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%I #22 Jan 06 2021 08:29:32
%S 10,80,350,1120,2940,6720,13860,26400,47190,80080,130130,203840,
%T 309400,456960,658920,930240,1288770,1755600,2355430,3116960,4073300,
%U 5262400,6727500,8517600,10687950,13300560,16424730,20137600
%N a(n) = 10*(n+1)*binomial(n+3,5)/3.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7, -21, 35, -35, 21, -7, 1).
%F Number of 9-subsequences of [ 1, n ] with just 3 contiguous pairs.
%F G.f.: 10*(1+x)*x^2/(1-x)^7.
%F a(n) = binomial(n+1, 3)*binomial(n+3, 3) = A000292(n-1)*A000292(n+1). - _Zerinvary Lajos_, May 13 2005
%F a(n) = 10*A040977(n). - _R. J. Mathar_, May 22 2013
%F From _Amiram Eldar_, Jan 06 2021: (Start)
%F Sum_{n>=2} 1/a(n) = 3*Pi^2/2 - 235/16.
%F Sum_{n>=2} (-1)^n/a(n) = 3*Pi^2/4 - 117/16. (End)
%t Table[10(n+1) Binomial[n+3,5]/3,{n,2,30}] (* or *) LinearRecurrence[{7,-21,35,-35,21,-7,1},{10,80,350,1120,2940,6720,13860},30] (* _Harvey P. Dale_, Jan 15 2015 *)
%Y Cf. A000292, A040977.
%K nonn,easy
%O 2,1
%A thi ngoc dinh (via _R. K. Guy_)
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