A116948 -id:A116948 - cp's OEIS frontend

A117571 Expansion of (1+2x^2)/((1-x)(1-x^3)).

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

Offset: 0

Paul Barry, Mar 29 2006

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)

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).

Cf. A001045, A116948, A273724.

Cf. A090858, A127002, A323055, A325242, A325243, A325244, A325270.

[1 + Floor(2*n/3) + Floor((n+1)/3) : n in [0..100]]; // Wesley Ivan Hurt, Jul 25 2016

A117571:=n->1 + floor(2*n/3) + floor((n+1)/3): seq(A117571(n), n=0..100); # Wesley Ivan Hurt, Jul 25 2016

CoefficientList[Series[(1 + 2 x^2)/((1 - x) (1 - x^3)), {x, 0, 71}], x] (* Michael De Vlieger, May 13 2016 *)

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

A116949 Riordan array ((1-x^3)/(1+2x^2),x).

Original entry on oeis.org

1, 0, 1, -2, 0, 1, -1, -2, 0, 1, 4, -1, -2, 0, 1, 2, 4, -1, -2, 0, 1, -8, 2, 4, -1, -2, 0, 1, -4, -8, 2, 4, -1, -2, 0, 1, 16, -4, -8, 2, 4, -1, -2, 0, 1, 8, 16, -4, -8, 2, 4, -1, -2, 0, 1, -32, 8, 16, -4, -8, 2, 4, -1, -2, 0, 1, -16, -32, 8, 16, -4, -8, 2, 4, -1, -2, 0, 1, 64, -16, -32, 8, 16, -4, -8, 2, 4, -1, -2, 0, 1

Offset: 0

Paul Barry, Mar 29 2006

Sequence array of (-2)^(n/2)(1+(-1)^n)/2-(-2)^((n-3)/2)(1-(-1)^n)/2-(comb(1,n)-comb(0,n))/2.
Inverse of A116948.
Row sums are A117575. Diagonal sums are A117576.

			Number triangle begins
1,
0, 1,
-2, 0, 1,
-1, -2, 0, 1,
4, -1, -2, 0, 1,
2, 4, -1, -2, 0, 1,
-8, 2, 4, -1, -2, 0, 1,
-4, -8, 2, 4, -1, -2, 0, 1,
16, -4, -8, 2, 4, -1, -2, 0, 1

Cf. A117575, A117576, A116948.

Entry revised by N. J. A. Sloane, Apr 10 2006

cp's OEIS Frontend

A117571 Expansion of (1+2x^2)/((1-x)(1-x^3)).

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A116949 Riordan array ((1-x^3)/(1+2x^2),x).

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cp's OEIS Frontend

A117571 Expansion of (1+2*x^2)/((1-x)*(1-x^3)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

A116949 Riordan array ((1-x^3)/(1+2x^2),x).

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A117571 Expansion of (1+2x^2)/((1-x)(1-x^3)).