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 A000969 Expansion of (1+x+2*x^2)/((1-x)^2*(1-x^3)). (Formerly M2630 N1042) 25
 1, 3, 7, 12, 18, 26, 35, 45, 57, 70, 84, 100, 117, 135, 155, 176, 198, 222, 247, 273, 301, 330, 360, 392, 425, 459, 495, 532, 570, 610, 651, 693, 737, 782, 828, 876, 925, 975, 1027, 1080, 1134, 1190, 1247, 1305, 1365, 1426, 1488, 1552, 1617, 1683, 1751, 1820, 1890 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Paul Curtz, Oct 07 2018: (Start) Terms that are on the x-axis of the following spiral (without 0):      28--29--29--30--31--31--32       |      27  13--14--15--15--16--17       |   |                   |      27  13   4---5---5---6  17       |   |   |           |   |      26  12   3   0---1   7  18       |   |   |       |   |   |      25  11   3---2---1   7  19       |   |               |   |      25  11--10---9---9---8  19       |                       |      24--23--23--22--21--21--20  (End) Diagonal 1, 4, 8, 13, 20, 28, ... (without 0) is A143978. - Bruno Berselli, Oct 08 2018 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS N. J. A. Sloane, Table of n, a(n) for n = 0..10000 David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980. P. A. Piza, Fermat coefficients, Math. Mag., 27 (1954), 141-146. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1). FORMULA a(n) = floor( (2*n+3)*(n+1)/3 ). Or, a(n) = (2*n+3)*(n+1)/3 but subtract 1/3 if n == 1 mod 3. - N. J. A. Sloane, May 05 2010. a(2^k-2) = A139250(2^k-1), k >= 1. - Omar E. Pol, Feb 13 2010 a(n) = Sum_{i=0..n} floor(4*i/3). - Enrique Pérez Herrero, Apr 21 2012 a(n) = +2*a(n-1) -1*a(n-2) +1*a(n-3) -2*a(n-4) +1*a(n-5). - Joerg Arndt, Apr 22 2012 a(n) = A014105(n+1) = A258708(n+3,n). - Reinhard Zumkeller, Jun 23 2015 MAPLE A000969:=-(1+z+2*z**2)/(z**2+z+1)/(z-1)**3; # Simon Plouffe in his 1992 dissertation MATHEMATICA f[x_, y_] := Floor[ Abs[ y/x - x/y]]; Table[ f[3, 2 n^2 + n + 2], {n, 53}] (* Robert G. Wilson v, Aug 11 2010 *) CoefficientList[Series[(1+x+2*x^2)/((1-x)^2*(1-x^3)), {x, 0, 50}], x] (* Stefano Spezia, Oct 08 2018 *) PROG (Haskell) a000969 = flip div 3 . a014105 . (+ 1)  -- Reinhard Zumkeller, Jun 23 2015 (PARI) a(n)=([0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1; 1, -2, 1, -1, 2]^n*[1; 3; 7; 12; 18])[1, 1] \\ Charles R Greathouse IV, May 10 2016 CROSSREFS Cf. A004773 (first differences), A092498 (partial sums). Cf. A139250, A160165, A258708, A014105. Sequence in context: A257941 A257944 A005228 * A194117 A122250 A169679 Adjacent sequences:  A000966 A000967 A000968 * A000970 A000971 A000972 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified October 23 19:18 EDT 2018. Contains 316530 sequences. (Running on oeis4.)