A130490 a(n) = Sum_{k=0..n} (k mod 12) (Partial sums of A010881).

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 66, 67, 69, 72, 76, 81, 87, 94, 102, 111, 121, 132, 132, 133, 135, 138, 142, 147, 153, 160, 168, 177, 187, 198, 198, 199, 201, 204, 208, 213, 219, 226, 234, 243, 253, 264, 264, 265, 267, 270, 274, 279, 285, 292, 300

Offset: 0

Author

Hieronymus Fischer, May 31 2007

Keywords

• nonn

Comments

Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 12, 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 (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Crossrefs

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A010878, A010879, A010880, A130481, A130482, A130483, A130484, A130485, A130486, A130487, A130488, A130489.

Programs

GAP

List([0..60], n-> Sum([0..n], k-> k mod 12 )); # G. C. Greubel, Sep 01 2019

Magma

[&+[(k mod 12): k in [0..n]]: n in [0..60]]; // G. C. Greubel, Sep 01 2019

Maple

seq(coeff(series(x*(1-12*x^11+11*x^12)/((1-x^12)*(1-x)^3), x, n+1), x, n), n = 0..60); # G. C. Greubel, Sep 01 2019

Mathematica

Sum[Mod[k, 12], {k, 0, Range[0, 60]}] (* G. C. Greubel, Sep 01 2019 *)
LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,0,1,-1},{0,1,3,6,10,15,21,28,36,45,55,66,66},60] (* Harvey P. Dale, Jan 16 2024 *)

PARI

a(n) = sum(k=0, n, k % 12); \\ Michel Marcus, Apr 29 2018

Sage

[sum(k%12 for k in (0..n)) for n in (0..60)] # G. C. Greubel, Sep 01 2019

Formula

a(n) = 66*floor(n/12) + A010881(n)*(A010881(n) + 1)/2.

G.f.: (Sum_{k=1..11} k*x^k)/((1-x^12)*(1-x)).

G.f.: x*(1 - 12*x^11 + 11*x^12)/((1-x^12)*(1-x)^3).