|
%I #13 Sep 08 2022 08:45:14
%S 1,11,111,1221,11221,133331,1133331,14466441,114466441,1561110551,
%T 11561110551,167672165661,1167672165661,17934888731771,
%U 117934888731771,1911423761908881,11911423761908881,203053799952796991
%N a(n) = Sum_{k=0..n} binomial(floor((n+1)/2), floor((k+1)/2)) * 10^k.
%C a(n) = (11/10)*{1, 101, 101, 10201, 10201, 1030301, ...} - 100*{0, 1, 0, 100, 0, 10000, ...} - (1/10)*{1, 1, 1, 1, 1, 1, ...}.
%C Partial sums of A097178.
%H G. C. Greubel, <a href="/A097177/b097177.txt">Table of n, a(n) for n = 0..980</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,201,-201,-10100,10100)
%F G.f.: (1+10*x-101*x^2-900*x^3)/((1-x)*(1-100*x^2)*(1-101*x^2)).
%F a(n) = (11/20)*( (1-sqrt(101))*(-sqrt(101))^n + (1+sqrt(101))*(sqrt(101))^n ) - 5*(10^n - (-10)^n) - 1/10.
%F a(n) = a(n-1) + 201*a(n-2) - 201*a(n-3) - 10100*a(n-4) + 10100*a(n-5).
%p seq(coeff(series((1+10*x-101*x^2-900*x^3)/((1-x)*(1-100*x^2)*(1-101*x^2)), x, n+1), x, n), n = 0 ..20); # _G. C. Greubel_, Sep 17 2019
%t Accumulate[LinearRecurrence[{0,201,0,-10100},{1,10,100,1110},20]] (* or *) LinearRecurrence[{1,201,-201,-10100,10100},{1,11,111,1221,11221},20] (* _Harvey P. Dale_, Dec 08 2018 *)
%o (PARI) my(x='x+O('x^20)); Vec((1+10*x-101*x^2-900*x^3)/((1-x)*(1-100*x^2)*(1-101*x^2))) \\ _G. C. Greubel_, Sep 17 2019
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 20); Coefficients(R!( (1+10*x-101*x^2-900*x^3)/((1-x)*(1-100*x^2)*(1-101*x^2)) )); // _G. C. Greubel_, Sep 17 2019
%o (Sage)
%o def A097177_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P((1+10*x-101*x^2-900*x^3)/((1-x)*(1-100*x^2)*(1-101*x^2))).list()
%o A097177_list(20) # _G. C. Greubel_, Sep 17 2019
%o (GAP) a:=[1, 11, 111, 1221, 11221];; for n in [6..20] do a[n]:=a[n-1] + 201*a[n-2] - 201*a[n-3] - 10100*a[n-4] + 10100*a[n-5]; od; a; # _G. C. Greubel_, Sep 17 2019
%Y Cf. A097175, A097176, A097178.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Jul 30 2004
|
