recurrences with constant coefficients, signature (4,-3).
-
[(3/2)*(1+3^n): n in [0..30]]; // Vincenzo Librandi, Jun 20 2011
-
LinearRecurrence[{4, -3}, {3, 6}, 26] (* or *)
CoefficientList[Series[3 (1 - 2 x)/((1 - x) (1 - 3 x)), {x, 0, 25}], x] (* Michael De Vlieger, Feb 02 2017 *)
Table[3/2 (3^n + 1), {n, 0, 20}] (* Eric W. Weisstein, Jan 14 2024 *)
3/2 (3^Range[0, 20] + 1) (* Eric W. Weisstein, Jan 14 2024 *)
A212084
Triangle T(n,k), n>=0, 0<=k<=2n, read by rows: row n gives the coefficients of the chromatic polynomial of the complete bipartite graph K_(n,n), highest powers first.
Original entry on oeis.org
1, 1, -1, 0, 1, -4, 6, -3, 0, 1, -9, 36, -75, 78, -31, 0, 1, -16, 120, -524, 1400, -2236, 1930, -675, 0, 1, -25, 300, -2200, 10650, -34730, 75170, -102545, 78610, -25231, 0, 1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, 5552680, -6796926, 4787174
Offset: 0
3 example graphs: +-----------+
. o o o o o o |
. | |\ /| |\ /|\ /|\ /
. | | X | | X | X | X
. | |/ \| |/ \|/ \|/ \
. o o o o o o |
. +-----------+
Graph: K_(1,1) K_(2,2) K_(3,3)
Vertices: 2 4 6
Edges: 1 4 9
The complete bipartite graph K_(2,2) is the cycle graph C_4 with chromatic polynomial q^4 -4*q^3 +6*q^2 -3*q => row 2 = [1, -4, 6, -3, 0].
Triangle T(n,k) begins:
1;
1, -1, 0;
1, -4, 6, -3, 0;
1, -9, 36, -75, 78, -31, 0;
1, -16, 120, -524, 1400, -2236, 1930, -675, ...
1, -25, 300, -2200, 10650, -34730, 75170, -102545, ...
1, -36, 630, -6915, 52080, -279142, 1074822, -2942445, ...
...
Row sums and last elements of rows give:
A000007.
Sums of absolute values of row elements give:
A048163(n+1).
-
P:= n-> add(Stirling2(n, k) *mul(q-i, i=0..k-1) *(q-k)^n, k=0..n):
T:= n-> seq(coeff(P(n), q, 2*n-k), k=0..2*n):
seq(T(n), n=1..8);
A193283
Triangle T(n,k), n>=1, 0<=k<=n*(n+1)/2, read by rows: row n gives the coefficients of the chromatic polynomial of the n X n X n triangular grid, highest powers first.
Original entry on oeis.org
1, 0, 1, -3, 2, 0, 1, -9, 32, -56, 48, -16, 0, 1, -18, 144, -672, 2016, -4031, 5368, -4584, 2272, -496, 0, 1, -30, 419, -3612, 21477, -93207, 304555, -761340, 1463473, -2152758, 2385118, -1929184, 1075936, -369824, 58976, 0
Offset: 1
4 example graphs: o
/ \
o o---o
/ \ / \ / \
o o---o o---o---o
/ \ / \ / \ / \ / \ / \
o o---o o---o---o o---o---o---o
n: 1 2 3 4
Vertices: 1 3 6 10
Edges: 0 3 9 18
The 2 X 2 X 2 triangular grid is equal to the cycle graph C_3 with chromatic polynomial q^3 -3*q^2 +2*q => [1, -3, 2, 0].
Triangle T(n,k) begins:
1, 0;
1, -3, 2, 0;
1, -9, 32, -56, 48, -16, 0;
1, -18, 144, -672, 2016, -4031, 5368, ...
1, -30, 419, -3612, 21477, -93207, 304555, ...
1, -45, 965, -13115, 126720, -925528, 5303300, ...
...
A370933
Number of pairs of antipodal vertices in the level n>1 Sierpiński triangle graph.
Original entry on oeis.org
6, 15, 42, 132, 456, 1680, 6432, 25152, 99456, 395520, 1577472, 6300672, 25184256, 100700160, 402726912, 1610760192, 6442745856, 25770393600, 103080394752, 412319219712, 1649272160256, 6597079203840, 26388297940992, 105553154015232, 422212540563456, 1688850011258880, 6755399743045632
Offset: 2
3 example graphs: o
/ \
o---o
/ \ / \
o o---o---o
/ \ / \ / \
o o---o o---o o---o
/ \ / \ / \ / \ / \ / \ / \
o---o o---o---o o---o---o---o---o
Graph: S_1 S_2 S_3
For S_2, there are 3 pairs of corners and 3 pairs of a corner and a middle vertex, so a(2) = 6.
- Paolo Xausa, Table of n, a(n) for n = 2..1000
- Index entries for linear recurrences with constant coefficients, signature (6,-8).
- Allan Bickle, Properties of Sierpinski Triangle Graphs, Springer PROMS 448 (2021) 295-303.
- A. Hinz, S. Klavzar, and S. Zemljic, A survey and classification of Sierpinski-type graphs, Discrete Applied Mathematics 217 3 (2017), 565-600.
- Eric Weisstein's World of Mathematics, Sierpiński Gasket Graph
Cf.
A375256 (antipodal pairs in Hanoi graphs).
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A370933[n_] := 3*2^(n - 3)*(2^(n - 2) + 3);
Array[A370933, 30, 2] (* or *)
LinearRecurrence[{6, -8}, {6, 15}, 30] (* Paolo Xausa, Sep 23 2024 *)
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a(n) = 3*2^(n-3)*(2^(n-2)+3); \\ Michel Marcus, Aug 08 2024
A295189
Chromatic invariant of the n-Sierpinski gasket graph.
Original entry on oeis.org
1, 1, 27, 20346417, 505719202397835049370570187
Offset: 1
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