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 A022266 a(n) = n*(9*n - 1)/2. 14
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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). 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 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 *) PROG (PARI) a(n)=n*(9*n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015 (Magma) [n*(9*n-1)/2 : n in [0..50]]; // Wesley Ivan Hurt, Dec 04 2016 CROSSREFS Cf. A000217, A000326, A022267, A049452, A051682, A218470, A235332. Cf. similar sequences listed in A022288. Sequence in context: A182868 A178947 A041859 * A273309 A145995 A275815 Adjacent sequences: A022263 A022264 A022265 * A022267 A022268 A022269 KEYWORD nonn,easy AUTHOR N. J. A. Sloane STATUS approved

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Last modified June 17 11:50 EDT 2024. Contains 373445 sequences. (Running on oeis4.)