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%I #37 Sep 08 2022 08:45:50
%S 1,2,5,11,25,55,121,263,569,1223,2617,5575,11833,25031,52793,111047,
%T 233017,487879,1019449,2126279,4427321,9204167,19107385,39612871,
%U 82021945,169636295,350457401,723284423,1491308089,3072094663,6323146297,13004206535,26724240953
%N a(n) = (3*n*2^n+2^(n+4)+2*(-1)^n)/18.
%C The binomial transform is in A126184.
%C An elephant sequence, see A175654 and A175655. There are 24 A[5] vectors, with decimal values between 7 and 448, that lead for the corner squares to this sequence. Its companion sequence for the central square is A175656. Furthermore there are 36 A[5] vectors, with decimal values between 15 and 480, that lead for the central square to four times this sequence for n >= -1. Its companion sequence for the corner squares is A059570. - _Johannes W. Meijer_, Aug 15 2010
%C a(n) is also the number of runs of weakly increasing parts in all compositions of n+1. a(2) = 5: (111), (12), (2)(1), (3). - _Alois P. Heinz_, Apr 30 2017
%H Vincenzo Librandi, <a href="/A172481/b172481.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-4)
%F G.f.: (1-x-x^2)/((1+x)*(1-2*x)^2).
%F a(n) = A001045(n-1)+2*a(n-1), n>0.
%F a(n)+A139790(n) = 2^(n+1) = A000079(n+1).
%F a(n) = A139790(n)+A140960(n).
%F a(n) = A001045(n)+(-1)^n*A084219(n).
%F a(n) = A127984(n) + 2^(n-1). Application: Problem 11623, AMM 119 (2012) 161. - _Stephen J. Herschkorn_, Feb 11 2012
%t Table[(3n 2^n+2^(n+4)+2(-1)^n)/18,{n,0,40}] (* or *)
%t CoefficientList[Series[(1-x-x^2)/((1+x)(1-2x)^2), {x,0,40}], x] (* _Harvey P. Dale_, Mar 28 2011 *)
%o (Magma) [(3*n*2^n+2^(n+4)+2*(-1)^n)/18: n in [0..40]]; // _Vincenzo Librandi_, Aug 04 2011
%o (PARI) a(n)=(3*n*2^n+2^(n+4)+2*(-1)^n)/18 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A059570, A151529, A127984.
%K nonn,easy
%O 0,2
%A _Paul Curtz_, Feb 04 2010
%E Definition replaced by explicit formula by _R. J. Mathar_, Feb 11 2010
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