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A283070
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Sierpinski tetrahedron or tetrix numbers: a(n) = 2*4^n + 2.
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3
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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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).
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=2*4^n+2 \\ Charles R Greathouse IV, Feb 28 2017
(PARI) Vec(2*(2 - 5*x) / ((1 - x)*(1 - 4*x)) + O(x^30)) \\ Colin Barker, Mar 02 2017
(Python)
def a(n): return 2*4**n + 2
print([a(n) for n in range(25)]) # Michael S. Branicky, Aug 29 2021
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CROSSREFS
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Subsequence of A016957.
Cf. A052539, A279511, A279512.
First bisection of A052548, A087288; second bisection of A049332, A133140, A135440.
Cf. A002023 (edge count).
Sequence in context: A006343 A149173 A149174 * A222631 A030003 A339845
Adjacent sequences: A283067 A283068 A283069 * A283071 A283072 A283073
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KEYWORD
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nonn,easy
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AUTHOR
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Peter M. Chema, Feb 28 2017
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EXTENSIONS
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Entry revised by Editors of OEIS, Mar 01 2017
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STATUS
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approved
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