Steven Beard - cp's OEIS frontend

A293144 Number of vertices in a Menger Sponge constructed from a cubic lattice: a(n) = 20a(n-1) - 24A293143(n-1).

8, 64, 896, 15616, 295808, 5789440, 114790784, 2287878400, 45694209920, 913377753856, 18263504780672, 365237697021184, 7304494763023232, 146087821875273472, 2921739850525976960, 58434664314989709568, 1168692224736473884544

Offset: 0

Steven Beard, Oct 01 2017

a(n) is the number of vertices of a (3^n+1)^3 cubic lattice minus the number of vertices missing for the openings within the sponge. The cubic honeycomb can be constructed by joining 20 cubes of the previous term and subtracting the overlapping vertices of 24 faces (see example).

			For a(0) we start with a simple cube, having a(0) = 8 corners.
For a(1), the cube is subdivided into 27 smaller cubes forming a lattice of 64 vertices. 7 cubes are removed (but the cubes have no facial or internal vertices to remove until the next stage).
Twenty a(1) cubes, each with 64 vertices, are then combined to form the lattice for a(2). The overlapped vertices of 24 faces (each with 16 vertices) are removed. Thus a(2) = (20*64) - (24*16) = 1280 - 384 = 896. The faces of the cubes are the Sierpinski Carpet grid of A293143.

M. F. Hasler, Table of n, a(n) for n = 0..800 (Terms n = 1 .. 768 computed by Colin Barker.)
Eric Weisstein's World of Mathematics, Menger Sponge.
Wikipedia, Menger sponge.
Index entries for linear recurrences with constant coefficients, signature (32,-275,724,-480).

Cf. A293143, A034472.

CoefficientList[Series[8 (1 - 24 x + 131 x^2 - 156 x^3)/((1 - x) (1 - 3 x) (1 - 8 x) (1 - 20 x)), {x, 0, 15}], x] (* Michael De Vlieger, Oct 09 2017 *)

Vec(8*(1 - 24*x + 131*x^2 - 156*x^3) / ((1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)) + O(x^30)) \\ Colin Barker, Oct 09 2017

A293144(n)=(255+133*3^(n+1)+63*4^n*5^(n+1)+3553*8^(n-1))*64/11305 \\ M. F. Hasler, Oct 16 2017

From Colin Barker, Oct 02 2017, adjusted for initial a(0) = 8 by M. F. Hasler, Oct 16 2017: (Start)
G.f.: 8*(1 - 24*x + 131*x^2 - 156*x^3) / ((1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)).
a(n) = 64*(3/133 + 3^(1+n)/85 + 11*8^(n-1)/35 + 9*20^n/323).
a(n) = 32*a(n-1) - 275*a(n-2) + 724*a(n-3) - 480*a(n-4) for n > 3.
(End)
a(n) = (64*(133*3^(n+1) + 63*4^n*5^(n+1) + 3553*8^(n-1) + 255)) / 11305.

Edited to include initial term 8 by M. F. Hasler, Oct 16 2017

A293143 Number of vertex points in a Sierpinski Carpet grid subdivided into squares: a(n+1) = 8a(n) - 8(3^n+1), a(0) = 4.

Original entry on oeis.org

4, 16, 96, 688, 5280, 41584, 330720, 2639920, 21101856, 168762352, 1349941344, 10799058352, 86391049632, 691124145520, 5528980409568, 44231805012784, 353854325311008, 2830834258114288, 22646673031792992, 181173381154980016

Offset: 0

Steven Beard, Oct 01 2017

Figurate number sequence for the Sierpinski Carpet lattice. See the faces of the cubes in "Image 2" in the Wikipedia link of A293144 for an example of the construction grid of the Sierpinski Carpet.

			The carpet is formed by squares within a square grid. The initial term is a(0) = 4 for the corners of the unit square. For n = 1 there are 3 X 3 squares, the middle one being open (empty), with 16 vertex points. At the next stage of recursion, these become eight squares with open center, now based on a square of 10 X 10 points. The remaining center square is empty, missing 4 points, thus the next term is 100 - 4 = 96 for a(2). In the next stage there are 8 squares missing 4 points and the new center is missing 64, thus the 28 square grid now has 784 - 32 - 64 = 688 for a(3). This carpet sequence becomes the faces for the cubes in the Menger Sponge recursion of A293144.

Colin Barker, Table of n, a(n) for n = 0..1000 (Recomputed by M. F. Hasler to include the initial term 4.)
Eric Weisstein's World of Mathematics, Sierpinski Carpet.
Wikipedia, Sierpinski carpet.
Index entries for linear recurrences with constant coefficients, signature (12,-35,24).

Cf. A293144, A034472.

FoldList[8 #1 - 8 (3^(#2-1) + 1) &, 4, Range@ 18] (* Michael De Vlieger, Oct 02 2017 *)

prev=4; concat(prev, vector(20, n, prev=8*prev-8*(3^(n-1)+1))) \\ Colin Barker, Oct 08 2017

Vec(4*(1 - 8*x + 11*x^2)/((1 - x)*(1 - 3*x)*(1 - 8*x)) + O(x^30)) \\ Colin Barker, Oct 09 2017

A293143(n)=8*(5+11*2^(3*n-1)+7*3^n)/35 \\ M. F. Hasler, Oct 16 2017

From Colin Barker, Oct 02 2017, corrected for a(0) = 4 by M. F. Hasler, Oct 16 2017: (Start)
G.f.: 4*(1 - 8*x + 11*x^2) / ((1 - x)*(1 - 3*x)*(1 - 8*x)).
a(n) = 8*(5 + 11*2^(3*n-1) + 7*3^n) / 35.
a(n) = 12*a(n-1) - 35*a(n-2) + 24*a(n-3) for n > 2. (End)

Edited to start with a(0) = 4 by M. F. Hasler, Oct 16 2017

A289999 Sierpinski cuboctahedral numbers: a(n) = 164^n - 122^n + 9.

Original entry on oeis.org

13, 49, 217, 937, 3913, 16009, 64777, 260617, 1045513, 4188169, 16764937, 67084297, 268386313, 1073643529, 4294770697, 17179475977, 68718690313, 274876334089, 1099508482057, 4398040219657, 17592173461513, 70368719011849, 281474926379017, 1125899806179337, 4503599426043913, 18014398106828809

Offset: 0

Steven Beard, Sep 03 2017

Sierpinski cuboctahedron constructed by joining eight Sierpinski tetrahedra of sequence 4, 10, 34, 130, 514, 2050, 8194... (4^n*2)+2 (the double of A052539). This sequence is also Sierpinski recursion for the octahemioctahedron A274974.

Colin Barker, Table of n, a(n) for n = 0..1000
Wikipedia, Sierpinski tetrahedron.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A005902, A290396, A274974, A281699, A067771, A279511.

CoefficientList[Series[(13 - 42 x + 56 x^2)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 25}], x] (* Michael De Vlieger, Sep 03 2017 *)
Table[16*4^n-12*2^n+9,{n,0,30}] (* or *) LinearRecurrence[{7,-14,8},{13,49,217},30] (* Harvey P. Dale, Dec 31 2018 *)

Vec((13 - 42*x + 56*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Sep 03 2017

a(n) = 16*4^n - 12*2^n + 9 \\ Charles R Greathouse IV, Nov 03 2017

a(n) = -3*2^(n + 2) + 2^(2n + 4) + 9.
From Colin Barker, Sep 03 2017: (Start)
G.f.: (13 - 42*x + 56*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
(End)

A290396 a(n) = 32^n + 34^n + 6^(n+1) + 1.

Original entry on oeis.org

13, 55, 277, 1513, 8593, 49825, 292417, 1729153, 10275073, 61254145, 365945857, 2189371393, 13111037953, 78565515265, 470990340097, 2824331231233, 16939544543233, 101611496669185, 609565899227137, 3656983075356673, 21940249178406913, 131634897988091905, 789782999624318977

Offset: 0

Steven Beard, Jul 29 2017

One of several types of cuboctahedra with Sierpinski recursion. This Sierpinski cuboctahedron is a truncation of the six corners of the Sierpinski octahedron (A279512). The resulting Sierpinski cuboctahedron contains six square pyramids joined along the edges thus it can also be constructed by joining Sierpinski square pyramids (A281698).
Each face of the Sierpinski octahedron is a Sierpinski sieve with all triangular spaces completely formed. The triangles opening within the sieve, on each face, become tetrahedral excavations (cf. A067771). The third term above, for example, begins with an octahedron of 489 counters from A005900. From each of the eight faces a tetrahedron of four counters is removed. Thus the corresponding Sierpinski octahedron has 457 counters at this stage of recursion. Next, to construct the Sierpinski cuboctahedron, the truncation of the six corners is 30 counters each, thus 457 minus 180 equals 277, the third term of the Sierpinski cuboctahedral sequence shown above. For the next term in the sequence, two different sizes of tetrahedra are removed (and not only seen on the outer eight faces but also on the newly opened internal faces, now excavated in like manner.
To arrive at the next term, 1513, there are eight tetrahedra of 56 counters AND forty-eight tetrahedra of 4 counters each excavated from the octahedron, and then the six corners of 188 counters each are truncated thus: (56*8) + (4*48) = 640 total counters removed from the octahedral number 3281 (in A005900) to yield the Sierpinski octahedral number 2641 (in A279512, note that for subsequent terms the number of different size excavations of the octahedron increases, thus the next term would require three different sized excavations for the associated octahedron: 8 tetrahedra of 560 counters, 48 tetrahedra of 56 counters, and 288 tetrahedra of 4 counters).
Next truncate the six corners of 188 counters each: (6*188) = 1128. Sierpinski octahedron 2641 - 1128 = 1513, the fourth term in the Sierpinski cuboctahedral sequence above. The square pyramid of 188 counters truncated from each corner is found in A281698 by, in this case, taking the fourth term of 269 and subtracting its base of 81 counters.
The base of the square pyramid that is not counted is actually the square face of the remaining cuboctahedron. It is therefore easier to construct this Sierpinski cuboctahedral geometry by taking the six square pyramids of 269 counters each, subtracting 5 for the overlapping apices and then subtracting 96 to account for the edges where the square pyramids are overlapped: 6*269 = 1614 - 5 = 1609, - 96 = 1513, the fourth term shown in the Sierpinski cuboctahedral sequence above.

Colin Barker, Table of n, a(n) for n = 0..1000
Wikipedia, , Octahedron flake and Sierpinski tetrahedron.
Index entries for linear recurrences with constant coefficients, signature (13,-56,92,-48).

Cf. A005902, A279512, A274973, A281698, A067771.

Table[3*2^n + 3*4^n + 6^(n + 1) + 1, {n, 0, 22}] (* Michael De Vlieger, Jul 29 2017 *)

Vec((13 - 114*x + 290*x^2 - 204*x^3) / ((1 - x)*(1 - 2*x)*(1 - 4*x)*(1 - 6*x)) + O(x^30)) \\ Colin Barker, Jul 29 2017

a(n) = 3*2^n + 3*4^n + 6^(n+1) + 1 \\ Charles R Greathouse IV, Nov 03 2017

a(n) = 3*2^n + 3*4^n + 6^(n+1) + 1.
From Colin Barker, Jul 29 2017: (Start)
G.f.: (13 - 114*x + 290*x^2 - 204*x^3) / ((1 - x)*(1 - 2*x)*(1 - 4*x)*(1 - 6*x)).
a(n) = 13*a(n-1) - 56*a(n-2) + 92*a(n-3) - 48*a(n-4) for n>3.
(End)

A281698 a(n) = 52^(n-1) + 2^(2n-1) + 6^n + 1.

Original entry on oeis.org

5, 14, 55, 269, 1465, 8369, 48865, 288449, 1713025, 10210049, 60993025, 364899329, 2185181185, 13094268929, 78498422785, 470721937409, 2823257554945, 16935249707009, 101594317062145, 609497180274689, 3656708198498305, 21939149668876289, 131630499945775105

Offset: 0

Steven Beard, Jan 27 2017

Similar to A279511 Sierpinski square-based pyramid but with tetrahedral openings as found in the structure of the Sierpinski octahedron A279512.

Colin Barker, Table of n, a(n) for n = 0..1000
Wikipedia, Sierpinski triangle, see section on higher dimensional analogs.
Index entries for linear recurrences with constant coefficients, signature (13,-56,92,-48).

Cf. A000330, A047999, A279511, A279512.

A281698:=n->5*2^(n-1) + 2^(2*n-1) + 6^n + 1: seq(A281698(n), n=0..30); # Wesley Ivan Hurt, Apr 09 2017

Table[5*2^(n - 1) + 2^(2 n - 1) + 6^n + 1, {n, 0, 22}] (* or *)
LinearRecurrence[{13, -56, 92, -48}, {5, 14, 55, 269}, 23] (* or *)
CoefficientList[Series[(5 - 51 x + 153 x^2 - 122 x^3)/((1 - x) (1 - 2 x) (1 - 4 x) (1 - 6 x)), {x, 0, 22}], x] (* Michael De Vlieger, Jan 28 2017 *)

Vec((5 - 51*x + 153*x^2 - 122*x^3) / ((1 - x)*(1 - 2*x)*(1 - 4*x)*(1 - 6*x)) + O(x^30)) \\ Colin Barker, Jan 28 2017

a(n) = 5*2^(n-1) + 2^(2*n-1) + 6^n + 1 \\ Charles R Greathouse IV, Jan 29 2017

From Colin Barker, Jan 28 2017: (Start)
a(n) = 13*a(n-1) - 56*a(n-2) + 92*a(n-3) - 48*a(n-4) for n>3.
G.f.: (5 - 51*x + 153*x^2 - 122*x^3) / ((1 - x)*(1 - 2*x)*(1 - 4*x)*(1 - 6*x)).
(End)

A281699 Sierpinski stellated octahedron numbers: a(n) = 2(-32^(n+1) + 2^(2n+3) + 5).

Original entry on oeis.org

14, 50, 218, 938, 3914, 16010, 64778, 260618, 1045514, 4188170, 16764938, 67084298, 268386314, 1073643530, 4294770698, 17179475978, 68718690314, 274876334090, 1099508482058, 4398040219658, 17592173461514, 70368719011850, 281474926379018, 1125899806179338, 4503599426043914, 18014398106828810

Offset: 0

Steven Beard, Jan 27 2017

Stella octangula with Sierpinski recursion.

Colin Barker, Table of n, a(n) for n = 0..1000
Wikipedia, Sierpinski triangle, see section on higher dimensional analogs.
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A007588, A027693, A052539, A052548, A067771, A178789, A233774, A279511.

Table[8 (2^(2 n + 1) + 2) - 6 (2^(n + 1) + 1), {n, 0, 25}] (* or *)
LinearRecurrence[{7, -14, 8}, {14, 50, 218}, 26] (* or *)
CoefficientList[Series[2 (7 - 24 x + 32 x^2)/((1 - x) (1 - 2 x) (1 - 4 x)), {x, 0, 25}], x] (* Michael De Vlieger, Jan 28 2017 *)

Vec(2*(7 - 24*x + 32*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Jan 28 2017

a(n) = 16*4^n - 12*2^n + 10 \\ Charles R Greathouse IV, Jan 29 2017

a(n) = 8*(2^(2*n+1)+2) - 6*(2^(n+1)+1).
From Colin Barker, Jan 28 2017: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>2.
G.f.: 2*(7 - 24*x + 32*x^2) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)

A279512 Sierpinski octahedron numbers a(n) = 26^n + 32^n + 1.

Original entry on oeis.org

6, 19, 85, 457, 2641, 15649, 93505, 560257, 3360001, 20156929, 120935425, 725600257, 4353576961, 26121412609, 156728377345, 940370067457, 5642220011521, 33853319282689, 203119914123265, 1218719481593857, 7312316883271681, 43873901287047169, 263243407697117185

Offset: 0

Steven Beard, Dec 14 2016

Sierpinski recursion applied to octahedron. Cf. A279511 for square pyramids.

Colin Barker, Table of n, a(n) for n = 0..1000
Wikipedia, Octahedron flake and Sierpinski tetrahedron.
Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Cf. A005900, A047999, A279511.

LinearRecurrence[{9, -20, 12}, {6, 19, 85}, 50] (* or *) Table[2*6^n + 3*2^n + 1, {n,0,50}] (* G. C. Greubel, Dec 22 2016 *)

Vec((6 - 35*x + 34*x^2) / ((1 - x)*(1 - 2*x)*(1 - 6*x)) + O(x^30)) \\ Colin Barker, Dec 15 2016

def a(n): return 2*6**n + 3*2**n + 1
print([a(n) for n in range(23)]) # Michael S. Branicky, Jun 19 2021

a(n) = 3*2^n + 2^(n+1)*3^n + 1.
a(n) = 6a(n-1) - 6(2^n+1) + 1.
a(n) = 6a(n-1) - (3*2^(n+1) + 5).
a(n) = 2*6^n + 3*2^n + 1.
From Colin Barker, Dec 15 2016: (Start)
a(n) = 9*a(n-1) - 20*a(n-2) + 12*a(n-3) for n>2.
G.f.: (6 - 35*x + 34*x^2) / ((1 - x)*(1 - 2*x)*(1 - 6*x)).
(End)

Incorrect terms corrected by Colin Barker, Dec 15 2016

A279511 Sierpinski square-based pyramid numbers: a(n) = 5*a(n-1) - (2^(n+1)+7).

Original entry on oeis.org

5, 14, 55, 252, 1221, 6034, 30035, 149912, 749041, 3744174, 18718815, 93589972, 467941661, 2339691914, 11698426795, 58492068432, 292460211081, 1462300793254, 7311503441975, 36557516161292, 182787578709301, 913937889352194, 4569689438372355, 22848447175084552

Offset: 0

Steven Beard, Dec 13 2016

Square pyramid where each face of the four triangular faces of the pyramid is a Sierpinski gasket. Similarly, a Sierpinski tetrahedron is sequence 4, 10, 34, 130, 514, 2050, 8194 (4^n*2)+2 (the double of A052539). The related octahedral form (creating tetrahedral openings), is A279512.

The sequence gives the number of vertices of this Sierpinski pyramid - see example. - M. F. Hasler, Oct 16 2017

			At iteration n=0, we simply have a square pyramid with 4+1 = 5 = a(0) vertices.
At iteration n=1, we have 5 copies of the elementary pyramid. However, some of the vertices coincide, and duplicate counts must be subtracted. The 4 vertices of the base of the top pyramid are also the top vertices of the 4 lower pyramids. The lower pyramids touch at the middle of the sides (these points were counted twice), and also in the very middle of the large square base (this point was counted 4 times). Thus a(1) = 25 - 4 - 4 - 3 = 14. - _M. F. Hasler_, Oct 16 2017

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Sierpinski Sieve
Wikipedia, Sierpinski triangle, see section "analogues in higher dimensions."
Index entries for linear recurrences with constant coefficients, signature (8,-17,10).

Cf. A000330, A047999, A279512.

LinearRecurrence[{8,-17,10},{5,14,55},30] (* Harvey P. Dale, May 24 2017 *)

Vec((5-26*x+28*x^2) / ((1-x)*(1-2*x)*(1-5*x)) + O(x^30)) \\ Colin Barker, Dec 15 2016

a(n) = 5*a(n-1) - (2^(n+1)+7).
From Colin Barker, Dec 15 2016: (Start)
a(n) = 8*a(n-1) - 17*a(n-2) + 10*a(n-3) for n > 2.
G.f.: (5-26*x+28*x^2) / ((1-x)*(1-2*x)*(1-5*x)). (End)
a(n) = 25*5^(n-1)+(2^(n+4)-37*5^n+21)/12. - Alan Michael Gómez Calderón, Oct 04 2023

A274974 Centered octahemioctahedral numbers: a(n) = (4n^3+24n^2+8*n+3)/3.

Original entry on oeis.org

1, 13, 49, 117, 225, 381, 593, 869, 1217, 1645, 2161, 2773, 3489, 4317, 5265, 6341, 7553, 8909, 10417, 12085, 13921, 15933, 18129, 20517, 23105, 25901, 28913, 32149, 35617, 39325, 43281, 47493, 51969, 56717, 61745, 67061, 72673, 78589, 84817, 91365, 98241

Offset: 0

Steven Beard, Jul 13 2016

Related to a faceting of the cuboctahedron, sharing the same triangular faces. The octahemioctahedron has the same edge and vertex arrangement as the cuboctahedron (as does A274973). Beginning with the third term, the six square faces are each now "missing" a square pyramid of size 1, 5, 14, 30, 55, 91...(A000330). See A274973 centered cubohemioctahedron for similar cuboctahedral faceting but without the triangular faces.

Steven Beard, Music track made with this sequence
Wikipedia, Octahemioctahedron

Cf. A005902 (centered cuboctahedral numbers), A274973 (centered cubohemioctahedral numbers).

CoefficientList[Series[(-5 x^3 + 3 x^2 + 9 x + 1)/(x - 1)^4, {x, 0, 40}], x] (* or *)
Table[(4 n^3 + 24 n^2 + 8 n+3)/3, {n, 41}] (* Michael De Vlieger, Jul 13 2016 *)

a(n)=(4*n^3+24*n^2+8*n+3)/3 \\ Charles R Greathouse IV, Nov 03 2017

a(n) = (4*n^3+24*n^2+8*n+3)/3.

G.f.: (-5*x^3+3*x^2+9*x+1)/(x-1)^4.

A274973 Centered cubohemioctahedral numbers: a(n) = 2n^3+9n^2+n+1.

Original entry on oeis.org

1, 13, 55, 139, 277, 481, 763, 1135, 1609, 2197, 2911, 3763, 4765, 5929, 7267, 8791, 10513, 12445, 14599, 16987, 19621, 22513, 25675, 29119, 32857, 36901, 41263, 45955, 50989, 56377, 62131, 68263, 74785, 81709, 89047, 96811, 105013, 113665, 122779, 132367

Offset: 0

Steven Beard, Jul 13 2016

A faceting of the cuboctahedron, sharing the same square faces. The cubohemioctahedron has the same edge and vertex arrangement as the cuboctahedron. Beginning with the fourth term, the eight tetrahedral faces are each now "missing" a tetrahedron of size 1,4,10,20,35...(A000292). See A274974 centered octahemioctahedron for similar cuboctahedral faceting but with the square faces "missing."

Wikipedia, Cubohemioctahedron
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A005902 (centered cuboctahedral numbers), A274974 (centered octahemioctahedral numbers).

Table[2 n^3 + 9 n^2 + n + 1, {n, 40}] (* Michael De Vlieger, Jul 13 2016 *)
LinearRecurrence[{4,-6,4,-1},{1,13,55,139},40] (* Harvey P. Dale, Feb 18 2024 *)

a(n)=2*n^3+9*n^2+n+1 \\ Charles R Greathouse IV, Jul 14 2016

a(n) = 2*n^3+9*n^2+n+1.

G.f.: (-7*x^3+9*x^2+9*x+1)/(x-1)^4.

cp's OEIS Frontend

User: Steven Beard

A293144 Number of vertices in a Menger Sponge constructed from a cubic lattice: a(n) = 20*a(n-1) - 24*A293143(n-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A293143 Number of vertex points in a Sierpinski Carpet grid subdivided into squares: a(n+1) = 8*a(n) - 8*(3^n+1), a(0) = 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

A289999 Sierpinski cuboctahedral numbers: a(n) = 16*4^n - 12*2^n + 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A290396 a(n) = 3*2^n + 3*4^n + 6^(n+1) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A281698 a(n) = 5*2^(n-1) + 2^(2*n-1) + 6^n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A281699 Sierpinski stellated octahedron numbers: a(n) = 2*(-3*2^(n+1) + 2^(2n+3) + 5).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A293144 Number of vertices in a Menger Sponge constructed from a cubic lattice: a(n) = 20a(n-1) - 24A293143(n-1).

A293143 Number of vertex points in a Sierpinski Carpet grid subdivided into squares: a(n+1) = 8a(n) - 8(3^n+1), a(0) = 4.

A289999 Sierpinski cuboctahedral numbers: a(n) = 164^n - 122^n + 9.

A290396 a(n) = 32^n + 34^n + 6^(n+1) + 1.

A281698 a(n) = 52^(n-1) + 2^(2n-1) + 6^n + 1.

A281699 Sierpinski stellated octahedron numbers: a(n) = 2(-32^(n+1) + 2^(2n+3) + 5).

A279512 Sierpinski octahedron numbers a(n) = 26^n + 32^n + 1.

A274974 Centered octahemioctahedral numbers: a(n) = (4n^3+24n^2+8*n+3)/3.

A274973 Centered cubohemioctahedral numbers: a(n) = 2n^3+9n^2+n+1.