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A067771
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Number of vertices in Sierpinski triangle of order n.
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13
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3, 6, 15, 42, 123, 366, 1095, 3282, 9843, 29526, 88575, 265722, 797163, 2391486, 7174455, 21523362, 64570083, 193710246, 581130735, 1743392202, 5230176603, 15690529806, 47071589415, 141214768242, 423644304723, 1270932914166
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history;
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OFFSET
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0,1
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COMMENTS
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This sequence represents another link from the product factor space Q X Q / {(1,1), (-1, -1)} to Sierpinski's triangle. The first "link" found was to sequence A048473. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Aug 05 2004
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REFERENCES
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Peter Wessendorf and Kristina Downing, personal communication.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..600
C. Lanius, Fractals
Eric Weisstein's World of Mathematics, Sierpinski Graph
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FORMULA
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a(n) = 3 + 3^1 + 3^2 + 3^3 + 3^4 +...+ 3^n= 3 + sum(k=1..n, 3^n).
a(1) = 3, a(n)=a(n-1)+3^n. a(n)=(3/2)*(1+3^n). - Zak Seidov, Mar 19 2007
a(n) = +4*a(n-1) -3*a(n-2)
G.f.: 3*(1-2*x)/((1-x)*(1-3*x)). [Colin Barker, Jan 10 2012]
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MATHEMATICA
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a=2; lst={}; Do[a=a*3-3; AppendTo[lst, a], {n, 0, 5!}]; lst [From Vladimir Joseph Stephan Orlovsky, Dec 25 2008]
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PROG
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(MAGMA) [(3/2)*(1+3^n): n in [0..30]]; // Vincenzo Librandi, Jun 20 2011
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CROSSREFS
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This is 3*A007051. Cf. A048473.
Cf. A003462, A007051, A034472, A024023 [From Vladimir Joseph Stephan Orlovsky, Dec 25 2008]
Sequence in context: A140824 A001433 A005368 * A182561 A056382 A028401
Adjacent sequences: A067768 A067769 A067770 * A067772 A067773 A067774
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KEYWORD
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nonn,easy
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AUTHOR
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Martin Wessendorf (martinw(AT)mail.ahc.umn.edu), Feb 09 2002
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EXTENSIONS
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More terms from Benoit Cloitre, Feb 22 2002
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STATUS
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approved
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