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A081267 Diagonal of triangular spiral in A051682. 9
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 (list; graph; refs; listen; history; text; internal format)
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 Janjic, Two Enumerative Functions

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

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, 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: A022421 A075395 A085367 * A052153 A154560 A249275

Adjacent sequences:  A081264 A081265 A081266 * A081268 A081269 A081270

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 15 2003

STATUS

approved

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Last modified December 5 22:37 EST 2019. Contains 329782 sequences. (Running on oeis4.)