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 A004798 Convolution of Fibonacci numbers 1,2,3,5,... with themselves. 11
 1, 4, 10, 22, 45, 88, 167, 310, 566, 1020, 1819, 3216, 5645, 9848, 17090, 29522, 50793, 87080, 148819, 253610, 431086, 731064, 1237175, 2089632, 3523225, 5930668, 9968122, 16730830, 28045221, 46954360, 78524159, 131181406, 218933030, 365044788, 608135635, 1012268592 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS From Emeric Deutsch, Feb 15 2010: (Start) a(n) is the number of subwords of the form 0000 in all binary words of length n+3 that have no pair of adjacent 1's. Example: a(2)=4 because in the 13 (=A000045(7)) binary words of length 5 that have no pair of adjacent 1's, namely 00000, 00001, 00010, 00100, 00101, 01000, 01001, 01010, 10000, 10001, 10010, 10100, 10101, we have 2 + 1 + 0 + 0 + 0 + 0 + 0 + 0 + 1 + 1 + 0 + 0 + 0 = 4 subwords of the form 0000. a(n) = Sum_{k>=0} k*A171855(n + 3,k). (End) a(n) is the total number of 0's in all binary words of length n that have no pair of adjacent 1's. Example: a(5) = 45 because in the binary words listed in the above example there are respectively 5 + 4 + 4 + 4 + 3 + 4 + 3 + 3 + 4 + 3 + 3 + 3 + 2 = 45. - Geoffrey Critzer, Jul 22 2013 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 É. Czabarka, R. Flórez, L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6. Bridget Eileen Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020. Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1). FORMULA O.g.f.: (x+1)^2*x/(1-x-x^2)^2. - Len Smiley, Dec 11 2001 a(n) = a(n-1) + a(n-2) + Fibonacci(n+2). - Philippe Deléham, Jan 22 2012 O.g.f. is the derivative of A(x,y) with respect to y and then evaluated at y = 1, where A(x,y) is the o.g.f. for A030528. - Geoffrey Critzer, Jul 22 2013 a(n) = A001629(n+1) + A001629(n-1) + 2*A001629(n). - R. J. Mathar, Oct 30 2015 a(n) = n*Fibonacci(n) + (2/5)*(n*Lucas(n) - Fibonacci(n)) = A045925(n) + 2*A001629(n), where Lucas = A000032, Fibonacci = A000045. - Vladimir Reshetnikov, Sep 27 2016 a(n) = Sum_{i=0..floor((n+1)/2)} binomial(n+1-i,i)*(n-i). - John M. Campbell, Apr 07 2018 a(n) = n*(hypergeom([-(n+1)/2, -n/2], [-n - 1], -4) - hypergeom([(1-n)/2, 1 - n/2], [-n], -4)). - Peter Luschny, Apr 10 2018 EXAMPLE a(6) = 45 + 22 + A000045(6+2) = 45 + 22 + 21 = 88. - Philippe Deléham, Jan 22 2012 MAPLE a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-1|-2|1|2>>^n. <<0, 1, 4, 10>>)[1, 1]: seq(a(n), n=1..40);  # Alois P. Heinz, Jul 04 2013 # Alternative: a := n -> n*(hypergeom([-(n+1)/2, -n/2], [-n-1], -4) - hypergeom([(1-n)/2, 1-n/2], [-n], -4)): seq(simplify(a(n)), n=1..40); # Peter Luschny, Apr 10 2018 MATHEMATICA nn=40; Drop[CoefficientList[Series[D[(1+x)/(1-y x -y x^2), y]/.y->1, {x, 0, nn}], x], 1] (* Geoffrey Critzer, Jul 22 2013 *) Table[n Fibonacci[n] + 2/5 (n LucasL[n] - Fibonacci[n]), {n, 40}] (* Vladimir Reshetnikov, Sep 27 2016 *) a[n_] := ListConvolve[f = Fibonacci[Range[2, n+1]], f][[1]]; Array[a, 40] (* Jean-François Alcover, Feb 15 2018 *) LinearRecurrence[{2, 1, -2, -1}, {1, 4, 10, 22}, 40] (* Vincenzo Librandi, Apr 08 2014 *) PROG (PARI) Vec(((1+x)/(1-x-x^2))^2+O(x^66)) \\ Joerg Arndt, Jul 04 2013 (MAGMA) I:=[1, 4, 10, 22]; [n le 4 select I[n] else 2*Self(n-1)+Self(n-2)-2*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Apr 08 2018 (Sage) [(n*lucas_number2(n+3, 1, -1) - 2*fibonacci(n))/5 for n in (1..40)] # G. C. Greubel, Jul 07 2019 (GAP) List([1..40], n-> (n*Lucas(1, -1, n+3)[2] - 2*Fibonacci(n))/5) # G. C. Greubel, Jul 07 2019 CROSSREFS Cf. A171855, A000045. - Emeric Deutsch, Feb 15 2010 Cf. A001629. - Alois P. Heinz, Jul 04 2013 a(n) = n*A000045(n+2) - A001629(n+1). - Peter Luschny, Apr 10 2018 Sequence in context: A294683 A265051 A266375 * A265052 A266372 A174622 Adjacent sequences:  A004795 A004796 A004797 * A004799 A004800 A004801 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified January 23 03:52 EST 2021. Contains 340384 sequences. (Running on oeis4.)