A081267 Diagonal of triangular spiral in A051682.

1, 9, 26, 52, 87, 131, 184, 246, 317, 397, 486, 584, 691, 807, 932, 1066, 1209, 1361, 1522, 1692, 1871, 2059, 2256, 2462, 2677, 2901, 3134, 3376, 3627, 3887, 4156, 4434, 4721, 5017, 5322, 5636, 5959, 6291, 6632, 6982, 7341, 7709, 8086, 8472, 8867, 9271

Offset: 0

Author

Paul Barry, Mar 15 2003

Keywords

• easy
• nonn

Comments

Binomial transform of (1, 8, 9, 0, 0, 0, ...).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Milan Janjić, Two Enumerative Functions
• Richard J. Mathar, Bivariate generating functions enumerating non-bonding dominoes on rectangular boards, arXiv:2404.18806 (2024) Table 2.
• Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 7.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000326, A050509, A051682, A081268.

Cf. A220083 for a list of numbers of the form n*P(s,n)-(n-1)*P(s,n-1), where P(s,n) is the n-th polygonal number with s sides.

Programs

Magma

[(9*n^2 + 7*n + 2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 14 2014

Mathematica

LinearRecurrence[{3,-3,1},{1,9,26},50] (* Harvey P. Dale, Aug 13 2014 *)
CoefficientList[Series[(1 + 6 x + 2 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 14 2014 *)

PARI

a(n)=(9*n^2+7*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = C(n, 0) + 8*C(n, 1) + 9*C(n, 2).

a(n) = (9*n^2 + 7*n + 2)/2.

G.f.: (1 + 6*x + 2*x^2)/(1 - x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), for n > 2. a(n) = right term in M^n * [1 1 1], where M = the 3 X 3 matrix [1 0 0 / 3 1 0 / 5 3 1]. M^n * [1 1 1] = [1 3n+1 a(n)]. - Gary W. Adamson, Dec 22 2004

a(n) = 9*n + a(n-1) - 1 with n > 0, a(0)=1. - Vincenzo Librandi, Aug 08 2010

a(n) = (n+1)*A000326(n+1) - (n)*A000326(n). - Bruno Berselli, Dec 10 2012

a(n) = A050509(n) - A050509(n-1). - Bill McEachen, Nov 01 2020

E.g.f.: exp(x)*(2 + 16*x + 9*x^2)/2. - Stefano Spezia, Dec 25 2022