OFFSET
0,3
COMMENTS
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
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
FORMULA
G.f.: x*(1 + 2*x + 3*x^2)/((1-x^4)*(1-x)).
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
a(n) = 3*n/2 + (n mod 2)* ( (n-1) mod 4 ) - (n mod 2)/2. - Ammar Khatab, Aug 27 2020
E.g.f.: (3*x*exp(x) - 2*sin(x) + sinh(x))/2. - Stefano Spezia, Apr 22 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)) + log(3)/4. - Amiram Eldar, Sep 17 2022
MAPLE
a:=n->add(chrem( [n, j], [1, 4] ), j=1..n):seq(a(n), n=0..70); # Zerinvary Lajos, Apr 07 2009
MATHEMATICA
Table[(6*n +(1-(-1)^n)*(1+2*I^(n+1)))/4, {n, 0, 70}] (* G. C. Greubel, Aug 31 2019 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 3, 6, 6}, 80] (* Harvey P. Dale, Feb 16 2024 *)
PROG
(PARI) a(n) = (1 - (-1)^n - (2*I)*(-I)^n + (2*I)*I^n + 6*n) / 4 \\ Colin Barker, Oct 15 2015
(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
(Sage)
def A130482_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P(x*(1+2*x+3*x^2)/((1-x^4)*(1-x))).list()
A130482_list(70) # G. C. Greubel, Aug 31 2019
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Hieronymus Fischer, May 29 2007
STATUS
approved