%I #56 Jan 28 2022 03:58:18

%S 4,30,120,350,840,1764,3360,5940,9900,15730,24024,35490,50960,71400,

%T 97920,131784,174420,227430,292600,371910,467544,581900,717600,877500,

%U 1064700,1282554,1534680,1824970,2157600,2537040,2968064,3455760,4005540,4623150,5314680

%N a(n) = 2*(n+1)*binomial(n+3,4).

%C Number of 8-subsequences of [ 1, n ] with just 3 contiguous pairs.

%C Also the number of 3-cycles in the n+3 tetrahedral graph. - _Eric W. Weisstein_, Jul 12 2017

%H Harvey P. Dale, <a href="/A027789/b027789.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F G.f.: 2*(2+3x)*x/(1-x)^6.

%F a(n) = 2*A006470(n).

%F a(n) = C(n+1, 2)*C(n+3, 3). - _Zerinvary Lajos_, May 10 2005, corrected by _R. J. Mathar_, Feb 13 2016

%F a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6). - _Harvey P. Dale_, Jan 20 2015

%F a(n) = Sum_{k=1..n+1} Sum_{i=1..n+1} (n-i+1) * C(k+1,k-1). - _Wesley Ivan Hurt_, Sep 21 2017

%F From _Amiram Eldar_, Jan 28 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 61/6 - Pi^2.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/2 - 8*log(2) + 5/6. (End)

%p A027789:=n->2*(n+1)*binomial(n+3,4): seq(A027789(n), n=1..60); # _Wesley Ivan Hurt_, Oct 23 2017

%t Table[2 (n + 1) Binomial[n + 3, 4], {n, 40}] (* _Harvey P. Dale_, Jan 20 2015 *)

%t LinearRecurrence[{6, -15, 20, -15, 6, -1}, {4, 30, 120, 350, 840, 1764},40] (* _Harvey P. Dale_, Jan 20 2015 *)

%t Table[n (1 + n)^2 (2 + n) (3 + n)/12, {n, 20}] (* _Eric W. Weisstein_, Jul 12 2017 *)

%t CoefficientList[Series[(2 (2 + 3 x))/(-1 + x)^6, {x, 0, 20}], x] (* _Eric W. Weisstein_, Jul 12 2017 *)

%o (Magma) [2*(n+1)*Binomial(n+3,4): n in [1..40]]; // _Vincenzo Librandi_, Jul 13 2017

%o (PARI) for(n=1,50, print1(2*(n+1)*binomial(n+3,4), ", ")) \\ _G. C. Greubel_, Oct 22 2017

%Y Cf. A006470, A289792 (4-cycles), A289793 (5-cycles), A289794 (6-cycles).

%K nonn,easy

%O 1,1

%A Thi Ngoc Dinh (via _R. K. Guy_)