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

0,2

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).

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

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 *)

Prog

(Magma) [(9*n^2 + 7*n + 2)/2: n in [0..50]]; // Vincenzo Librandi, Aug 14 2014
(PARI) a(n)=(9*n^2+7*n+2)/2 \\ Charles R Greathouse IV, Jun 17 2017

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.

Sequence in context: A085367 A343560 A330395 * A052153 A154560 A338548

Adjacent sequences: A081264 A081265 A081266 * A081268 A081269 A081270

Keyword

easy,nonn,changed

Author

Paul Barry, Mar 15 2003

Status

approved