A005920 - OEIS

%I M4611 #57 Sep 08 2022 08:44:34

%S 1,9,33,82,165,291,469,708,1017,1405,1881,2454,3133,3927,4845,5896,

%T 7089,8433,9937,11610,13461,15499,17733,20172,22825,25701,28809,32158,

%U 35757,39615,43741,48144,52833,57817,63105,68706,74629,80883,87477,94420

%N Tricapped prism numbers.

%C a(n) = (n+1)*A000326(n+1) - Sum_{i=0...n} A001477(i) = (n+1)*((n+1)*(3*n+2)/2) - A000217(n) = (n+1)*(3*n^2+4n+2)/2. - _Bruno Berselli_, Apr 25 2010

%C Also central terms of triangle A093445: a(n) = A093445(2*n+1,n+1). - _Reinhard Zumkeller_, Oct 03 2012

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Reinhard Zumkeller, <a href="/A005920/b005920.txt">Table of n, a(n) for n = 0..10000</a>

%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H B. K. Teo and N. J. A. Sloane, <a href="http://neilsloane.com/doc/magic1/magic1.html">Magic numbers in polygonal and polyhedral clusters</a>, Inorgan. Chem. 24 (1985),4545-4558.

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

%F a(n) = (1/2) * (3*n^3 + 7*n^2 + 6*n + 2). - _Ralf Stephan_, Apr 20 2004

%F a(0)=1, a(1)=9, a(2)=33, a(3)=82, a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Harvey P. Dale_, Sep 25 2012

%F E.g.f.: exp(x)*(2 + 16*x + 16*x^2 + 3*x^3)/2. - _Stefano Spezia_, Jun 10 2022

%p a:=n->(3*n^3+7*n^2+6*n+2)/2: seq(a(n),n=0..60);

%p A005920:=(1+5*z+3*z**2)/(z-1)**4; # conjectured by _Simon Plouffe_ in his 1992 dissertation

%t CoefficientList[ Series[(1+5x+3x^2)/(1-x)^4, {x, 0, 39}], x] (* _Jean-François Alcover_, Dec 02 2011, after _Simon Plouffe_ *)

%t LinearRecurrence[{4,-6,4,-1},{1,9,33,82},40] (* _Harvey P. Dale_, Sep 25 2012 *)

%o (Haskell)

%o a005920 n = (n * (n * (3 * n + 7) + 6) + 2) `div` 2

%o -- _Reinhard Zumkeller_, Oct 03 2012

%o (PARI) a(n)=n*(3*n^2+7*n+6)/2+1 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Magma) [(3*n^3+7*n^2+6*n+2)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, May 05 2021

%Y Cf. for recursive method [Ar(m) is the m-th term of a sequence in the OEIS] a(n) = n*Ar(n) - A000217(n-1) or a(n) = (n+1)*Ar(n+1) - A000217(n) or similar: A081436, A005945, A006003 and the terms T(2, n) or T(3, n) in the sequence A125860. - _Bruno Berselli_, Apr 25 2010

%Y Cf. A000326, A001477, A093445.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Emeric Deutsch_, May 09 2004