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 A029907 a(n+1) = a(n) + a(n-1) + Fibonacci(n). 29
 0, 1, 2, 4, 8, 15, 28, 51, 92, 164, 290, 509, 888, 1541, 2662, 4580, 7852, 13419, 22868, 38871, 65920, 111556, 188422, 317689, 534768, 898825, 1508618, 2528836, 4233872, 7080519, 11828620, 19741179, 32916068, 54835556, 91276202, 151814645, 252318312 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of matchings of the fan graph on n vertices, n>0 (a fan is the join of the path graph with one extra vertex). a(n+1) gives row sums of A054450. - Paul Barry, Oct 23 2004 Number of parts in all compositions of n into odd parts. Example: a(5)=15 because the compositions 5, 311, 131, 113, and 11111 have a total of 1+3+3+3+5=15 parts. a(n-1) is the number of compositions of n that contain one even part; for example, a(5-1)=a(4)=8 counts the compositions 1112, 1121, 1211, 14, 2111, 23, 32, 41. - Joerg Arndt, May 21 2013 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 0..1000 Jia Huang, Compositions with restricted parts, arXiv:1812.11010 [math.CO], 2018. Mengmeng Liu, Andrew Yezhou Wang, The Number of Designated Parts in Compositions with Restricted Parts, J. Int. Seq., Vol. 23 (2020), Article 20.1.8. Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1). FORMULA G.f.: x*(1-x^2)/(1-x-x^2)^2. a(n) = ((n+4)*Fibonacci(n) + 2*n*Fibonacci(n-1))/5. a(n+1) = Sum_{k=0..n} Sum_{j=0..floor(k/2)} binomial(n-j, j). - Paul Barry, Oct 23 2004 a(n) = A010049(n+1) + A152163(n+1). - R. J. Mathar, Dec 10 2011 a(n) = F(n) + Sum_{k=1..n-1} F(k)*F(n-k), where F=Fibonacci. - Reinhard Zumkeller, Nov 01 2013 EXAMPLE a(4)=8 because matchings of fan graph with edges {OA,OB,OC,AB,AC} are: {},{OA},{OB},{OC},{AB},{AC},{OA,BC},{OC,AB}. MAPLE with(combinat); A029907 := proc(n) options remember; if n <= 1 then n else procname(n-1)+procname(n-2)+fibonacci(n-1); fi; end; MATHEMATICA CoefficientList[Series[x(1-x^2)/(1-x-x^2)^2, {x, 0, 37}], x] (* or *) a[n_]:= a[n]= a[n-1] +a[n-2] +Fibonacci[n-1]; a[0]=0; a[1]=1; Array[a, 37] (* or *) LinearRecurrence[{2, 1, -2, -1}, {0, 1, 2, 4}, 38] (* Robert G. Wilson v, Jun 22 2014 *) PROG (PARI) alias(F, fibonacci); a(n)=((n+4)*F(n)+2*n*F(n-1))/5; (Haskell) a029907 n = a029907_list !! n a029907_list = 0 : 1 : zipWith (+) (tail a000045_list)                       (zipWith (+) (tail a029907_list) a029907_list) -- Reinhard Zumkeller, Nov 01 2013 (MAGMA) [((n+4)*Fibonacci(n)+2*n*Fibonacci(n-1))/5: n in [0..40]]; // Vincenzo Librandi, Feb 25 2018 CROSSREFS Cf. A000045, A001629, A010049, A240847. Sequence in context: A006727 A007673 A182725 * A005682 A114833 A065617 Adjacent sequences:  A029904 A029905 A029906 * A029908 A029909 A029910 KEYWORD nonn,easy AUTHOR EXTENSIONS Additional formula from Wolfdieter Lang, May 02 2000 Additional comments from Michael Somos, Jul 23 2002 STATUS approved

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Last modified August 13 06:22 EDT 2020. Contains 336442 sequences. (Running on oeis4.)