A130485 a(n) = Sum_{k=0..n} (k mod 7) (Partial sums of A010876).

0, 1, 3, 6, 10, 15, 21, 21, 22, 24, 27, 31, 36, 42, 42, 43, 45, 48, 52, 57, 63, 63, 64, 66, 69, 73, 78, 84, 84, 85, 87, 90, 94, 99, 105, 105, 106, 108, 111, 115, 120, 126, 126, 127, 129, 132, 136, 141, 147, 147, 148, 150, 153, 157, 162, 168, 168, 169, 171, 174, 178, 183

Offset

0,3

Comments

Let A be the Hessenberg n X n matrix defined by A[1,j] = j mod 7, 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 (1,0,0,0,0,0,1,-1).

Formula

a(n) = 21*floor(n/7) + A010876(n)*(A010876(n) + 1)/2.

G.f.: (Sum_{k=1..6} k*x^k)/((1-x^7)*(1-x)).

G.f.: x*(1 - 7*x^6 + 6*x^7)/((1-x^7)*(1-x)^3).

Maple

a:=n->add(chrem( [n, j], [1, 7] ), j=1..n):seq(a(n), n=1..70); # Zerinvary Lajos, Apr 07 2009

Mathematica

LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 6, 10, 15, 21, 21}, 70] (* Harvey P. Dale, Jul 30 2017 *)

Prog

(PARI) concat(0, Vec((1-7*x^6+6*x^7)/(1-x^7)/(1-x)^3+O(x^70))) \\ Charles R Greathouse IV, Dec 22 2011
(Magma) I:=[0, 1, 3, 6, 10, 15, 21, 21]; [n le 8 select I[n] else Self(n-1) + Self(n-7) - Self(n-8): n in [1..71]]; // G. C. Greubel, Aug 31 2019
(Sage)
def A130485_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P(x*(1-7*x^6+6*x^7)/((1-x^7)*(1-x)^3)).list()
A130485_list(70) # G. C. Greubel, Aug 31 2019
(GAP) a:=[0, 1, 3, 6, 10, 15, 21, 21];; for n in [9..71] do a[n]:=a[n-1]+a[n-7]-a[n-8]; od; a; # G. C. Greubel, Aug 31 2019

Crossrefs

Cf. A010872, A010873, A010874, A010875, A010877, A130481, A130482, A130483, A130484.

Sequence in context: A368825 A306698 A139131 * A115015 A231676 A056150

Adjacent sequences: A130482 A130483 A130484 * A130486 A130487 A130488

Keyword

nonn,easy

Author

Hieronymus Fischer, May 31 2007

Status

approved