OFFSET
0,2
COMMENTS
One of a family of sequences with palindromic generators.
Also as A(n) = (1/6)*(6*n^3 - 3*n^2 + 3*n), n>0: structured pentagonal diamond numbers (vertex structure 5). (Cf. A004068 = alternate vertex; A000447 = structured diamonds; A100145 for more on structured numbers.) - James A. Record (james.record(AT)gmail.com), Nov 07 2004
Sequence of the absolute values of the z^1 coefficients of the polynomials in the GF4 denominators of A156933. See A157705 for background information. - Johannes W. Meijer, Mar 07 2009
Partial sums of A056109. - J. M. Bergot, Jun 22 2013
Number of ordered pairs of intersecting multisets of size 2, each chosen with repetition from {1,...,n}. - Robin Whitty, Feb 12 2014
Row sums of A244418. - L. Edson Jeffery, Jan 10 2015
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..5000
J. A. Dias da Silva and P. J. Freitas, Counting Spectral Radii of Matrices with Positive Entries, arXiv:1305.1139 [math.CO], 2013.
Theorem of the Day, Lovász Local Lemma example involving intersecting pairs of multisets
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = (n+1)*(2*n^2 + 3*n + 2)/2.
a(n) = A110449(n, n-1), for n>1.
a(n) = (n+1)*T(n+1) + n*T(n), where T( ) are triangular numbers. Binomial transform of [1, 6, 11, 6, 0, 0, 0, ...]. - Gary W. Adamson, Dec 28 2007
E.g.f.: exp(x)*(2 + 12*x + 11*x^2 + 2*x^3)/2. - Stefano Spezia, Apr 13 2021
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Apr 14 2021
MAPLE
A081436 := proc(n)
(n+1)*(2*n^2+3*n+2)/2 ;
end proc:
seq(A081436(n), n=0..60) ; # R. J. Mathar, Jun 26 2013
MATHEMATICA
LinearRecurrence[{4, -6, 4, -1}, {1, 7, 24, 58}, 40] (* Jean-François Alcover, Sep 21 2017 *)
PROG
(Magma) [(2*n^3+5*n^2+5*n+2)/2: n in [0..40]]; // Vincenzo Librandi, Jul 19 2011
(PARI) a(n)=n^3+5/2*n*(n+1)+1 \\ Charles R Greathouse IV, Jun 20 2013
(Sage) [(n+1)*(2*(n+1)^2-n)/2 for n in (0..40)] # G. C. Greubel, Aug 14 2019
(GAP) List([0..40], n-> (n+1)*(2*(n+1)^2-n)/2); # G. C. Greubel, Aug 14 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 21 2003
EXTENSIONS
G.f. simplified and crossrefs added by Johannes W. Meijer, Mar 07 2009
STATUS
approved