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A279511 Sierpinski square-based pyramid numbers: a(n) = 5*a(n-1) - (2^(n+1)+7). 6
5, 14, 55, 252, 1221, 6034, 30035, 149912, 749041, 3744174, 18718815, 93589972, 467941661, 2339691914, 11698426795, 58492068432, 292460211081, 1462300793254, 7311503441975, 36557516161292, 182787578709301, 913937889352194, 4569689438372355, 22848447175084552 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

LINKS

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).

FORMULA

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)

EXAMPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A000330, A047999, A279512.

Sequence in context: A177049 A127922 A262247 * A281698 A268814 A165517

Adjacent sequences:  A279508 A279509 A279510 * A279512 A279513 A279514

KEYWORD

nonn,easy,changed

AUTHOR

Steven Beard, Dec 13 2016

STATUS

approved

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Last modified October 23 00:06 EDT 2017. Contains 293782 sequences.