A130482 a(n) = Sum_{k=0..n} (k mod 4) (Partial sums of A010873).

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, 37, 39, 42, 42, 43, 45, 48, 48, 49, 51, 54, 54, 55, 57, 60, 60, 61, 63, 66, 66, 67, 69, 72, 72, 73, 75, 78, 78, 79, 81, 84, 84, 85, 87, 90, 90, 91, 93, 96, 96, 97, 99, 102, 102, 103, 105

Offset: 0

Author

Hieronymus Fischer, May 29 2007

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A010872, A010874, A010875, A010876, A010877. A130481, A130483, A130484, A130485.

Programs

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

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

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 *)

PARI

a(n) = (1 - (-1)^n - (2*I)*(-I)^n + (2*I)*I^n + 6*n) / 4 \\ Colin Barker, Oct 15 2015

Sage

def A130482_list(prec):
    P. = 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

Formula

a(n) = 6*floor(n/4) + A010873(n)*(A010873(n)+1)/2.

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