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 A117571 Expansion of (1+2*x^2)/((1-x)*(1-x^3)). 17
 1, 1, 3, 4, 4, 6, 7, 7, 9, 10, 10, 12, 13, 13, 15, 16, 16, 18, 19, 19, 21, 22, 22, 24, 25, 25, 27, 28, 28, 30, 31, 31, 33, 34, 34, 36, 37, 37, 39, 40, 40, 42, 43, 43, 45, 46, 46, 48, 49, 49, 51, 52, 52, 54, 55, 55, 57, 58, 58, 60, 61, 61, 63, 64, 64, 66, 67, 67, 69, 70, 70, 72 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Row sums of A116948. Place n+2 equally-spaced points around a circle, labeled 0,1,2,...,n+1. For each i = 0..n+1 such that 2i != i mod n+2, draw an (undirected) chord from i to 2i mod n+2. Then a(n) is the number of distinct chords. - Kival Ngaokrajang, May 13 2016 (Edited by N. J. A. Sloane, Jun 23 2016) From Gus Wiseman, Apr 19 2019: (Start) Also the number of integer partitions of n + 2 with 1 fewer distinct multiplicities than (not necessarily distinct) parts. These are partitions of the form (x,x), (x,y), (x,x,y), or (x,y,y). For example, the a(0) = 1 through a(8) = 9 partitions are the following. The Heinz numbers of these partitions are given by A325270.   (11)  (21)  (22)   (32)   (33)   (43)   (44)   (54)   (55)               (31)   (41)   (42)   (52)   (53)   (63)   (64)               (211)  (221)  (51)   (61)   (62)   (72)   (73)                      (311)  (411)  (322)  (71)   (81)   (82)                                    (331)  (332)  (441)  (91)                                    (511)  (422)  (522)  (433)                                           (611)  (711)  (442)                                                         (622)                                                         (811) (End) LINKS Kival Ngaokrajang, Illustration of initial terms Burkard Polster, Times Tables, Mandelbrot and the Heart of Mathematics, Mathologer video (2015). Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1). FORMULA G.f.: (1+2*x^2)/((1-x)*(1-x^3)). a(n) = a(n-1) + a(n-3) - a(n-4) for n>3. a(n) = cos(2*Pi*n/3+Pi/6)/sqrt(3)-sin(2*Pi*n/3+Pi/6)/3+(3n+2)/3. a(n) = Sum_{k=0..n} 2*A001045(L((n-k+2)/3)) where L(j/p) is the Legendre symbol of j and p. a(n) = 1 + floor((n+1)/3) + floor(2*n/3). - Wesley Ivan Hurt, Jul 25 2016 a(n) = n+sign((n-1) mod 3). - Wesley Ivan Hurt, Sep 25 2017 MAPLE A117571:=n->1 + floor(2*n/3) + floor((n+1)/3): seq(A117571(n), n=0..100); # Wesley Ivan Hurt, Jul 25 2016 MATHEMATICA CoefficientList[Series[(1 + 2 x^2)/((1 - x) (1 - x^3)), {x, 0, 71}], x] (* Michael De Vlieger, May 13 2016 *) PROG [1 + Floor(2*n/3) + Floor((n+1)/3) : n in [0..100]]; // Wesley Ivan Hurt, Jul 25 2016 CROSSREFS Cf. A001045, A116948, A273724. Cf. A090858, A127002, A323055, A325242, A325243, A325244, A325270. Sequence in context: A240728 A164326 A317645 * A008474 A111611 A100478 Adjacent sequences:  A117568 A117569 A117570 * A117572 A117573 A117574 KEYWORD easy,nonn AUTHOR Paul Barry, Mar 29 2006 STATUS approved

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Last modified May 27 21:12 EDT 2020. Contains 334671 sequences. (Running on oeis4.)