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A005915 Hexagonal prism numbers: a(n) = (n + 1)*(3*n^2 + 3*n + 1).
(Formerly M4933)
16
1, 14, 57, 148, 305, 546, 889, 1352, 1953, 2710, 3641, 4764, 6097, 7658, 9465, 11536, 13889, 16542, 19513, 22820, 26481, 30514, 34937, 39768, 45025, 50726, 56889, 63532, 70673, 78330, 86521, 95264, 104577, 114478, 124985, 136116, 147889 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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
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
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985), 4545-4558.
LINKS
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
FORMULA
a(n)=(n+1)^3 + 6* (n*(n+1)*(2*n+1)/6). - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)
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
G.f.: (1+10*x+7*x^2)/(1-x)^4. - Harvey P. Dale, Jun 25 2011
Equals row sums of triangle A143804 and binomial transform of [1, 13, 30, 18, 0, 0, 0, ...]. - Gary W. Adamson, Sep 01 2008
2*a(n+1) = A213829(n). - Clark Kimberling, Jul 04 2012
MAPLE
A005915:=(1+10*z+7*z**2)/(z-1)**4; # Conjectured by Simon Plouffe in his 1992 dissertation
MATHEMATICA
Table[(n+1)(3n^2+3n+1), {n, 0, 50}] (* Harvey P. Dale, Mar 31 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 14, 57, 148}, 50] (* Harvey P. Dale, Jun 25 2011 *)
PROG
(Magma) [(n + 1)*(3*n^2 + 3*n + 1): n in [0..50]]; // Vincenzo Librandi, May 16 2011
(PARI) a(n) = (n + 1)*(3*n^2 + 3*n + 1);
CROSSREFS
Cf. A143804.
Cf. A260260 (comment). - Bruno Berselli, Jul 22 2015
Sequence in context: A041374 A070741 A022286 * A211069 A212675 A041376
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from James A. Sellers, Dec 24 1999
STATUS
approved

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Last modified March 29 05:48 EDT 2024. Contains 371265 sequences. (Running on oeis4.)