A022267 a(n) = n*(9*n + 1)/2.

0, 5, 19, 42, 74, 115, 165, 224, 292, 369, 455, 550, 654, 767, 889, 1020, 1160, 1309, 1467, 1634, 1810, 1995, 2189, 2392, 2604, 2825, 3055, 3294, 3542, 3799, 4065, 4340, 4624, 4917, 5219, 5530, 5850, 6179

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, 5, ... . 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) is the sum of n consecutive integers starting from 4*n+1: (5), (9+10), (13+14+15), ... - Klaus Purath, Jul 07 2020

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

Links

• Lei Zhou, Table of n, a(n) for n = 0..10000
• Milan Janjic, Two Enumerative Functions
• Leo Tavares, Illustration: X Triangles
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A051682, A110449, A235332.

Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A045944, A186029, A007742, A033429, A022268, A049452, A186030, A135703, A152734, A139273.

Cf. similar sequences listed in A254963.

Cf. similar sequences listed in A022289.

Cf. A060544, A016813.

Programs

Maple

seq(binomial(9*n+1,2)/9, n=0..37); # Zerinvary Lajos, Jan 21 2007

Mathematica

Table[ n (9 n + 1)/2, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 19}, 40] (* Harvey P. Dale, Jul 01 2013 *)

PARI

vector(100,n,(n-1)*(9*n-8)/2) \\ Derek Orr, Feb 06 2015

Formula

a(n) = A110449(n, 4) for n>3.

From Bruno Berselli, Feb 11 2011: (Start)

G.f.: x*(5 + 4*x)/(1 - x)^3.

a(n) = 4*A000217(n) + A000566(n). (End)

a(n) = 9*n + a(n-1) - 4 with n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010

a(n) = A218470(9*n+4). - Philippe Deléham, Mar 27 2013

a(n) = A000217(5*n) - A000217(4*n). - Bruno Berselli, Oct 13 2016

E.g.f.: (1/2)*(9*x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017

a(n) = A060544(n+1) - A016813(n). - Leo Tavares, Mar 20 2022