

A048473


a(0)=1, a(n) = 3*a(n1) + 2; a(n) = 2*3^n  1.


51



1, 5, 17, 53, 161, 485, 1457, 4373, 13121, 39365, 118097, 354293, 1062881, 3188645, 9565937, 28697813, 86093441, 258280325, 774840977, 2324522933, 6973568801, 20920706405, 62762119217, 188286357653, 564859072961
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OFFSET

0,2


COMMENTS

The number of triangles (of all sizes, including holes) in Sierpiński's triangle after n inscriptions.  Lee Reeves, May 10 2004
The sequence is not only related to Sierpiński's triangle, but also to "Floret's cube" and the quaternion factor space Q X Q / {(1,1), (1,1)}. It can be written as a_n = ves((A+1)x)^n) as described at the Math Forum Discussions link.  Creighton Dement, Jul 28 2004
Relation to C(n) = Collatz function iteration using only odd steps: If we look for record subsequences where C(n) > n, this subsequence starts at 2^n  1 and stops at the local maximum of 2*3^n  1. Examples: [3,5], [7,11,17], [15,23,35,53], ..., [127,191,287,431,647,971,1457].  Lambert Klasen, Mar 11 2005
Group the natural numbers so that the (2n1)th group sum is a multiple of the (2n)th group containing one term. (1,2),(3),(4,5,6,7,8,9,10,11),(12),(13,14,15,16,17,18,19,...38),(39),(40,41,...,118,119),(120), (121,122,123,...) ... a(n) = {the sum of the terms of (2n1)th group}/{the term of (2n)th group}. The first term of the odd numbered group is given by A003462. The only term of even numbered group is given by A029858.  Amarnath Murthy, Aug 01 2005
a(n)+1 = A008776(n); it appears that this gives the number of terms in the (n+1)th "gap" of numbers missing in A171884.  M. F. Hasler, May 09 2013
Sum of nth row of triangle of powers of 3: 1; 1 3 1; 1 3 9 3 1; 1 3 9 27 9 3 1; ...  Philippe Deléham, Feb 23 2014
For n >= 3, also the number of dominating sets in the nhelm graph.  Eric W. Weisstein, May 28 2017
The number of elements of length <= n in the free group on two generators.  Anton Mellit, Aug 10 2017


REFERENCES

Theoni Pappas, Math Stuff, Wide World Publ/Tetra, San Carlos CA, page 15, 2002


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Dement, A paper on floretions and quaternions, some questions, The Math Forum.
Eric Weisstein's World of Mathematics, Dominating Set
Eric Weisstein's World of Mathematics, Helm Graph
Index entries for linear recurrences with constant coefficients, signature (4,3).


FORMULA

nth difference of a(n), a(n1), ..., a(0) is 2^(n+1) for n=1, 2, 3, ...
a(0)=1, a(n) = a(n1) + 3^n + 3^(n1).  Lee Reeves, May 10 2004
a(n) = (3^n + 3^(n+1)  2)/2.  Creighton Dement, Jul 31 2004
(1, 5, 17, 53, 161, ...) = Ternary (1, 12, 122, 1222, 12222, ...).  Gary W. Adamson, May 02 2005
Row sums of triangle A134347. Also, binomial transform of A046055: (1, 4, 8, 16, 32, 64, ...); and double binomial transform of A010684: (1, 3, 1, 3, 1, 3, ...).  Gary W. Adamson, Oct 21 2007
G.f.:(1+x)/((13*x)(1x)).  Zerinvary Lajos, Jan 11 2009, R. J. Mathar, Jan 21 2009
a(0)=1, a(1)=5, a(n) = 4*a(n1)  3*a(n2).  Harvey P. Dale, Mar 06 2012
a(n) = Sum_{k=0..n} A112468(n,k)*4^k.  Philippe Deléham, Feb 23 2014


EXAMPLE

a(0) = 1;
a(1) = 1 + 3 + 1 = 5;
a(2) = 1 + 3 + 9 + 3 + 1 = 17;
a(3) = 1 + 3 + 9 + 27 + 9 + 3 + 1 = 53; etc.  Philippe Deléham, Feb 23 2014


MAPLE

g:=x*((1+x)/(13*x)/(1x)): gser:=series(g, x=0, 43): seq(coeff(gser, x, n), n=1..30); # Zerinvary Lajos, Jan 11 2009


MATHEMATICA

NestList[3 # + 2 &, 1, 30] (* Harvey P. Dale, Mar 06 2012 *)
LinearRecurrence[{4, 3}, {1, 5}, 30] (* Harvey P. Dale, Mar 06 2012 *)
Table[2 3^n  1, {n, 20}] (* Eric W. Weisstein, May 28 2017 *)
2 3^Range[20]  1 (* Eric W. Weisstein, May 28 2017 *)


PROG

(MAGMA) [2*3^n  1: n in [0..30]]; // Vincenzo Librandi, Sep 23 2011
(PARI) first(m)=vector(m, n, n; 2*3^n  1) \\ Anders Hellström, Dec 11 2015


CROSSREFS

a(n)=T(2, n), array T given by A048471.
Cf. A003462, A029858. A column of A119725.
Cf. A134347, A046055, A010684.
Cf. A112468, A112739.
Sequence in context: A222160 A027028 A176086 * A154992 A178828 A242429
Adjacent sequences: A048470 A048471 A048472 * A048474 A048475 A048476


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling


EXTENSIONS

Better description from Amarnath Murthy, May 27 2001
Divided g.f. by x to match the offset  R. J. Mathar, Jan 21 2009
Typo in Maple program fixed by Marko Mihaily, Mar 07 2009


STATUS

approved



