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 A097472 Number of different candle trees having a total of m edges. 4
 1, 3, 10, 31, 96, 296, 912, 2809, 8651, 26642, 82047, 252672, 778128, 2396320, 7379697, 22726483, 69988378, 215535903, 663763424, 2044122936, 6295072048, 19386276329, 59701891739, 183857684514, 566207320575, 1743689586432 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A candle tree is a graph on the plane square lattice Z X Z whose edges have length one with the following properties: (a) It contains a line segment ("trunk") of length from 0 to m on the vertical axis, its lowest node is at the origin. (b) It contains horizontal line segments ("branches"); each of them intersects the trunk. (c) Each branch is allowed to have "candles", which are vertical edges of length 1, whose lower node is on a branch. Row sums of triangle in A238241. - Philippe Deléham, Feb 21 2014 REFERENCES Alexander Malkis, "Polyedges, polyominoes and the 'Digit' game", diploma thesis in computer science, Universitaet des Saarlandes, Germany LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Alexander Malkis, Polyedges, polyominoes and the 'Digit' game Index entries for linear recurrences with constant coefficients, signature (3, 1, -2, -1). FORMULA a(n) = sum_{s, d, k>=0 with s+d+k=m} binom(s+2d+1, s)*binom(s, k); generating function = 1/(x^4 + 2x^3 - x^2 - 3x + 1); a(n+4) = 3a(n+3)+a(n+2)-2a(n+1)-a(n) a(n) = sum(m=1..n, sum(k=1..n-m+1, binomial(k,n-m-k+1)*binomial(k+2*m-1,2*m-1)))+1. - Vladimir Kruchinin, May 12 2011 a(0)=1, a(1)=3, a(2)=10, a(3)=31, a(n) = 3*a(n-1)+a(n-2)-2*a(n-3)-a(n-4). - Harvey P. Dale, Jun 14 2011 a(n) = Sum_{k=0..n} A238241(n,k). - Philippe Deléham, Feb 21 2014 MATHEMATICA CoefficientList[Series[1/(x^4+2x^3-x^2-3x+1), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 1, -2, -1}, {1, 3, 10, 31}, 30] (* Harvey P. Dale, Jun 14 2011 *) PROG (Maxima) a(n):=sum(sum(binomial(k, n-m-k+1)*binomial(k+2*m-1, 2*m-1), k, 1, n-m+1), m, 1, n)+1; /* Vladimir Kruchinin, May 12 2011 */ (PARI) a(n)=sum(m=1, n, sum(k=1, n-m+1, binomial(k, n-m-k+1)*binomial(k+2*m-1, 2*m-1))) \\ Charles R Greathouse IV, Jun 17 2013 CROSSREFS Bisection of A060945 and |A077930|. Sequence in context: A192337 A106517 A055217 * A068094 A100058 A002160 Adjacent sequences:  A097469 A097470 A097471 * A097473 A097474 A097475 KEYWORD easy,nice,nonn AUTHOR Alexander Malkis, Sep 18 2004 STATUS approved

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Last modified June 20 08:06 EDT 2019. Contains 324229 sequences. (Running on oeis4.)