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A000340 a(0)=1, a(n)=3*a(n-1)+n+1.
(Formerly M3882 N1592)
25
1, 5, 18, 58, 179, 543, 1636, 4916, 14757, 44281, 132854, 398574, 1195735, 3587219, 10761672, 32285032, 96855113, 290565357, 871696090, 2615088290, 7845264891, 23535794695, 70607384108, 211822152348, 635466457069 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Johannes W. Meijer, Feb 20 2009: (Start)

Second right hand column (n-m=1) of the A156920 triangle.

The generating function of this sequence enabled the analysis of the polynomials A156921 and A156925.

(End)

REFERENCES

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 389

Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

FORMULA

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

a(n)=(3^(n+2)-2*n-5)/4.

a(n)=sum{k=0..n+1, (n-k+1)*3^k}=sum{k=0..n+1, k*3^(n-k+1)}. - Paul Barry, Jul 30 2004

a(n)=sum{k=0..n, binomial(n+2, k+2)*2^k}. - Paul Barry, Jul 30 2004

a(-1)=0, a(0)=1, a(n)=4*a(n-1)-3*a(n-2)+1. - Miklos Kristof, Mar 09 2005

a(n) = right term of M^(n+1) * [1,0,0], where M = the 3 X 3 matrix [1,0,0; 1,1,0; 1,1,3]. E.g., a(4) = 179 since M^5 = [1, 5, 179]. - Gary W. Adamson, Dec 28 2006

a(n)=5*a(n-1)-7*a(n-2)+3*a(n-3) with a(0)=1, a(1)=5 and a(2)=18. - Johannes W. Meijer, Feb 20 2009

a(-2 - n) = 3^-n * A014915(n). - Michael Somos, May 28 2014

EXAMPLE

G.f. = 1 + 5*x + 18*x^2 + 58*x^3 + 179*x^4 + 543*x^5 + 1636*x^6 + ...

MAPLE

a[ -1]:=0:a[0]:=1:for n from 1 to 50 do a[n]:=4*a[n-1]-3*a[n-2]+1 od: seq(a[n], n=0..50); # (Kristof)

A000340:=-1/(3*z-1)/(z-1)**2; # Conjectured by Simon Plouffe in his 1992 dissertation

a:=n->sum(3^(n-j)*j, j=0..n): seq(a(n), n=1..25); # Zerinvary Lajos, Jun 07 2008

MATHEMATICA

lst={}; s=0; Do[s+=(s+(n+=s)); AppendTo[lst, s], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2008 *)

a[ n_] := MatrixPower[ {{1, 0, 0}, {1, 1, 0}, {1, 1, 3}}, n + 1][[3, 1]]; (* Michael Somos, May 28 2014 *)

RecurrenceTable[{a[0]==1, a[n]==3a[n-1]+n+1}, a, {n, 30}] (* or *) LinearRecurrence[{5, -7, 3}, {1, 5, 18}, 30] (* Harvey P. Dale, Jan 31 2017 *)

PROG

(MAGMA) [(3^(n+2)-2*n-5)/4: n in [0..30]]; // Vincenzo Librandi, Aug 15 2011

CROSSREFS

From Johannes W. Meijer, Feb 20 2009: (Start)

Cf. A156921, A156925, A156927, A156933. Other columns A156922, A156923, A156924.

Equals A156920 second right hand column.

Equals A142963 second right hand column divided by 2^n

Equals A156919 second right hand column divided by 2.

(End)

Cf. A014915.

Sequence in context: A128553 A190163 A235612 * A034567 A133648 A284968

Adjacent sequences:  A000337 A000338 A000339 * A000341 A000342 A000343

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified August 17 20:59 EDT 2017. Contains 290655 sequences.