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%I #21 Jun 03 2020 12:14:10
%S 1,10,83,668,5349,42798,342391,2739136,21913097,175304786,1402438299,
%T 11219506404,89756051245,718048409974,5744387279807,45955098238472,
%U 367640785907793,2941126287262362,23529010298098915,188232082384791340,1505856659078330741
%N a(1)=1; for n>1, a(n) = 8*a(n-1)+n.
%H Colin Barker, <a href="/A014831/b014831.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (10,-17,8).
%F a(n) = (8^(n+1)-7*n-8)/49. - _Rolf Pleisch_, Oct 21 2010
%F a(n) = Sum_{i=0..n-1} 7^i*binomial(n+1,n-1-i). [_Bruno Berselli_, Nov 13 2015]
%F From _Colin Barker_, Jun 03 2020: (Start)
%F G.f.: x / ((1 - x)^2*(1 - 8*x)).
%F a(n) = 10*a(n-1) - 17*a(n-2) + 8*a(n-3) for n>3.
%F (End)
%e For n=5, a(5) = 1*15 + 7*20 + 7^2*15 + 7^3*6 + 7^4*1 = 5349. [_Bruno Berselli_, Nov 13 2015]
%p a:=n->sum((8^(n-j)-1)/7,j=0..n): seq(a(n), n=1..19); # _Zerinvary Lajos_, Jan 15 2007
%p a:= n-> (Matrix ([[1, 0, 1], [1, 1, 1], [0, 0, 8]])^n)[2, 3]: seq (a(n), n=1..25); # _Alois P. Heinz_, Aug 06 2008
%t Table[(8^(n + 1) - 7 n - 8)/49, {n, 1, 25}] (* _Bruno Berselli_, Nov 13 2015 *)
%o (PARI) Vec(x / ((1 - x)^2*(1 - 8*x)) + O(x^25)) \\ _Colin Barker_, Jun 03 2020
%Y Cf. A000420, A104712.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
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