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%I M4933 #60 Sep 08 2022 08:44:34
%S 1,14,57,148,305,546,889,1352,1953,2710,3641,4764,6097,7658,9465,
%T 11536,13889,16542,19513,22820,26481,30514,34937,39768,45025,50726,
%U 56889,63532,70673,78330,86521,95264,104577,114478,124985,136116,147889
%N Hexagonal prism numbers: a(n) = (n + 1)*(3*n^2 + 3*n + 1).
%C Also as a(n)=(1/6)*(18*n^3-18*n^2+6*n), n>0: structured rhombic dodecahedral numbers (vertex structure 7) (A100157 = alternate vertex); structured tetrakis hexahedral numbers (vertex structure 7) (Cf. A100174 = alternate vertex); and structured hexagonal anti-diamond numbers (vertex structure 7) (Cf. A007588 = alternate vertex) (Cf. A100188 = structured anti-diamonds). Cf. A100145 for more on structured polyhedral numbers. - James A. Record (james.record(AT)gmail.com), Nov 07 2004
%C a(n) is the number of 4-tuples (w,x,y,z) with all terms in {0,...,n} and w=x or x=y or y=z. - _Clark Kimberling_, May 31 2012
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
%H Vincenzo Librandi, <a href="/A005915/b005915.txt">Table of n, a(n) for n = 0..1000</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 <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)^3 + 6* (n*(n+1)*(2*n+1)/6). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=14, a(2)=57, a(3)=148. - _Harvey P. Dale_, Jun 25 2011
%F G.f.: (1+10*x+7*x^2)/(1-x)^4. - _Harvey P. Dale_, Jun 25 2011
%F Equals row sums of triangle A143804 and binomial transform of [1, 13, 30, 18, 0, 0, 0, ...]. - _Gary W. Adamson_, Sep 01 2008
%F 2*a(n+1) = A213829(n). - _Clark Kimberling_, Jul 04 2012
%p A005915:=(1+10*z+7*z**2)/(z-1)**4; # Conjectured by _Simon Plouffe_ in his 1992 dissertation
%t Table[(n+1)(3n^2+3n+1),{n,0,50}] (* _Harvey P. Dale_, Mar 31 2011 *)
%t LinearRecurrence[{4,-6,4,-1},{1,14,57,148},50] (* _Harvey P. Dale_, Jun 25 2011 *)
%o (Magma) [(n + 1)*(3*n^2 + 3*n + 1): n in [0..50]]; // _Vincenzo Librandi_, May 16 2011
%o (PARI) a(n) = (n + 1)*(3*n^2 + 3*n + 1);
%Y Cf. A143804.
%Y Cf. A260260 (comment). - _Bruno Berselli_, Jul 22 2015
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Dec 24 1999
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