

A084849


a(n) = 1 + n + 2*n^2.


38



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
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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
Partial sums are A174723.  Wesley Ivan Hurt, Apr 16 2016
Also the number of irredundant sets in the ncocktail party graph.  Eric W. Weisstein, Aug 09 2017


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000
W. Burrows, C. Tuffley, Maximising common fixtures in a round robin tournament with two divisions, arXiv preprint arXiv:1502.06664 [math.CO], 2015.
GuoNiu Han, Enumeration of Standard Puzzles
GuoNiu 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.
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Irredundant Set
Index entries for 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(2n1) + 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(n1) + 3*a(n2)  a(n3) = 0, with n>2.  Bruno Berselli, May 24 2010
a(n) = 4*n + a(n1)  1 (with a(0)=1).  Vincenzo Librandi, Aug 08 2010
With an offset of 1 the generating function is 2*t^23*t+2, which is the Alexander polynomial (with negative powers cleared) of the 3twist knot. The associated Seifert matrix S is [[1,1],[0,2]]. a(n1) = det(transpose(S)n*S). Cf. A060884.  Peter Bala, Mar 14 2012
E.g.f.: (1 + 3*x + 2*x^2)*exp(x).  Ilya Gutkovskiy, Apr 16 2016


MAPLE

A084849:=n>1+n+2*n^2: seq(A084849(n), n=0..100); # Wesley Ivan Hurt, Apr 15 2016


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 *)
Table[1 + n + 2 n^2, {n, 0, 20}] (* Eric W. Weisstein, Aug 09 2017 *)
LinearRecurrence[{3, 3, 1}, {4, 11, 22}, {0, 20}] (* Eric W. Weisstein, Aug 09 2017 *)
CoefficientList[Series[(1  x  2 x^2)/(1 + x)^3, {x, 0, 20}], x] (* Eric W. Weisstein, Aug 09 2017 *)


PROG

(PARI) a(n)=1+n+2*n^2 \\ Charles R Greathouse IV, Sep 24 2015
(MAGMA) [1+n+2*n^2 : n in [0..100]]; // Wesley Ivan Hurt, Apr 15 2016


CROSSREFS

Cf. A100040, A100041, A100036, A100037, A100038, A100039, A131901, A134082.
Cf. A004767 (first differences), A060884, A174723.
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



