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)
a(n) = 25*5^(n-1)+(2^(n+4)-37*5^n+21)/12. - Alan Michael Gómez Calderón, Oct 04 2023
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
KEYWORD
nonn,easy
AUTHOR
Steven Beard, Dec 13 2016
STATUS
approved