A093485 a(n) = (27*n^2 + 9*n + 2)/2.

1, 19, 64, 136, 235, 361, 514, 694, 901, 1135, 1396, 1684, 1999, 2341, 2710, 3106, 3529, 3979, 4456, 4960, 5491, 6049, 6634, 7246, 7885, 8551, 9244, 9964, 10711, 11485, 12286, 13114, 13969, 14851, 15760, 16696, 17659, 18649, 19666, 20710, 21781

Offset: 0

Author

Michael Joseph Halm, May 13 2004

Keywords

• nonn
• easy

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

Examples

			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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A000330, A003215, A005901, A006656.

Programs

Haskell

a093485 n = (9 * n * (3 * n + 1) + 2) `div` 2
-- Reinhard Zumkeller, Jun 16 2013

Magma

[(27*n^2 + 9*n + 2)/2 : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011

PARI

a(n)=(27*n^2+9*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

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

Extensions

New definition from Ralf Stephan, Dec 01 2004

Name corrected and initial term added by Arkadiusz Wesolowski, Aug 15 2011