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

%I A279511 #37 Feb 16 2025 08:33:38
%S A279511 5,14,55,252,1221,6034,30035,149912,749041,3744174,18718815,93589972,
%T A279511 467941661,2339691914,11698426795,58492068432,292460211081,
%U A279511 1462300793254,7311503441975,36557516161292,182787578709301,913937889352194,4569689438372355,22848447175084552
%N A279511 Sierpinski square-based pyramid numbers: a(n) = 5*a(n-1) - (2^(n+1)+7).
%C A279511 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.
%C A279511 The sequence gives the number of vertices of this Sierpinski pyramid - see example. - _M. F. Hasler_, Oct 16 2017
%H A279511 Colin Barker, <a href="/A279511/b279511.txt">Table of n, a(n) for n = 0..1000</a>
%H A279511 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SierpinskiSieve.html">Sierpinski Sieve</a>
%H A279511 Wikipedia, <a href="https://en.wikipedia.org/wiki/Sierpinski_triangle">Sierpinski triangle</a>, see section "analogues in higher dimensions."
%H A279511 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-17,10).
%F A279511 a(n) = 5*a(n-1) - (2^(n+1)+7).
%F A279511 From _Colin Barker_, Dec 15 2016: (Start)
%F A279511 a(n) = 8*a(n-1) - 17*a(n-2) + 10*a(n-3) for n > 2.
%F A279511 G.f.: (5-26*x+28*x^2) / ((1-x)*(1-2*x)*(1-5*x)). (End)
%F A279511 a(n) = 25*5^(n-1)+(2^(n+4)-37*5^n+21)/12. - _Alan Michael Gómez Calderón_, Oct 04 2023
%e A279511 At iteration n=0, we simply have a square pyramid with 4+1 = 5 = a(0) vertices.
%e A279511 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
%t A279511 LinearRecurrence[{8,-17,10},{5,14,55},30] (* _Harvey P. Dale_, May 24 2017 *)
%o A279511 (PARI) Vec((5-26*x+28*x^2) / ((1-x)*(1-2*x)*(1-5*x)) + O(x^30)) \\ _Colin Barker_, Dec 15 2016
%Y A279511 Cf. A000330, A047999, A279512.
%K A279511 nonn,easy
%O A279511 0,1
%A A279511 _Steven Beard_, Dec 13 2016