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A084849 a(n) = 1 + n + 2*n^2. 33
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). - 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. - Vaclav Kotesovec, Jan 29 2010

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]

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.

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. - Enrique Pérez Herrero, Apr 02 2010

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

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

a(n) = 4*n + a(n-1) - 1 (with a(0)=1). - 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 (* Zerinvary Lajos, Jul 11 2009 *)

f[n_]:=(n*(2*n+1)+1); Table[f[n], {n, 5!}] (* 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 April 24 20:26 EDT 2015. Contains 257064 sequences.