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Fifth subdiagonal in array of n-gonal numbers A081422.
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%I #70 Sep 08 2022 08:45:09

%S 1,7,24,58,115,201,322,484,693,955,1276,1662,2119,2653,3270,3976,4777,

%T 5679,6688,7810,9051,10417,11914,13548,15325,17251,19332,21574,23983,

%U 26565,29326,32272,35409,38743,42280,46026,49987,54169,58578,63220

%N Fifth subdiagonal in array of n-gonal numbers A081422.

%C One of a family of sequences with palindromic generators.

%C 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

%C 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

%C Row 1 of the convolution arrays A213831 and A213833. - _Clark Kimberling_, Jul 04 2012

%C Partial sums of A056109. - _J. M. Bergot_, Jun 22 2013

%C Number of ordered pairs of intersecting multisets of size 2, each chosen with repetition from {1,...,n}. - _Robin Whitty_, Feb 12 2014

%C Row sums of A244418. - _L. Edson Jeffery_, Jan 10 2015

%H Vincenzo Librandi, <a href="/A081436/b081436.txt">Table of n, a(n) for n = 0..5000</a>

%H J. A. Dias da Silva and P. J. Freitas, <a href="http://arxiv.org/abs/1305.1139">Counting Spectral Radii of Matrices with Positive Entries</a>, arXiv:1305.1139 [math.CO], 2013.

%H Theorem of the Day, <a href="http://www.theoremoftheday.org/Resources/TheoremNotes.htm#120">Lovász Local Lemma example involving intersecting pairs of multisets</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = (n+1)*(2*n^2 + 3*n + 2)/2.

%F G.f.: (1+x)*(1+2*x)/(1-x)^4. (Convolution of A005408 and A016777.)

%F a(n) = A110449(n, n-1), for n>1.

%F 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

%F E.g.f.: exp(x)*(2 + 12*x + 11*x^2 + 2*x^3)/2. - _Stefano Spezia_, Apr 13 2021

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Wesley Ivan Hurt_, Apr 14 2021

%p A081436 := proc(n)

%p (n+1)*(2*n^2+3*n+2)/2 ;

%p end proc:

%p seq(A081436(n),n=0..60) ; # _R. J. Mathar_, Jun 26 2013

%t LinearRecurrence[{4, -6, 4, -1}, {1, 7, 24, 58}, 40] (* _Jean-François Alcover_, Sep 21 2017 *)

%o (Magma) [(2*n^3+5*n^2+5*n+2)/2: n in [0..40]]; // _Vincenzo Librandi_, Jul 19 2011

%o (PARI) a(n)=n^3+5/2*n*(n+1)+1 \\ _Charles R Greathouse IV_, Jun 20 2013

%o (Sage) [(n+1)*(2*(n+1)^2-n)/2 for n in (0..40)] # _G. C. Greubel_, Aug 14 2019

%o (GAP) List([0..40], n-> (n+1)*(2*(n+1)^2-n)/2); # _G. C. Greubel_, Aug 14 2019

%Y Cf. A081434, A081435, A081437, A156933, A157705, A244418.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 21 2003

%E G.f. simplified and crossrefs added by _Johannes W. Meijer_, Mar 07 2009