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A033113 Base-3 digits are, in order, the first n terms of the periodic sequence with initial period 1,0. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Partial sums of round(3^n/4). - Mircea Merca, Dec 28 2010

Written in base 3, this yields A033113. - 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

Index entries for linear recurrences with constant coefficients, signature (3,1,-3).

FORMULA

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) = (1/8)*(3^(n+1) - (-1)^n - 2), with n >= 1. - Paolo P. Lava, Jan 19 2009, simplified by M. F. Hasler, Oct 06 2018

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, n >= 1. - Wolfdieter Lang, Apr 13 2012

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 *)

PROG

(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

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

CROSSREFS

Pairwise sums seem to be in A003462.

Equals A039300 - 1.

Sequence in context: A014531 A062107 A269800 * A290718 A300421 A302289

Adjacent sequences:  A033110 A033111 A033112 * A033114 A033115 A033116

KEYWORD

nonn,base,easy

AUTHOR

Clark Kimberling

EXTENSIONS

Duplicate of a recurrence replaced with another recurrence by R. J. Mathar, Jun 28 2010

STATUS

approved

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Last modified May 24 04:25 EDT 2019. Contains 323528 sequences. (Running on oeis4.)