|
%I #43 Feb 16 2024 15:01:20
%S 0,1,3,6,6,7,9,12,12,13,15,18,18,19,21,24,24,25,27,30,30,31,33,36,36,
%T 37,39,42,42,43,45,48,48,49,51,54,54,55,57,60,60,61,63,66,66,67,69,72,
%U 72,73,75,78,78,79,81,84,84,85,87,90,90,91,93,96,96,97,99,102,102,103,105
%N a(n) = Sum_{k=0..n} (k mod 4) (Partial sums of A010873).
%C Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 4, A[i,i]:=1, A[i,i-1]=-1. Then, for n>=1, a(n)=det(A). - _Milan Janjic_, Jan 24 2010
%H Colin Barker, <a href="/A130482/b130482.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F a(n) = 6*floor(n/4) + A010873(n)*(A010873(n)+1)/2.
%F G.f.: x*(1 + 2*x + 3*x^2)/((1-x^4)*(1-x)).
%F a(n) = (1 - (-1)^n - (2*i)*(-i)^n + (2*i)*i^n + 6*n) / 4 where i = sqrt(-1). - _Colin Barker_, Oct 15 2015
%F a(n) = 3*n/2 + (n mod 2)* ( (n-1) mod 4 ) - (n mod 2)/2. - _Ammar Khatab_, Aug 27 2020
%F E.g.f.: (3*x*exp(x) - 2*sin(x) + sinh(x))/2. - _Stefano Spezia_, Apr 22 2021
%F Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + log(3)/4. - _Amiram Eldar_, Sep 17 2022
%p a:=n->add(chrem( [n,j], [1,4] ),j=1..n):seq(a(n), n=0..70); # _Zerinvary Lajos_, Apr 07 2009
%t Table[(6*n +(1-(-1)^n)*(1+2*I^(n+1)))/4, {n,0,70}] (* _G. C. Greubel_, Aug 31 2019 *)
%t LinearRecurrence[{1,0,0,1,-1},{0,1,3,6,6},80] (* _Harvey P. Dale_, Feb 16 2024 *)
%o (PARI) a(n) = (1 - (-1)^n - (2*I)*(-I)^n + (2*I)*I^n + 6*n) / 4 \\ _Colin Barker_, Oct 15 2015
%o (Magma) I:=[0,1,3,6,6]; [n le 5 select I[n] else Self(n-1) + Self(n-4) - Self(n-5): n in [1..71]]; // _G. C. Greubel_, Aug 31 2019
%o (Sage)
%o def A130482_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P(x*(1+2*x+3*x^2)/((1-x^4)*(1-x))).list()
%o A130482_list(70) # _G. C. Greubel_, Aug 31 2019
%o (GAP) a:=[0,1,3,6,6];; for n in [6..71] do a[n]:=a[n-1]+a[n-4]-a[n-5]; od; a; # _G. C. Greubel_, Aug 31 2019
%Y Cf. A010872, A010874, A010875, A010876, A010877. A130481, A130483, A130484, A130485.
%K nonn,easy
%O 0,3
%A _Hieronymus Fischer_, May 29 2007
|
