A000340 a(0)=1, a(n) = 3*a(n-1) + n + 1. (Formerly M3882 N1592)

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

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)

Partial sums of A003462, and thus the second partial sums of A000244 (3^n). Also column k=2 of A106516. - John Keith, Jan 04 2022

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 389

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

László Tóth, On Schizophrenic Patterns in b-ary Expansions of Some Irrational Numbers, arXiv:2002.06584 [math.NT], 2020. Mentions this sequence. See also Proc. Amer. Math. Soc. 148 (2020), 461-469.

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) = 5*a(n-1) - 7*a(n-2) + 3*a(n-3). - 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); # Miklos Kristof, Mar 09 2005
A000340:=-1/(3*z-1)/(z-1)**2; # conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

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.

Equals column k=1 of A008971 (shifted). - Jeremy Dover, Jul 11 2021

Cf. A000340, A003462 (first differences), A106516.

Sequence in context: A364553 A335551 A235612 * A301880 A034567 A133648

Adjacent sequences: A000337 A000338 A000339 * A000341 A000342 A000343

Keyword

nonn,easy

Author

N. J. A. Sloane, Simon Plouffe

Status

approved