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 A033484 a(n) = 3*2^n - 2. 61
 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; text; internal format)
 OFFSET 0,2 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, 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 i10. a(n) = A007283(n) - 2. G.f. is equivalent to (1-2x-3x^2)/((1-x)(1-2x)(1-3x)). - Paul Barry, Apr 28 2004 A099257(a(n))=A099258(a(n))=a(n); a(n)=2*A055010(n)=(A068156(n)-1)/2. - Reinhard Zumkeller, Oct 09 2004 Row sums of triangle A130452. - Gary W. Adamson, May 26 2007 Row sums of triangle A131110. - Gary W. Adamson, Jun 15 2007 Binomial transform of (1, 3, 3, 3,...). - Gary W. Adamson, Oct 17 2007 Row sums of triangle A051597 (a triangle generated from Pascal's rule given right and left borders = 1, 2, 3,...). - Gary W. Adamson, Nov 04 2007 Equals A132776 * [1/1, 1/2, 1/3,...]. - Gary W. Adamson, Nov 16 2007 a(n) = Sum_{k=0..n} A112468(n,k)*3^k. - Philippe Deléham, Feb 23 2014 a(n) = -(2^n) * A036563(1-n) for all n in Z. - Michael Somos, Jul 04 2017 EXAMPLE Binary: 1, 100, 1010, 10110, 101110, 1011110, 10111110, 101111110, 1011111110, 10111111110, 101111111110, 1011111111110, 10111111111110, G.f. = 1 + 4*x + 10*x^2 + 22*x^3 + 46*x^4 + 94*x^5 + 190*x^6 + 382*x^7 + ... MAPLE with(combinat):a:=n->stirling2(n, 2)+stirling2(n+1, 2): seq(a(n), n=1..28); # Zerinvary Lajos, Oct 07 2007 a:=0:a:=1:for n from 2 to 50 do a[n]:=(a[n-1]+1)*2 od: seq(a[n], n=1..28); # Zerinvary Lajos, Feb 22 2008 MATHEMATICA Table[3 2^n - 2, {n, 0, 20}] (* Vladimir Joseph Stephan Orlovsky, Dec 16 2008 *) (* Start from Eric W. Weisstein, Sep 21 2017 *) 3 2^Range[0, 20] - 2 LinearRecurrence[{3, -2}, {1, 4}, 20] CoefficientList[Series[(1 + x)/(1 - 3 x + 2 x^2), {x, 0, 20}], x] (* End *) PROG (MAGMA)[3*2^n-2: n in [1..50]] // Vincenzo Librandi, Nov 22 2010] (PARI) a(n) = 3<

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Last modified October 23 14:45 EDT 2019. Contains 328345 sequences. (Running on oeis4.)