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A000340
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a(0)=1, a(n)=3*a(n-1)+n+1.
(Formerly M3882 N1592)
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19
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| Contribution 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)
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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).
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 389
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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]
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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 S. Plouffe in his 1992 dissertation.]
a:=n->sum(3^(n-j)*j, j=0..n): seq(a(n), n=1..25); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 07 2008
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MATHEMATICA
| lst={}; s=0; Do[s+=(s+(n+=s)); AppendTo[lst, s], {n, 5!}]; lst [From Vladimir Orlovsky, Oct 10 2008]
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PROG
| (MAGMA) [(3^(n+2)-2*n-5)/4: n in [0..30]]; // Vincenzo Librandi, Aug 15 2011
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CROSSREFS
| Contribution 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)
Sequence in context: A000745 A128553 A190163 * A034567 A133648 A099449
Adjacent sequences: A000337 A000338 A000339 * A000341 A000342 A000343
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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