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

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, 160322, 173433, 187240

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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), pp. 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).

Crossrefs

Cf. A143804.

Cf. A260260 (comment). - Bruno Berselli, Jul 22 2015

Cf. A007588, A100145, A100157, A100174, A100188, A143804, A213829.

Programs

Magma

[(n + 1)*(3*n^2 + 3*n + 1): n in [0..50]]; // Vincenzo Librandi, May 16 2011

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 *)

PARI

a(n) = (n + 1)*(3*n^2 + 3*n + 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

E.g.f.: exp(x)*(1 + x)*(1 + 12*x + 3*x^2). - Elmo R. Oliveira, Aug 04 2025

Extensions

More terms from James Sellers, Dec 24 1999