A062783 a(n) = 3*n*(4*n-1).

0, 9, 42, 99, 180, 285, 414, 567, 744, 945, 1170, 1419, 1692, 1989, 2310, 2655, 3024, 3417, 3834, 4275, 4740, 5229, 5742, 6279, 6840, 7425, 8034, 8667, 9324, 10005, 10710, 11439, 12192, 12969, 13770, 14595, 15444, 16317, 17214, 18135, 19080

Offset: 0

Author

Floor van Lamoen, Jul 21 2001

Keywords

• easy
• nonn

Comments

Write 1, 2, 3, 4, ... counterclockwise in a hexagonal spiral around 0 starting left down, then a(n) is the sequence found by reading from 0 in the vertical downward direction.

Polygonal number connection: 2He_n + 7S_n where He_n is the n-th Heptagonal number and S_n is the n-th Square number. - William A. Tedeschi, Sep 12 2010

Examples

			The spiral begins:
......16..15..14
....17..5...4...13
..18..6...0...3...12
19..7...1...2...11..26
..20..8...9...10..25
....21..22..23..24
		

References

• L. Berzolari, Allgemeine Theorie der Höeren Ebenen Algebraischen Kurven, Encyclopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Band III_2. Heft 3, Leipzig: B. G. Teubner, 1906. p. 341.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Equals 3*A033991.

Cf. A000567, A063436.

Programs

Mathematica

s=0;lst={s};Do[s+=n++ +9;AppendTo[lst, s], {n, 0, 8!, 24}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 17 2008 *)

PARI

a(n)=3*n*(4*n-1) \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 24*n + a(n-1) - 15 with n > 0, a(0)=0. - Vincenzo Librandi, Aug 07 2010

G.f.: 3*x*(3 + 5*x)/(1-x)^3. - Colin Barker, Feb 28 2012

From Elmo R. Oliveira, Oct 31 2024: (Start)

E.g.f.: 3*x*(3 + 4*x)*exp(x).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)