A130481 a(n) = Sum_{k=0..n} (k mod 3) (i.e., partial sums of A010872).

0, 1, 3, 3, 4, 6, 6, 7, 9, 9, 10, 12, 12, 13, 15, 15, 16, 18, 18, 19, 21, 21, 22, 24, 24, 25, 27, 27, 28, 30, 30, 31, 33, 33, 34, 36, 36, 37, 39, 39, 40, 42, 42, 43, 45, 45, 46, 48, 48, 49, 51, 51, 52, 54, 54, 55, 57, 57, 58, 60, 60, 61, 63, 63, 64, 66, 66, 67, 69, 69, 70, 72, 72

Offset: 0

Author

Hieronymus Fischer, May 29 2007

Keywords

• nonn
• easy

Comments

Essentially the same as A092200. - R. J. Mathar, Jun 13 2008

Let A be the Hessenberg n X n matrix defined by: A[1,j]=j mod 3, A[i,i]:=1, A[i,i-1]=-1. Then, for n>=1, a(n)=det(A). - Milan Janjic, Jan 24 2010

2-adic valuation of A104537(n+1). - Gerry Martens, Jul 14 2015

Conjecture: a(n) is the exponent of the largest power of 2 that divides all the entries of the matrix {{3,1},{1,-1}}^n. - Greg Dresden, Sep 09 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A010872, A010873, A010874, A010875, A010876, A010877, A130482, A130483, A130484.

Cf. A092200, A104537.

Programs

GAP

List([0..80], n-> Int((n+1)/3) + Int(2*(n+1)/3)); # G. C. Greubel, Aug 31 2019

Magma

[Floor((n+1)/3) + Floor(2*(n+1)/3): n in [0..80]]; // G. C. Greubel, Aug 31 2019

Maple

seq(coeff(series(x*(1+2*x)/((1-x^3)*(1-x)), x, n+1), x, n), n = 0..80); # G. C. Greubel, Aug 31 2019

Mathematica

a[n_]:= Floor[(n+1)/3] + Floor[2(n+1)/3]; Table[a[n], {n, 0, 80}] (* Clark Kimberling, May 28 2012 *)
a[n_]:= IntegerExponent[A104537[n + 1], 2];
Table[a[n], {n, 0, 80}]  (* Gerry Martens, Jul 14 2015 *)
CoefficientList[Series[x(1+2x)/((1-x^3)(1-x)), {x, 0, 80}], x] (* Stefano Spezia, Sep 09 2018 *)
LinearRecurrence[{1,0,1,-1},{0,1,3,3},100] (* Harvey P. Dale, Jun 14 2021 *)

PARI

main(size)=my(n,k);vector(size,n,sum(k=0,n,k%3)) \\ Anders Hellström, Jul 14 2015

PARI

first(n)=my(s); concat(0, vector(n,k,s+=k%3)) \\ Charles R Greathouse IV, Jul 14 2015

PARI

a(n)=n\3*3+[0,1,3][n%3+1] \\ Charles R Greathouse IV, Jul 14 2015

Sage

def A130481_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x*(1+2*x)/((1-x^3)*(1-x))).list()
A130481_list(80) # G. C. Greubel, Aug 31 2019

Formula

a(n) = 3*floor(n/3) + A010872(n)*(A010872(n) + 1)/2.

G.f.: x*(1 + 2*x)/((1-x^3)*(1-x)).

a(n) = n + 1 - (Fibonacci(n+1) mod 2). - Gary Detlefs, Mar 13 2011

a(n) = floor((n+1)/3) + floor(2*(n+1)/3). - Clark Kimberling, May 28 2010

a(n) = n when n+1 is not a multiple of 3, and a(n) = n+1 when n+1 is a multiple of 3. - Dennis P. Walsh, Aug 06 2012

a(n) = n + 1 - sign((n+1) mod 3). - Wesley Ivan Hurt, Sep 25 2017

a(n) = n + (1-cos(2*(n+2)*Pi/3))/3 + sin(2*(n+2)*Pi/3)/sqrt(3). - Wesley Ivan Hurt, Sep 27 2017

a(n) = n + 1 - (n+1)^2 mod 3. - Ammar Khatab, Aug 14 2020

E.g.f.: ((1 + 3*x)*cosh(x) - (cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))*(cosh(x/2) - sinh(x/2)) + (1 + 3*x)*sinh(x))/3. - Stefano Spezia, May 28 2021

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) + log(2)/3. - Amiram Eldar, Sep 17 2022