%I M1609 #49 Dec 28 2021 00:02:24

%S 2,6,16,30,54,84,128,180,250,330,432,546,686,840,1024,1224,1458,1710,

%T 2000,2310,2662,3036,3456,3900,4394,4914,5488,6090,6750,7440,8192,

%U 8976,9826,10710,11664,12654,13718

%N G.f.: 2*(1-x^3)/((1-x)^5*(1+x)^2).

%C a(n) is also the number of triples (w,x,y) having all terms in {0,...,n} and w<R<=x, where R=max(w,x,y)-min(w,x,y). [Clark Kimberling, Jun 10 2012]

%C a(n) is also the sum of all elements of the square matrix M(n-1) = M1(n-1) x M2(n-1), where M1(n) is the square matrix with elements m1(i,j)= (1+(-1)^(i+j+1))/2, A057212; and M2(n) is the square matrix given by m2(i,j)= (1+(-1)^(i+j))/2, A057212. - _Enrique Pérez Herrero_, Jun 15 2013

%C Also the number of longest paths in the (n+1)-web graph for n > 2. - _Eric W. Weisstein_, Mar 27 2018

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Enrique Pérez Herrero, <a href="/A005996/b005996.txt">Table of n, a(n) for n = 1..1000</a>

%H S. M. Losanitsch, <a href="https://doi.org/10.1002/cber.189703002144">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), pp. 1917-1926.

%H S. M. Losanitsch, <a href="/A000602/a000602_1.pdf">Die Isomerie-Arten bei den Homologen der Paraffin-Reihe</a>, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LongestPathProblem.html">Longest Path Problem</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WebGraph.html">Web Graph</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).

%F a(n) = 2*(A006918(n) + A006918(n-1) + A006918(n-2)), n>1. - _Ralf Stephan_, Apr 26 2003

%F a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6), with a(1)=2, a(2)=6, a(3)=16, a(4)=30, a(5)=54, a(6)=84. - _Harvey P. Dale_, Apr 08 2013

%F From _Ayoub Saber Rguez_, Nov 20 2021: (Start)

%F a(n) = A143785(n) - A002620(n+1).

%F a(n) = A128624(n) + A002620(n+1).

%F a(n) = (n^3 + 3*n^2 + 2*n + 1 + n*(n mod 2) - ((n+1) mod 2))/4. (End)

%t Table[(1/4)*(1 + n)*(-2 + 5*n + n^2 + 2*Ceiling[1/2 - n/2] - 4*Floor[n/2]), {n, 1, 200}] (* _Enrique Pérez Herrero_, Aug 03 2012 *)

%t CoefficientList[Series[2 (1 - x^3)/((1 - x)^5 (1 + x)^2), {x, 0, 40}], x] (* _Harvey P. Dale_, Apr 08 2013 *)

%t LinearRecurrence[{2, 1, -4, 1, 2, -1}, {2, 6, 16, 30, 54, 84}, 40] (* _Harvey P. Dale_, Apr 08 2013 *)

%t Table[(n + 1) (2 n (n + 2) + 1 - (-1)^n)/8, {n, 20}] (* _Eric W. Weisstein_, Mar 27 2018 *)

%Y Essentially twice A034828.

%K nonn,easy

%O 1,1

%A _N. J. A. Sloane_

%E Edited by _N. J. A. Sloane_, Aug 03 2012