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%I #39 May 04 2023 19:32:59
%S 0,1,3,9,27,80,234,677,1941,5523,15615,43906,122868,342409,950727,
%T 2631165,7260579,19982612,54865566,150316973,411015705,1121818311,
%U 3056773383,8316416134,22593883752,61301547025,166118284299,449639574897,1215751720491,3283883157848
%N a(n) = (3*n*F(2n-1) + (3-n)*F(2n))/5 where F() = Fibonacci numbers A000045.
%C Substituting x(1-x)/(1-2x) into x/(1-x)^2 yields g.f. of sequence.
%C 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).
%H Harry J. Smith, <a href="/A059502/b059502.txt">Table of n, a(n) for n = 0..200</a>
%H Sergi Elizalde, Rigoberto Flórez, and José Luis Ramírez, <a href="https://doi.org/10.26493/1855-3974.2478.d1b">Enumerating symmetric peaks in non-decreasing Dyck paths</a>, Ars Mathematica Contemporanea (2021).
%H <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6,-1).
%F a(n) = 2*a(n-1) + Sum{m<=n-2}a(m) + A001519(n-2).
%F G.f.: x(1-x)(1-2x)/(1-3x+x^2)^2. - _Emeric Deutsch_, Oct 07 2002
%F a(n) = A147703(n,1). - _Philippe Deléham_, Nov 29 2008
%F a(n) = A001871(n-1)-3*A001871(n-2)+2*A001871(n-3). - _R. J. Mathar_, Apr 09 2019
%e The array (see A059503) begins
%e 1 3 9 27 80 ...
%e 2 5 14 40 ...
%e 3 7 19 ...
%e 4 9 5 ...
%t 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 *)
%o (PARI) a(n)=(3*n*fibonacci(2*n-1)+(3-n)*fibonacci(2*n))/5
%o (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
%o (Magma) [(3*n*Fibonacci(2*n-1)+(3-n)*Fibonacci(2*n))/5: n in [0..100]]; // _Vincenzo Librandi_, Apr 23 2011
%Y Cf. A000045, A000667, A059216, A059219, A027994, A059512, A059503.
%K easy,nonn,nice
%O 0,3
%A _Floor van Lamoen_, Jan 19 2001
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