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 A035608 Expansion of x*(1 + 3*x)/((1 + x)*(1 - x)^3). 47
 0, 1, 5, 10, 18, 27, 39, 52, 68, 85, 105, 126, 150, 175, 203, 232, 264, 297, 333, 370, 410, 451, 495, 540, 588, 637, 689, 742, 798, 855, 915, 976, 1040, 1105, 1173, 1242, 1314, 1387, 1463, 1540, 1620, 1701, 1785, 1870, 1958, 2047, 2139, 2232, 2328, 2425 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Maximum value of Voronoi's principal quadratic form of the first type when variables restricted to {-1,0,1}. - Michael Somos, Mar 10 2004 Row sums of triangle A133983. - Gary W. Adamson, Sep 30 2007 This is the main row of a version of the "square spiral" when read alternatively from left to right (see link). See also A001107, A007742, A033954, A033991. It is easy to see that the only prime in the sequence is 5. - Emilio Apricena (emilioapricena(AT)yahoo.it), Feb 08 2009 From Mitch Phillipson, Manda Riehl, Tristan Williams, Mar 06 2009: (Start) a(n) gives the number of elements of S_2 \wr C_k that avoid the pattern 12, using the following ordering: In S_j, a permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a3, a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Harvey P. Dale, Feb 21 2013 For n>1: a(n) = a(n-2) + 4*n - 3; see also row sums of triangle A253146. - Reinhard Zumkeller, Dec 27 2014 a(n) = 3*A002620(n) + A002620(n+1). - R. J. Mathar, Jul 18 2015 MAPLE A035608:=n->floor((n + 1/4)^2): seq(A035608(n), n=0..100); # Wesley Ivan Hurt, Oct 29 2017 MATHEMATICA Table[n^2 + Floor[n/2], {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Apr 12 2011 *) CoefficientList[Series[x (1 + 3 x)/((1 + x) (1 - x)^3), {x, 0, 60}], x] (* or *) LinearRecurrence[{2, 0, -2, 1}, {0, 1, 5, 10}, 60] (* Harvey P. Dale, Feb 21 2013 *) PROG (PARI) a(n)=n^2+n-1-(n-1)\2 (MAGMA) [n^2 + n - 1 - Floor((n-1)/2): n in [0..25]]; // G. C. Greubel, Oct 29 2017 CROSSREFS Partial sums of A042948. Cf. A133983. Cf. A253146. Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951. Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754. Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335. Sequence in context: A313991 A208953 A092390 * A091386 A164004 A025010 Adjacent sequences:  A035605 A035606 A035607 * A035609 A035610 A035611 KEYWORD nonn,easy AUTHOR STATUS approved

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Last modified April 10 08:09 EDT 2021. Contains 342845 sequences. (Running on oeis4.)