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 A059502 a(n) = (3*n*F(2n-1) + (3-n)*F(2n))/5 where F() = Fibonacci numbers A000045. 11
 0, 1, 3, 9, 27, 80, 234, 677, 1941, 5523, 15615, 43906, 122868, 342409, 950727, 2631165, 7260579, 19982612, 54865566, 150316973, 411015705, 1121818311, 3056773383, 8316416134, 22593883752, 61301547025, 166118284299 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Substituting x(1-x)/(1-2x) into x/(1-x)^2 yields g.f. of sequence. Variation of A059216 (and of Boustrophedon transform) applied to 1,2,3,4,...: fill an array by diagonals, each time in the same direction, say the 'up' direction. The first column is 1,2,3,4,... For the next element of a diagonal, add to the previous element the elements of the row the new element is in. The first row gives a(n). LINKS Harry J. Smith, Table of n, a(n) for n = 0..200 Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021). Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1). FORMULA a(n) = 2*a(n-1) + Sum{m<=n-2}a(m) + A001519(n-2). G.f.: x(1-x)(1-2x)/(1-3x+x^2)^2. - Emeric Deutsch, Oct 07 2002 a(n) = A147703(n,1). - Philippe Deléham, Nov 29 2008 a(n) = A001871(n-1)-3*A001871(n-2)+2*A001871(n-3). - R. J. Mathar, Apr 09 2019 EXAMPLE The array (see A059503) begins 1 3 9 27 80 ... 2 5 14 40 ... 3 7 19 ... 4 9 5 ... MATHEMATICA Table[(3n Fibonacci[2n-1]+(3-n)Fibonacci[2n])/5, {n, 0, 30}] (* or *) CoefficientList[Series[x(1-x)(1-2x)/(1-3x+x^2)^2, {x, 0, 30}], x] (* Harvey P. Dale, Apr 23 2011 *) PROG (PARI) a(n)=(3*n*fibonacci(2*n-1)+(3-n)*fibonacci(2*n))/5 (PARI) { for (n = 0, 200, write("b059502.txt", n, " ", (3*n*fibonacci(2*n - 1) + (3 - n)*fibonacci(2*n))/5); ) } \\ Harry J. Smith, Jun 27 2009 (Magma) [(3*n*Fibonacci(2*n-1)+(3-n)*Fibonacci(2*n))/5: n in [0..100]]; // Vincenzo Librandi, Apr 23 2011 CROSSREFS Cf. A000667, A059216, A059219, A027994, A059512, A059503. Sequence in context: A287898 A129770 A134396 * A291006 A289781 A209239 Adjacent sequences: A059499 A059500 A059501 * A059503 A059504 A059505 KEYWORD easy,nonn,nice AUTHOR Floor van Lamoen, Jan 19 2001 STATUS approved

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Last modified December 9 19:36 EST 2022. Contains 358703 sequences. (Running on oeis4.)