A130486 - OEIS

%I #28 Oct 21 2022 21:59:47

%S 0,1,3,6,10,15,21,28,28,29,31,34,38,43,49,56,56,57,59,62,66,71,77,84,

%T 84,85,87,90,94,99,105,112,112,113,115,118,122,127,133,140,140,141,

%U 143,146,150,155,161,168,168,169,171,174,178,183,189,196,196,197,199,202,206

%N a(n) = Sum_{k=0..n} (k mod 8) (Partial sums of A010877).

%C Let A be the Hessenberg n X n matrix defined by A[1,j] = j mod 8, 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="/A130486/b130486.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,1,-1).

%F a(n) = 28*floor(n/8) + A010877(n)*(A010877(n) + 1)/2.

%F G.f.: (Sum_{k=1..7} k*x^k)/((1-x^8)*(1-x)).

%F G.f.: x*(1 - 8*x^7 + 7*x^8)/((1-x^8)*(1-x)^3).

%p seq(coeff(series(x*(1-8*x^7+7*x^8)/((1-x^8)*(1-x)^3), x, n+1), x, n), n = 0 .. 40); # _G. C. Greubel_, Aug 31 2019

%t Array[28 Floor[#1/8] + #2 (#2 + 1)/2 & @@ {#, Mod[#, 8]} &, 61, 0] (* _Michael De Vlieger_, Apr 28 2018 *)

%t Accumulate[PadRight[{},100,Range[0,7]]] (* _Harvey P. Dale_, Dec 21 2018 *)

%o (PARI) a(n) = sum(k=0, n, k % 8); \\ _Michel Marcus_, Apr 28 2018

%o (Magma) I:=[0,1,3,6,10,15,21,28,28]; [n le 9 select I[n] else Self(n-1) + Self(n-8) - Self(n-9): n in [1..71]]; // _G. C. Greubel_, Aug 31 2019

%o (Sage)

%o def A130486_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P(x*(1-8*x^7+7*x^8)/((1-x^8)*(1-x)^3)).list()

%o A130486_list(70) # _G. C. Greubel_, Aug 31 2019

%o (GAP) a:=[0,1,3,6,10,15,21,28,28];; for n in [10..71] do a[n]:=a[n-1]+a[n-8]-a[n-9]; od; a; # _G. C. Greubel_, Aug 31 2019

%Y Cf. A010872, A010873, A010874, A010875, A010876, A010878. A130481, A130482, A130483, A130484, A130485, A130487.

%K nonn,easy

%O 0,3

%A _Hieronymus Fischer_, May 31 2007