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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

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

Cf. similar sequences listed in A022288.

Programs

Magma

[n*(9*n-1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Dec 04 2016

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 *)

PARI

a(n)=n*(9*n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

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