A130483 a(n) = Sum_{k=0..n} (k mod 5) (Partial sums of A010874).

0, 1, 3, 6, 10, 10, 11, 13, 16, 20, 20, 21, 23, 26, 30, 30, 31, 33, 36, 40, 40, 41, 43, 46, 50, 50, 51, 53, 56, 60, 60, 61, 63, 66, 70, 70, 71, 73, 76, 80, 80, 81, 83, 86, 90, 90, 91, 93, 96, 100, 100, 101, 103, 106, 110, 110, 111, 113, 116, 120, 120, 121, 123, 126, 130, 130

Offset: 0

Author

Hieronymus Fischer, May 29 2007

Keywords

• nonn
• easy

Comments

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

Links

• Shawn A. Broyles, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Crossrefs

Cf. A010872, A010873, A010875, A010876, A010877, A130481, A130482, A130484, A130485.

Programs

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

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

Maple

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

Mathematica

Accumulate[Mod[Range[0,70],5]] (* or *) Accumulate[PadRight[{},70,{0,1,2,3,4}]] (* Harvey P. Dale, Nov 11 2016 *)

PARI

a(n) = sum(k=0, n, k % 5); \\ Michel Marcus, Apr 28 2018

Sage

def A130483_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1+2*x+3*x^2+4*x^3)/((1-x^5)*(1-x))).list()
A130483_list(70) # G. C. Greubel, Aug 31 2019

Formula

a(n) = 10*floor(n/5) + A010874(n)*(A010874(n)+1)/2.

G.f.: x*(1 + 2*x + 3*x^2 + 4*x^3)/((1-x^5)*(1-x)).

From Wesley Ivan Hurt, Jul 23 2016: (Start)

a(n) = a(n-5) - a(n-6) for n>5; a(n) = a(n-5) + 10 for n>4.

a(n) = 10 + Sum_{k=1..4} k*floor((n-k)/5). (End)

a(n) = ((n mod 5)^2 - 3*(n mod 5) + 4*n)/2. - Ammar Khatab, Aug 13 2020