|
%I #45 Sep 08 2022 08:45:30
%S 0,1,3,6,10,10,11,13,16,20,20,21,23,26,30,30,31,33,36,40,40,41,43,46,
%T 50,50,51,53,56,60,60,61,63,66,70,70,71,73,76,80,80,81,83,86,90,90,91,
%U 93,96,100,100,101,103,106,110,110,111,113,116,120,120,121,123,126,130,130
%N a(n) = Sum_{k=0..n} (k mod 5) (Partial sums of A010874).
%C Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 5, A[i,i]=1, A[i,i-1]=-1. Then, for n>=1, a(n)=det(A). - _Milan Janjic_, Jan 24 2010
%H Shawn A. Broyles, <a href="/A130483/b130483.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F a(n) = 10*floor(n/5) + A010874(n)*(A010874(n)+1)/2.
%F G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3)/((1-x^5)*(1-x)).
%F From _Wesley Ivan Hurt_, Jul 23 2016: (Start)
%F a(n) = a(n-5) - a(n-6) for n>5; a(n) = a(n-5) + 10 for n>4.
%F a(n) = 10 + Sum_{k=1..4} k*floor((n-k)/5). (End)
%F a(n) = ((n mod 5)^2 - 3*(n mod 5) + 4*n)/2. - _Ammar Khatab_, Aug 13 2020
%p seq(coeff(series(x*(1+2*x+3*x^2+4*x^3)/((1-x^5)*(1-x)), x, n+1), x, n), n = 0..70); # _G. C. Greubel_, Aug 31 2019
%t Accumulate[Mod[Range[0,70],5]] (* or *) Accumulate[PadRight[{},70,{0,1,2,3,4}]] (* _Harvey P. Dale_, Nov 11 2016 *)
%o (PARI) a(n) = sum(k=0, n, k % 5); \\ _Michel Marcus_, Apr 28 2018
%o (Magma) I:=[0,1,3,6,10,10]; [n le 6 select I[n] else Self(n-1) + Self(n-5) - Self(n-6): n in [1..71]]; // _G. C. Greubel_, Aug 31 2019
%o (Sage)
%o def A130483_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P(x*(1+2*x+3*x^2+4*x^3)/((1-x^5)*(1-x))).list()
%o A130483_list(70) # _G. C. Greubel_, Aug 31 2019
%o (GAP) a:=[0,1,3,6,10,10];; for n in [7..71] do a[n]:=a[n-1]+a[n-5]-a[n-6]; od; a; # _G. C. Greubel_, Aug 31 2019
%Y Cf. A010872, A010873, A010875, A010876, A010877, A130481, A130482, A130484, A130485.
%K nonn,easy
%O 0,3
%A _Hieronymus Fischer_, May 29 2007
|
