A130484 a(n) = Sum_{k=0..n} (k mod 6) (Partial sums of A010875).

0, 1, 3, 6, 10, 15, 15, 16, 18, 21, 25, 30, 30, 31, 33, 36, 40, 45, 45, 46, 48, 51, 55, 60, 60, 61, 63, 66, 70, 75, 75, 76, 78, 81, 85, 90, 90, 91, 93, 96, 100, 105, 105, 106, 108, 111, 115, 120, 120, 121, 123, 126, 130, 135, 135, 136, 138, 141, 145, 150, 150, 151, 153

Offset

0,3

Comments

Let A be the Hessenberg n X n matrix defined by A[1,j] = j mod 6, 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 (1,0,0,0,0,1,-1).

Formula

a(n) = 15*floor(n/6) + A010875(n)*(A010875(n) + 1)/2.

G.f.: (Sum_{k=1..5} k*x^k)/((1-x^6)*(1-x)) = x*(1 - 6*x^5 + 5*x^6)/((1-x^6)*(1-x)^3).

Maple

seq(coeff(series(x*(1-6*x^5+5*x^6)/((1-x^6)*(1-x)^3), x, n+1), x, n), n = 0 .. 70); # G. C. Greubel, Aug 31 2019

Mathematica

Accumulate[Mod[Range[0, 70], 6]] (* or *) Accumulate[PadRight[ {}, 70, Range[0, 5]]] (* Harvey P. Dale, Jul 12 2016 *)

Prog

(PARI) a(n) = sum(k=0, n, k % 6); \\ Michel Marcus, Apr 28 2018
(PARI) a(n)=n\6*15 + binomial(n%6+1, 2) \\ Charles R Greathouse IV, Jan 24 2022
(Magma) I:=[0, 1, 3, 6, 10, 15, 15]; [n le 7 select I[n] else Self(n-1) + Self(n-6) - Self(n-7): n in [1..71]]; // G. C. Greubel, Aug 31 2019
(Sage)
def A130484_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P(x*(1-6*x^5+5*x^6)/((1-x^6)*(1-x)^3)).list()
A130484_list(70) # G. C. Greubel, Aug 31 2019
(GAP) a:=[0, 1, 3, 6, 10, 15, 15];; for n in [8..71] do a[n]:=a[n-1]+a[n-6]-a[n-7]; od; a; # G. C. Greubel, Aug 31 2019

Crossrefs

Cf. A010872, A010873, A010874, A010876, A010877, A130481, A130482, A130483, A130485.

Sequence in context: A105333 A126234 A259604 * A074374 A109804 A231672

Adjacent sequences: A130481 A130482 A130483 * A130485 A130486 A130487

Keyword

nonn,easy

Author

Hieronymus Fischer, May 31 2007

Status

approved