A064226 a(n) = (9*n^2 + 13*n + 6)/2.

3, 14, 34, 63, 101, 148, 204, 269, 343, 426, 518, 619, 729, 848, 976, 1113, 1259, 1414, 1578, 1751, 1933, 2124, 2324, 2533, 2751, 2978, 3214, 3459, 3713, 3976, 4248, 4529, 4819, 5118, 5426, 5743, 6069, 6404, 6748, 7101, 7463, 7834, 8214, 8603, 9001, 9408, 9824

Offset: 0

Author

N. J. A. Sloane, Sep 22 2001

Keywords

• nonn
• easy

Comments

Diagonal of triangular spiral in A051682. - Paul Barry, Mar 15 2003

Ehrhart polynomial of open quadrilateral with vertices (0,2),(2,3),(3,1),(2,0). - Michael Somos, Jul 22 2006

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Milan Janjic, Two Enumerative Functions.
• National Security Agency, Intrigued? (advertisement), Notices of the American Mathematical Society, Vol. 49 (2002), p. 216.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A051682, A064225, A081267, A081268, A235332.

Programs

Magma

I:=[3,14,34]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jul 19 2015

Maple

A064226:=n-> (9*n^2 + 13*n + 6) / 2; seq(A064226(n), n=0..50); # Wesley Ivan Hurt, May 08 2014

Mathematica

Table[(9 n^2 + 13 n + 6)/2, {n, 0, 50}] (* Wesley Ivan Hurt, May 08 2014 *)
LinearRecurrence[{3, -3, 1}, {3, 14, 34}, 50] (* Vincenzo Librandi, Jul 19 2015 *)

PARI

{a(n) = 3 + n * (9*n + 13) / 2}; /* Michael Somos, Jul 22 2006 */

Formula

From Paul Barry, Mar 15 2003: (Start)

a(n) = 3*C(n,0) + 11*C(n,1) + 9*C(n,2); binomial transform of (3, 11, 9, 0, 0, 0, ...).

G.f.: (3 + 5*x + x^2)/(1-x)^3.

a(n) = A081268(n) + 2. (End)

A064225(n) = a(-1-n). - Michael Somos, Jul 22 2006

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Wesley Ivan Hurt, Apr 16 2023

E.g.f.: (3 + 11*x + 9*x^2/2)*exp(x). - Elmo R. Oliveira, Oct 21 2024