A005915 Hexagonal prism numbers: a(n) = (n + 1)*(3*n^2 + 3*n + 1). (Formerly M4933)

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

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

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

Adjacent sequences: A005912 A005913 A005914 * A005916 A005917 A005918

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Dec 24 1999

Status

approved