A279512 -id:A279512 - cp's OEIS frontend

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

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

A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

Original entry on oeis.org

4, 10, 34, 130, 514, 2050, 8194, 32770, 131074, 524290, 2097154, 8388610, 33554434, 134217730, 536870914, 2147483650, 8589934594, 34359738370, 137438953474, 549755813890, 2199023255554, 8796093022210, 35184372088834, 140737488355330, 562949953421314

Offset: 0

Peter M. Chema, Feb 28 2017

Number of vertices required to make a Sierpinski tetrahedron or tetrix of side length 2^n. The sum of the vertices (balls) plus line segments (rods) of one tetrix equals the vertices of its larger, adjacent iteration. See formula.
Equivalently, the number of vertices in the (n+1)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 17 2017
Also the independence number of the (n+2)-Sierpinski tetrahedron graph. - Eric W. Weisstein, Aug 29 2021
Final digit alternates 4 and 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Sierpinski Tetrahedron Graph
Eric Weisstein's World of Mathematics, Tetrix
Eric Weisstein's World of Mathematics, Vertex Count
Index entries for linear recurrences with constant coefficients, signature (5,-4).

Subsequence of A016957.
Cf. A052539, A279511, A279512.
First bisection of A052548, A087288; second bisection of A049332, A133140, A135440.
Cf. A002023 (edge count).

Table[2 4^n + 2, {n, 0, 30}] (* Bruno Berselli, Feb 28 2017 *)
2 (4^Range[0, 20] + 1) (* Eric W. Weisstein, Aug 17 2017 *)
LinearRecurrence[{5, -4}, {4, 10}, 20] (* Eric W. Weisstein, Aug 17 2017 *)
CoefficientList[Series[-((2 (-2 + 5 x))/(1 - 5 x + 4 x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, Aug 17 2017 *)

a(n)=2*4^n+2 \\ Charles R Greathouse IV, Feb 28 2017

Vec(2*(2 - 5*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Mar 02 2017

def a(n): return 2*4**n + 2
print([a(n) for n in range(25)]) # Michael S. Branicky, Aug 29 2021

G.f.: 2*(2 - 5*x)/((1 - x)*(1 - 4*x)).
a(n) = 5*a(n-1) - 4*a(n-2) for n > 1.
a(n+1) = a(n) + A002023(n).
a(n) = 2*A052539(n) = A188161(n) - 1 = A087289(n) + 1 = A056469(2*n+2) = A261723(4*n+1).
E.g.f.: 2*(exp(4*x) + exp(x)). - G. C. Greubel, Aug 17 2017

Entry revised by Editors of OEIS, Mar 01 2017

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)

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)

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

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A283070 Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.

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

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

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

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