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A005915
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Hexagonal prism numbers: (n + 1)*(3*n^2 + 3*n + 1).
(Formerly M4933)
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10
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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; internal format)
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OFFSET
| 0,2
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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. 7, 2004.
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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.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| a(n)=(n+1)^3 + 6* (n*(n+1)*(2*n+1)/6) - Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de).
a(0)=1, a(1)=14, a(2)=57, a(3)=148, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4) [From Harvey P. Dale, June 25 2011]
G.f.: (1+10*x+7*x^2)/(1-x)^4. [From Harvey P. Dale, June 25 2011]
Equals row sums of triangle A143804 and binomial transform of [1, 13, 30, 18, 0, 0, 0,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2008]
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MAPLE
| A005915:=(1+10*z+7*z**2)/(z-1)**4; [Conjectured by S. Plouffe in his 1992 dissertation.]
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MATHEMATICA
| Table[(n+1)(3n^2+3n+1), {n, 0, 50}] (* From Harvey P. Dale, Mar 31 2011 *)
LinearRecurrence[{4, -6, 4, -1}, {1, 14, 57, 148}, 50] (* From Harvey P. Dale, June 25 2011 *)
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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);
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CROSSREFS
| A143804 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 01 2008]
Sequence in context: A041374 A070741 A022286 * A041376 A063537 A084195
Adjacent sequences: A005912 A005913 A005914 * A005916 A005917 A005918
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KEYWORD
| nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 24 1999
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