A130490 - OEIS

%I #28 Jan 16 2024 17:29:10

%S 0,1,3,6,10,15,21,28,36,45,55,66,66,67,69,72,76,81,87,94,102,111,121,

%T 132,132,133,135,138,142,147,153,160,168,177,187,198,198,199,201,204,

%U 208,213,219,226,234,243,253,264,264,265,267,270,274,279,285,292,300

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

%C 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

%H Shawn A. Broyles, <a href="/A130490/b130490.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

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

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

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

%p 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

%t Sum[Mod[k, 12], {k, 0, Range[0, 60]}] (* _G. C. Greubel_, Sep 01 2019 *)

%t 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 *)

%o (PARI) a(n) = sum(k=0, n, k % 12); \\ _Michel Marcus_, Apr 29 2018

%o (Magma) [&+[(k mod 12): k in [0..n]]: n in [0..60]]; // _G. C. Greubel_, Sep 01 2019

%o (Sage) [sum(k%12 for k in (0..n)) for n in (0..60)] # _G. C. Greubel_, Sep 01 2019

%o (GAP) List([0..60], n-> Sum([0..n], k-> k mod 12 )); # _G. C. Greubel_, Sep 01 2019

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

%K nonn

%O 0,3

%A _Hieronymus Fischer_, May 31 2007