OFFSET
0,2
COMMENTS
Dodecahedral gnomon numbers: first differences of dodecahedral numbers.
The sequence is related to other gnomon numbers of polyhedra, known by other more familiar names: triangular numbers (tetrahedral gnomon numbers), hexagonal numbers (cubic gnomon numbers), square pyramidal numbers (octahedral gnomon numbers).
A124388 = first differences; second differences = 27. - Reinhard Zumkeller, Oct 30 2006
Sums of the triangular numbers from A000217(3*n-1) to A000217(3*n+1), with A000217(-1) = 0. - Bruno Berselli, Sep 04 2018
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
FORMULA
a(n) = (n+1)*(3*(n+1)-1)*(3*(n+1)-2)/2-n*(3*n-1)*(3*n-2)/2.
G.f.: (1 + 16*x + 10*x^2)/(1 - x)^3. - Colin Barker, Mar 28 2012
EXAMPLE
a(1) = 19 because (1+1)*(3*(1+1)-1)*(3*(1+1)-2)/2-1*(3*1-1)*(3*1-2)/2 = 2*(6-1)*(6-2)/2 - 1*(3-1)*(3-2)/2 = 20-1 = 19.
PROG
(Magma) [(27*n^2 + 9*n + 2)/2 : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011
(Haskell)
a093485 n = (9 * n * (3 * n + 1) + 2) `div` 2
-- Reinhard Zumkeller, Jun 16 2013
(PARI) a(n)=(27*n^2+9*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael Joseph Halm, May 13 2004
EXTENSIONS
New definition from Ralf Stephan, Dec 01 2004
Name corrected and initial term added by Arkadiusz Wesolowski, Aug 15 2011
STATUS
approved