A033113 Base-3 digits are, in order, the first n terms of the periodic sequence with initial period 1,0.

1, 3, 10, 30, 91, 273, 820, 2460, 7381, 22143, 66430, 199290, 597871, 1793613, 5380840, 16142520, 48427561, 145282683, 435848050, 1307544150, 3922632451, 11767897353, 35303692060, 105911076180, 317733228541, 953199685623

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Written in base 3, this yields A056830. - M. F. Hasler, Oct 05 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 906
• R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 6.
• Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

Crossrefs

Cf. A003462, A039300.

Programs

Magma

[Round(3^(n+1)/8): n in [1..30]]; // Vincenzo Librandi, Jun 25 2011

Maple

a[0]:=0: a[1]:=1: for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2]+1 od: seq(a[n], n=1..33);# Zerinvary Lajos, Dec 14 2008
g:=x*(1/(1-3*x)/(1-x))/(1+x): gser:=series(g, x=0, 43): seq(coeff(gser, x, n), n=1..30);# Zerinvary Lajos, Jan 11 2009
A033113 := proc(n) (9*3^(n-1)-(-1)^n-2)/8 ; end proc: # R. J. Mathar, Jan 08 2011

Mathematica

Join[{a=1,b=3},Table[c=2*b+3*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
Module[{nn=30,d},d=PadRight[{},nn,{1,0}];Table[FromDigits[Take[d,n],3],{n,nn}]] (* or *) LinearRecurrence[{3,1,-3},{1,3,10},30] (* Harvey P. Dale, May 24 2014 *)

PARI

a(n)=3^n*3\8 \\ Simplified by M. F. Hasler, Oct 06 2018

PARI

A033113(n)=3^(n+1)>>3 \\ M. F. Hasler, Oct 05 2018

Formula

a(n) = A039300(n)-1.

a(n)+a(n+1) = A003462(n+1).

a(n) = 3*a(n-1) + a(n-2) -3*a(n-3). - R. J. Mathar, Jun 28 2010

From Paul Barry, Nov 12 2003: (Start)

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

a(n) = 2*a(n-1) + 3*a(n-2) + 1.

Partial sums of A015518. (End)

E.g.f.: (1/2)*exp(x)*(sinh(x))^2. - Paul Barry, Mar 12 2003

a(n) = Sum_{k=0..floor(n/2)} C(n+2, 2k+2)*4^k. - Paul Barry, Aug 24 2003

a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k); a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^(j+k)*3^j. - Paul Barry, Nov 12 2003

Convolution of A000244 and A059841 (3^n and periodic{1, 0}). a(n) = Sum_{k=0..n} (1 + (-1)^(n-k))*3^k/2. - Paul Barry, Jul 19 2004

a(n) = round(3^(n+1)/8) = floor((3^(n+1)-1)/8) = ceiling((3^(n+1)-3)/8) = round((3^(n+1)-3)/8). a(n) = a(n-2) + 3^(n-1), n > 2. - Mircea Merca, Dec 27 2010

a(n) = floor((3^(n+1))/4) / 2 = A081251(n)/2, n >= 1. - Wolfdieter Lang, Apr 13 2012