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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, 449639574897, 1215751720491, 3283883157848 (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
Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, Enumerating symmetric peaks in non-decreasing Dyck paths, Ars Mathematica Contemporanea (2021).
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
Sequence in context: A287898 A129770 A134396 * A291006 A289781 A209239
KEYWORD
easy,nonn,nice
AUTHOR
Floor van Lamoen, Jan 19 2001
STATUS
approved

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Last modified May 29 08:50 EDT 2024. Contains 372926 sequences. (Running on oeis4.)