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1, 4, 10, 22, 46, 94, 190, 382, 766, 1534, 3070, 6142, 12286, 24574, 49150, 98302, 196606, 393214, 786430, 1572862, 3145726, 6291454, 12582910, 25165822, 50331646, 100663294, 201326590, 402653182, 805306366, 1610612734, 3221225470
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of nodes in rooted tree of height n in which every node (including the root) has valency 3.
Pascal diamond numbers : reflect Pascal's n-th triangle vertically and sum all elements. E.g. a(3)=1+(1+1)+(1+2+1)+(1+1)+1. - Paul Barry (pbarry(AT)wit.ie), Jun 23 2003
Number of 2 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (11;0). An occurrence of a ranpp (xy;z) in a matrix A=(a(i,j)) is a triple (a(i1,j1), a(i1,j2), a(i2,j1)) where i1<i2 and j1<j2 and these elements are in same relative order as those in the triple (x,y,z). - Sergey Kitaev (kitaev(AT)ms.uky.edu), Nov 11 2004
Binomial and inverse binomial transform are in A001047 (shifted) and A122553. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 02 2008]
a(n) = (SUM_{n=0..(n-1)} a(n)) + (2*n + 1); e.g. a(3) = 22 = (1 + 4 + 10) + 7. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 21 2009]
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is a proper subset of y or y is a proper subset of x and x and y are disjoint, or 1) x equals y. Then a(n) = |R|. [From Ross La Haye (rlahaye(AT)new.rr.com), Mar 19 2009]
Equals the Jacobsthal sequence A001045 convolved with (1, 3, 4, 4, 4, 4, 4, ...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 24 2009]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 24 2010: (Start)
Equals the eigensequence of a triangle with the odd integers as the left
border and the rest 1's. (End)
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 15 2010: (Start)
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 58, 154, 178 and 184, lead to this sequence. For the corner squares these vectors lead to the companion sequence A097813.
(End)
a(n+2) is the integer with bit string "10" * "1"^n * "10";
a(n) = A027383(2n). - Jason Kimberley, Nov 03 2011
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REFERENCES
| J. Riordan, Series-parallel realization of the sum modulo 2 of n switching variables, in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 877-878.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From Ross La Haye (rlahaye(AT)new.rr.com), Mar 19 2009]
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-2).
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, Integers: Electronic Journal of Combinatorial Number Theory 4 (2004), A21, 20pp.
S. Kitaev, On multi-avoidance of right angled numbered polyomino patterns, University of Kentucky Research Reports (2004).
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FORMULA
| G.f.: (1+x)/(1-3*x+2*x^2). a(0)=1, a(n) = 2*{a(n-1) + 1}.
G.f. is equivalent to (1-2x-3x^2)/((1-x)(1-2x)(1-3x)). - Paul Barry (pbarry(AT)wit.ie), Apr 28 2004
A099257(a(n))=A099258(a(n))=a(n); a(n)=2*A055010(n)=(A068156(n)-1)/2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Oct 09 2004
Row sums of triangle A130452. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 26 2007
Row sums of triangle A131110. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2007
Binomial transform of (1, 3, 3, 3,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 17 2007
Binomial transform of [1, 3, 3, 3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 21 2007
Row sums of triangle A051597 (a triangle generated from Pascal's rule given right and left borders = 1, 2, 3,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 04 2007
Equals A132776 * [1/1, 1/2, 1/3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 16 2007
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EXAMPLE
| Binary: 1, 100, 1010, 10110, 101110, 1011110, 10111110, 101111110, 1011111110, 10111111110, 101111111110, 1011111111110, 10111111111110,
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MAPLE
| with (combinat):a:=n->stirling2(n, 2)+stirling2(n+1, 2): seq(a(n), n=1..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 07 2007
a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=(a[n-1]+1)*2 od: seq(a[n], n=1..28); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 22 2008
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MATHEMATICA
| a[n_]:=3*2^n-2; ...and/or...a=4; lst={1, a}; k=6; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 16 2008]
a=1; lst={1}; k=3; Do[a+=k; AppendTo[lst, a]; k+=k, {n, 0, 6!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 15 2008]
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PROG
| (MAGMA)[3*2^n-2: n in [1..50]] [From Vincenzo Librandi Nov 22 2010]
(PARI) a(n)=3<<n-2 \\ Charles R Greathouse IV, Nov 02 2011
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CROSSREFS
| Cf. A033484, A131110, A051597, A132776, A001045.
Sequence in context: A038621 A078407 * A099018 A008267 A056112 A118430
Adjacent sequences: A033481 A033482 A033483 * A033485 A033486 A033487
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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