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%I A022267 #82 Mar 24 2022 03:41:30
%S A022267 0,5,19,42,74,115,165,224,292,369,455,550,654,767,889,1020,1160,1309,
%T A022267 1467,1634,1810,1995,2189,2392,2604,2825,3055,3294,3542,3799,4065,
%U A022267 4340,4624,4917,5219,5530,5850,6179
%N A022267 a(n) = n*(9*n + 1)/2.
%C A022267 From _Floor van Lamoen_, Jul 21 2001: (Start)
%C A022267 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:
%C A022267 .
%C A022267             15
%C A022267             / \
%C A022267           16  14
%C A022267           /     \
%C A022267         17   3  13
%C A022267         /   / \   \
%C A022267       18   4   2  12
%C A022267       /   /     \   \
%C A022267     19   5   0---1  11
%C A022267     /   /             \
%C A022267   20   6---7---8---9--10
%C A022267 .
%C A022267 (End)
%C A022267 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
%C A022267 a(n) with n>0 are the numbers with the periodic length 3 in the Bulgarian and Mancala solitaire. - _Paul Weisenhorn_, Jan 29 2022
%H A022267 Lei Zhou, <a href="/A022267/b022267.txt">Table of n, a(n) for n = 0..10000</a>
%H A022267 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H A022267 Leo Tavares, <a href="/A022267/a022267.jpg">Illustration: X Triangles</a>
%H A022267 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A022267 a(n) = A110449(n, 4) for n>3.
%F A022267 From _Bruno Berselli_, Feb 11 2011: (Start)
%F A022267 G.f.: x*(5 + 4*x)/(1 - x)^3.
%F A022267 a(n) = 4*A000217(n) + A000566(n). (End)
%F A022267 a(n) = 9*n + a(n-1) - 4 with n>0, a(0)=0. - _Vincenzo Librandi_, Aug 04 2010
%F A022267 a(n) = A218470(9*n+4). - _Philippe Deléham_, Mar 27 2013
%F A022267 a(n) = A000217(5*n) - A000217(4*n). - _Bruno Berselli_, Oct 13 2016
%F A022267 E.g.f.: (1/2)*(9*x^2 + 10*x)*exp(x). - _G. C. Greubel_, Jul 17 2017
%F A022267 a(n) = A060544(n+1) - A016813(n). - _Leo Tavares_, Mar 20 2022
%p A022267 seq(binomial(9*n+1,2)/9, n=0..37); # _Zerinvary Lajos_, Jan 21 2007
%t A022267 Table[ n (9 n + 1)/2, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 19}, 40] (* _Harvey P. Dale_, Jul 01 2013 *)
%o A022267 (PARI) vector(100,n,(n-1)*(9*n-8)/2) \\ _Derek Orr_, Feb 06 2015
%Y A022267 Cf. A000217, A051682, A110449, A235332.
%Y A022267 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.
%Y A022267 Cf. similar sequences listed in A254963.
%Y A022267 Cf. similar sequences listed in A022289.
%Y A022267 Cf. A060544, A016813.
%K A022267 nonn,easy
%O A022267 0,2
%A A022267 _N. J. A. Sloane_

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