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 A067331 Convolution of Fibonacci F(n+1), n >= 0, with F(n+3), n >= 0. 19
 2, 5, 12, 25, 50, 96, 180, 331, 600, 1075, 1908, 3360, 5878, 10225, 17700, 30509, 52390, 89664, 153000, 260375, 442032, 748775, 1265832, 2136000, 3598250, 6052061, 10164540, 17048641, 28559450, 47786400, 79870428, 133359715, 222457608, 370747675, 617363100 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Third diagonal of A067330. Third column of A067418. From Emeric Deutsch, Jun 15 2010: (Start) a(n) is the external path length of the Fibonacci tree of order n+3. A Fibonacci tree of order n (n >= 2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. The external path length of a tree is the sum of the levels of its external nodes (i.e., leaves). a(n) = Sum_{k>=0} k*A178524(n+2,k). (End) a(n) equals the penultimate immanant of the (n+3) X (n+3) tridiagonal matrix with ones along the main diagonal, the superdiagonal, and the subdiagonal. - John M. Campbell, Jan 01 2016 a(n) is the sum of the eccentricities of the vertices of the Fibonacci cube G(n+1). Example: a(1)=5; indeed, the Fibonacci cube G(2) is the path graph P(3), the vertices of which have eccentricities 2, 1, 2. - Emeric Deutsch, May 28 2017 REFERENCES D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417. LINKS Robert Israel, Table of n, a(n) for n = 0..4720 Matthew Blair, Rigoberto Flórez, Antara Mukherjee, and José L. Ramírez, Matrices in the determinant Hosoya triangle, Fibonacci Quart. 58 (2020), no. 5, 34-54. Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Geometric Patterns in The Determinant Hosoya Triangle, INTEGERS, A90, 2021. J. Bodeen, S. Butler, T. Kim, X. Sun, and S. Wang, Tiling a strip with triangles, Electron. J. Combin. 21 (1) (2014), P1.7. John M. Campbell, On the external path length of a Fibonacci tree. Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20(2) (1982), 168-178. S. Klavzar and M. Mollard, Asymptotic properties of Fibonacci cubes and Lucas cubes, HAL Id: hal-00836788, 2013. S. Klavzar and M. Mollard, Asymptotic properties of Fibonacci cubes and Lucas cubes, Ann. Comb. 18 (2014), 447-457. Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1). FORMULA a(n) = A067330(n+2, n) = A067418(n+2, 2) = Sum_{k=0..n} F(k+1)*F(n+3-k), n >= 0. a(n) = ((7*n + 10)*F(n + 1) + 4*(n + 1)*F(n))/5, with F(n) = A000045(n) (Fibonacci). G.f.: (2 + x)/(1 - x - x^2)^2. a(n) = Sum_{i=0..floor((n+3)/2)} binomial(n+3-i, i)*(n + 2 - 2*i). - John M. Campbell, Jan 04 2016 E.g.f.: exp(x/2)*(50 + 55*x)*cosh(sqrt(5)*x/2) + sqrt(5)*(18 + 25*x)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 04 2023 EXAMPLE From John M. Campbell, Jan 03 2016: (Start) Letting n=2, the external path length of the Fibonacci tree T(5) of order n+3=5 illustrated below is 12 = a(2) = F(1)*F(5) + F(2)*F(4) + F(3)*F(3). . / \ /\ /\ /\ (End) MAPLE f:= gfun:-rectoproc({a(n) = 2*a(n-1)+a(n-2) - 2*a(n-3)-a(n-4), a(0)=2, a(1)=5, a(2)=12, a(3)=25}, a(n), remember): map(f, [\$0..50]); # Robert Israel, Jan 06 2016 MATHEMATICA LinearRecurrence[{2, 1, -2, -1}, {2, 5, 12, 25}, 70] (* Vincenzo Librandi, Jan 02 2016 *) Table[SeriesCoefficient[(2 + x)/(1 - x - x^2)^2, {x, 0, n}], {n, 0, 34}] (* Michael De Vlieger, Jan 02 2016 *) Print[Table[Sum[Binomial[n + 3 - i, i]*(n + 2 - 2*i), {i, 0, Floor[(n + 3)/2]}], {n, 0, 100}]] (* John M. Campbell, Jan 04 2016 *) Module[{nn=40, fibs}, fibs=Fibonacci[Range[nn]]; Table[ListConvolve[Take[ fibs, n], Take[fibs, {2, n+2}]], {n, nn-2}]][[All, 2]] (* Harvey P. Dale, Aug 03 2019 *) PROG (Magma) [((7*n+10)*Fibonacci(n+1)+4*(n+1)*Fibonacci(n))/5: n in [0..40]]; // Vincenzo Librandi, Jan 02 2016 (PARI) Vec((2+x)/(1-x-x^2)^2 + O(x^100)) \\ Altug Alkan, Jan 04 2016 CROSSREFS Row sums of A108038. Cf. A000045, A004070, A067330, A067418, A178523, A178524. Sequence in context: A116730 A240847 A166106 * A116734 A101836 A262667 Adjacent sequences: A067328 A067329 A067330 * A067332 A067333 A067334 KEYWORD nonn,easy,nice AUTHOR Wolfdieter Lang, Feb 15 2002 STATUS approved

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Last modified April 23 07:34 EDT 2024. Contains 371905 sequences. (Running on oeis4.)