A022266 a(n) = n*(9*n - 1)/2.

0, 4, 17, 39, 70, 110, 159, 217, 284, 360, 445, 539, 642, 754, 875, 1005, 1144, 1292, 1449, 1615, 1790, 1974, 2167, 2369, 2580, 2800, 3029, 3267, 3514, 3770, 4035, 4309, 4592, 4884, 5185, 5495, 5814, 6142, 6479, 6825, 7180, 7544, 7917, 8299, 8690, 9090, 9499

Offset

0,2

Comments

From Floor van Lamoen, Jul 21 2001: (Start)

Write 0,1,2,3,4,... in a triangular spiral, then a(n) is the sequence found by reading the line from 0 in the direction 0,4,...

The spiral begins:

15

/ \

16 14

/ \

17 3 13

/ / \ \

18 4 2 12

/ / \ \

19 5 0---1 11

/ / \

20 6---7---8---9--10

(End)

a(n) with n>0 are the numbers with period length 3 in Bulgarian and Mancala solitaire. - Paul Weisenhorn Jan 29 2022

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].

Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).

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

Formula

a(n) = binomial(9*n,2)/9 for n >= 0. - Zerinvary Lajos, Jan 02 2007

a(n) = A049452(n) - A000326(n). - Zerinvary Lajos, Jun 12 2007

a(n) = 9*n + a(n-1) - 5 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 04 2010

G.f.: x*(4 + 5*x)/(1 - x)^3. - Colin Barker, Feb 14 2012

a(n) = A218470(9*n+3). - Philippe Deléham, Mar 27 2013

a(n) = A000217(5*n-1) - A000217(4*n-1). - Bruno Berselli, Oct 17 2016

From Wesley Ivan Hurt, Dec 04 2016: (Start)

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

a(n) = (1/7) * Sum_{i=n..(8*n-1)} i. (End)

E.g.f.: (x/2)*(9*x + 8)*exp(x). - G. C. Greubel, Aug 24 2017

a(n) = A000326(3*n) / 3. - Joerg Arndt, May 04 2021

Maple

[seq(binomial(9*n, 2)/9, n=0..37)]; # Zerinvary Lajos, Jan 02 2007
seq(n*(6*n-1)-n*(3*n-1)/2, n=0..37); # Zerinvary Lajos, Jun 12 2007

Mathematica

Table[n (9 n - 1)/2, {n, 0, 40}] (* Bruno Berselli, Oct 17 2016 *)
LinearRecurrence[{3, -3, 1}, {0, 4, 17}, 50] (* Harvey P. Dale, Aug 06 2023 *)

Prog

(PARI) a(n)=n*(9*n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015
(Magma) [n*(9*n-1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Dec 04 2016

Crossrefs

Cf. A000217, A000326, A022267, A049452, A051682, A218470, A235332.

Cf. similar sequences listed in A022288.

Sequence in context: A182868 A178947 A041859 * A273309 A145995 A275815

Adjacent sequences: A022263 A022264 A022265 * A022267 A022268 A022269

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved