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a(n) = n*(9*n - 1)/2.
14

%I #94 Aug 06 2023 16:54:57

%S 0,4,17,39,70,110,159,217,284,360,445,539,642,754,875,1005,1144,1292,

%T 1449,1615,1790,1974,2167,2369,2580,2800,3029,3267,3514,3770,4035,

%U 4309,4592,4884,5185,5495,5814,6142,6479,6825,7180,7544,7917,8299,8690,9090,9499

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

%C From _Floor van Lamoen_, Jul 21 2001: (Start)

%C 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 The spiral begins:

%C 15

%C / \

%C 16 14

%C / \

%C 17 3 13

%C / / \ \

%C 18 4 2 12

%C / / \ \

%C 19 5 0---1 11

%C / / \

%C 20 6---7---8---9--10

%C (End)

%C a(n) with n>0 are the numbers with period length 3 in Bulgarian and Mancala solitaire. - _Paul Weisenhorn_ Jan 29 2022

%H G. C. Greubel, <a href="/A022266/b022266.txt">Table of n, a(n) for n = 0..5000</a>

%H 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 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 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = binomial(9*n,2)/9 for n >= 0. - _Zerinvary Lajos_, Jan 02 2007

%F a(n) = A049452(n) - A000326(n). - _Zerinvary Lajos_, Jun 12 2007

%F a(n) = 9*n + a(n-1) - 5 for n > 0, a(0)=0. - _Vincenzo Librandi_, Aug 04 2010

%F G.f.: x*(4 + 5*x)/(1 - x)^3. - _Colin Barker_, Feb 14 2012

%F a(n) = A218470(9*n+3). - _Philippe Deléham_, Mar 27 2013

%F a(n) = A000217(5*n-1) - A000217(4*n-1). - _Bruno Berselli_, Oct 17 2016

%F From _Wesley Ivan Hurt_, Dec 04 2016: (Start)

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

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

%F E.g.f.: (x/2)*(9*x + 8)*exp(x). - _G. C. Greubel_, Aug 24 2017

%F a(n) = A000326(3*n) / 3. - _Joerg Arndt_, May 04 2021

%p [seq(binomial(9*n,2)/9, n=0..37)]; # _Zerinvary Lajos_, Jan 02 2007

%p seq(n*(6*n-1)-n*(3*n-1)/2, n=0..37); # _Zerinvary Lajos_, Jun 12 2007

%t Table[n (9 n - 1)/2, {n, 0, 40}] (* _Bruno Berselli_, Oct 17 2016 *)

%t LinearRecurrence[{3,-3,1},{0,4,17},50] (* _Harvey P. Dale_, Aug 06 2023 *)

%o (PARI) a(n)=n*(9*n-1)/2 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Magma) [n*(9*n-1)/2 : n in [0..50]]; // _Wesley Ivan Hurt_, Dec 04 2016

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

%Y Cf. similar sequences listed in A022288.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_