A081271 Vertical of triangular spiral in A051682.

1, 13, 34, 64, 103, 151, 208, 274, 349, 433, 526, 628, 739, 859, 988, 1126, 1273, 1429, 1594, 1768, 1951, 2143, 2344, 2554, 2773, 3001, 3238, 3484, 3739, 4003, 4276, 4558, 4849, 5149, 5458, 5776, 6103, 6439, 6784, 7138, 7501, 7873, 8254, 8644, 9043, 9451

Offset: 0

Author

Paul Barry, Mar 15 2003

Keywords

• nonn
• easy
• changed

Comments

Lies to the right of the y-axis of the triangle.

Binomial transform of (1, 12, 9, 0, 0, 0, ...).

Links

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

Crossrefs

Cf. A062741, A283394 (see Crossrefs section).

Programs

Mathematica

LinearRecurrence[{3,-3,1},{1,13,34},50] (* Harvey P. Dale, Aug 30 2025 *)

PARI

a(n)=(9*n^2+15*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: (1 + 10*x - 2*x^2)/(1 - x)^3.

a(n) = binomial(n,0) + 12*binomial(n,1) + 9*binomial(n,2).

a(n) = (9*n^2 + 15*n + 2)/2.

a(0) = 1, a(n) = a(n-1) + 9*n + 3 for n > 0 - Gerald McGarvey, Aug 18 2004

From Elmo R. Oliveira, Oct 25 2024: (Start)

E.g.f.: exp(x)*(1 + 12*x + 9*x^2/2).

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