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%I #35 Sep 08 2022 08:45:35
%S 0,0,2,6,18,46,114,270,626,1422,3186,7054,15474,33678,72818,156558,
%T 334962,713614,1514610,3203982,6757490,14214030,29826162,62448526,
%U 130489458,272163726,566697074,1178133390,2445745266,5070447502,10498808946,21713445774,44858547314
%N a(n) = (2*(-1)^n - 2^(n+1) + 3*n*2^n)/9.
%C Specify that a triangle has T(n,0) = T(n,n) = A001045(n), and T(r,c) = T(r-1,c-1) + T(r-1,c). The sum of the terms in the first n rows is a(n+1). - _J. M. Bergot_, May 21 2013
%C a(n) is the difference between the total number of runs of equal parts in the compositions of n+1, and the compositions of n+1. - _Gregory L. Simay_, May 04 2017
%H Vincenzo Librandi, <a href="/A140960/b140960.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 a(n+1) - 2*a(n) = A078008(n+1) = 2*A001045(n).
%F G.f.: 2*x^2/((1+x)*(1-2*x)^2).
%F a(n) = 2*A045883(n-1).
%F a(n) = 3*a(n-1) - 4*a(n-3), n > 2.
%F a(n) = A059570(n+1) - A011782(n+1). - _Gregory L. Simay_, May 04 2017
%t LinearRecurrence[{3,0,-4},{0,0,2},40] (* _Harvey P. Dale_, Apr 14 2015 *)
%o (Magma) [( 2*(-1)^n-2^(n+1)+3*n*2^n)/9: n in [0..40]]; // Vincenzo Librandi, Aug 08 2011
%o (PARI) a(n)=(2*(-1)^n-2^(n+1)+3*n*2^n)/9 \\ _Charles R Greathouse IV_, Oct 16 2015
%K nonn,easy
%O 0,3
%A _Paul Curtz_, Jul 26 2008
%E Definition replaced with Lava's closed form of August 2008 by _R. J. Mathar_, Feb 11 2010
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