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A130485
a(n) = Sum_{k=0..n} (k mod 7) (Partial sums of A010876).
21
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
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
KEYWORD
nonn,easy
AUTHOR
Hieronymus Fischer, May 31 2007
STATUS
approved