cp's OEIS Frontend

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

%I A022266 #94 Aug 06 2023 16:54:57
%S A022266 0,4,17,39,70,110,159,217,284,360,445,539,642,754,875,1005,1144,1292,
%T A022266 1449,1615,1790,1974,2167,2369,2580,2800,3029,3267,3514,3770,4035,
%U A022266 4309,4592,4884,5185,5495,5814,6142,6479,6825,7180,7544,7917,8299,8690,9090,9499
%N A022266 a(n) = n*(9*n - 1)/2.
%C A022266 From _Floor van Lamoen_, Jul 21 2001: (Start)
%C A022266 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,...
%C A022266 The spiral begins:
%C A022266             15
%C A022266             / \
%C A022266           16  14
%C A022266           /     \
%C A022266         17   3  13
%C A022266         /   / \   \
%C A022266       18   4   2  12
%C A022266       /   /     \   \
%C A022266     19   5   0---1  11
%C A022266     /   /             \
%C A022266   20   6---7---8---9--10
%C A022266 (End)
%C A022266 a(n) with n>0 are the numbers with period length 3 in Bulgarian and Mancala solitaire. - _Paul Weisenhorn_ Jan 29 2022
%H A022266 G. C. Greubel, <a href="/A022266/b022266.txt">Table of n, a(n) for n = 0..5000</a>
%H A022266 Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3471358">The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences</a>, Politecnico di Torino, Italy (2019), [math.NT].
%H A022266 Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3470205">The groupoid of the Triangular Numbers and the generation of related integer sequences</a>, Politecnico di Torino, Italy (2019).
%H A022266 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A022266 a(n) = binomial(9*n,2)/9 for n >= 0. - _Zerinvary Lajos_, Jan 02 2007
%F A022266 a(n) = A049452(n) - A000326(n). - _Zerinvary Lajos_, Jun 12 2007
%F A022266 a(n) = 9*n + a(n-1) - 5 for n > 0, a(0)=0. - _Vincenzo Librandi_, Aug 04 2010
%F A022266 G.f.: x*(4 + 5*x)/(1 - x)^3. - _Colin Barker_, Feb 14 2012
%F A022266 a(n) = A218470(9*n+3). - _Philippe Deléham_, Mar 27 2013
%F A022266 a(n) = A000217(5*n-1) - A000217(4*n-1). - _Bruno Berselli_, Oct 17 2016
%F A022266 From _Wesley Ivan Hurt_, Dec 04 2016: (Start)
%F A022266 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
%F A022266 a(n) = (1/7) * Sum_{i=n..(8*n-1)} i. (End)
%F A022266 E.g.f.: (x/2)*(9*x + 8)*exp(x). - _G. C. Greubel_, Aug 24 2017
%F A022266 a(n) = A000326(3*n) / 3. - _Joerg Arndt_, May 04 2021
%p A022266 [seq(binomial(9*n,2)/9, n=0..37)]; # _Zerinvary Lajos_, Jan 02 2007
%p A022266 seq(n*(6*n-1)-n*(3*n-1)/2, n=0..37); # _Zerinvary Lajos_, Jun 12 2007
%t A022266 Table[n (9 n - 1)/2, {n, 0, 40}] (* _Bruno Berselli_, Oct 17 2016 *)
%t A022266 LinearRecurrence[{3,-3,1},{0,4,17},50] (* _Harvey P. Dale_, Aug 06 2023 *)
%o A022266 (PARI) a(n)=n*(9*n-1)/2 \\ _Charles R Greathouse IV_, Oct 07 2015
%o A022266 (Magma) [n*(9*n-1)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Dec 04 2016
%Y A022266 Cf. A000217, A000326, A022267, A049452, A051682, A218470, A235332.
%Y A022266 Cf. similar sequences listed in A022288.
%K A022266 nonn,easy
%O A022266 0,2
%A A022266 _N. J. A. Sloane_