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%I #16 Feb 04 2022 11:06:57
%S 210,2772,19404,96096,378378,1261260,3699696,9801792,23891868,
%T 54318264,116396280,237025152,461705244,864913896,1565082288,
%U 2745758400,4684950270,7795127340,12676924260,20190250080,31547265750,48432564180,73156880160,108851783040
%N a(n) = 42*(n+1)*binomial(n+6,10).
%C Number of 17-subsequences of [ 1, n ] with just 6 contiguous pairs.
%H T. D. Noe, <a href="/A027822/b027822.txt">Table of n, a(n) for n = 4..1000</a>
%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).
%F G.f.: 42*(5+6*x)*x^4/(1-x)^12.
%F a(n) = C(n+1,5)*C(n+6,6). - _Zerinvary Lajos_, Jun 08 2005
%F From _Amiram Eldar_, Feb 04 2022: (Start)
%F Sum_{n>=4} 1/a(n) = 5*Pi^2 - 10444891/211680.
%F Sum_{n>=4} (-1)^n/a(n) = 5*Pi^2/2 - 512*log(2)/21 - 1644749/211680. (End)
%t Table[42*(n + 1)*Binomial[n + 6, 10], {n, 4, 30}] (* _Amiram Eldar_, Feb 04 2022*)
%K nonn,easy
%O 4,1
%A Thi Ngoc Dinh (via _R. K. Guy_)