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A084849 1+n+2*n^2. 31
1, 4, 11, 22, 37, 56, 79, 106, 137, 172, 211, 254, 301, 352, 407, 466, 529, 596, 667, 742, 821, 904, 991, 1082, 1177, 1276, 1379, 1486, 1597, 1712, 1831, 1954, 2081, 2212, 2347, 2486, 2629, 2776, 2927, 3082, 3241, 3404, 3571, 3742, 3917, 4096, 4279, 4466 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n)=A058331(n)+A000027(n).

a(n) = A014105(n) + 1; A100035(a(n)) = 1. - Reinhard Zumkeller, Oct 31 2004

Equals (1, 2, 3,...) convolved with (1, 2, 4, 4, 4,...). a(3) = 22 = (1, 2, 3, 4) dot (4, 4, 2, 1) = (4 + 8 + 6 + 4). [From Gary W. Adamson, May 01 2009]

a(n) is also the number of ways to place 2 nonattacking bishops on a 2 X (n+1) board. [From Vaclav Kotesovec, Jan 29 2010]

REFERENCES

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf

LINKS

Table of n, a(n) for n=0..47.

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

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

FORMULA

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

a(n)=ceiling((2n+1)^2/2)-n=A001844(n)-n; - Paul Barry, Jul 16 2006

Row sums of triangle A131901. A084849 = binomial transform of (1, 3, 4, 0, 0, 0,...). - Gary W. Adamson, Jul 26 2007

Equals A134082 * [1,2,3,...]. - Gary W. Adamson, Oct 07 2007

a(n)=(1+A000217(2n-1)+A000217(2n+1))/2 [From Enrique Pérez Herrero, Apr 02 2010]

a(n)=(A177342(n+1)-A177342(n))/2, with n>0. [From Bruno Berselli, May 19 2010]

a(n)-3*a(n-1)+3*a(n-2)-a(n-3)=0, with n>2. [From Bruno Berselli, May 24 2010]

a(n)=4*n+a(n-1)-1 (with a(0)=1) [From Vincenzo Librandi, Aug 08 2010]

With an offset of 1 the generating function is 2*t^2-3*t+2, which is the Alexander polynomial (with negative powers cleared) of the 3-twist knot. The associated Seifert matrix S is [[-1,-1],[0,-2]]. a(n-1) = det(transpose(S)-n*S). Cf. A060884. - Peter Bala, Mar 14 2012.

MATHEMATICA

s = 1; lst = {s}; Do[s += n + 2; AppendTo[lst, s], {n, 1, 200, 4}]; lst [From Zerinvary Lajos, Jul 11 2009]

f[n_]:=(n*(2*n+1)+1); Table[f[n], {n, 5!}] [From Vladimir Joseph Stephan Orlovsky, Feb 07 2010]

CROSSREFS

Cf. A100040, A100041, A100036, A100037, A100038, A100039, A131901, A134082.

Cf. A004767 (first differences). A060884.

Sequence in context: A038414 A008154 A008162 * A008265 A160424 A008229

Adjacent sequences:  A084846 A084847 A084848 * A084850 A084851 A084852

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Jun 09 2003

STATUS

approved

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Last modified July 30 19:06 EDT 2014. Contains 245074 sequences.