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

Author

Steven Beard, Dec 13 2016

Keywords

• nonn
• easy

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

Examples

			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
		

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).

Crossrefs

Cf. A000330, A047999, A279512.

Programs

Mathematica

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

PARI

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

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)

a(n) = 25*5^(n-1)+(2^(n+4)-37*5^n+21)/12. - Alan Michael Gómez Calderón, Oct 04 2023