A124072 First differences of A129819.

0, 1, 0, 2, 1, 3, 1, 4, 2, 5, 2, 6, 3, 7, 3, 8, 4, 9, 4, 10, 5, 11, 5, 12, 6, 13, 6, 14, 7, 15, 7, 16, 8, 17, 8, 18, 9, 19, 9, 20, 10, 21, 10, 22, 11, 23, 11, 24, 12, 25, 12, 26

Offset

0,4

Comments

A129819 and its repeated differences are

0.0.1..1..3..4..7...8..12..14.19..21.27....

..0.1..0..2..1..3...1...4...2..5...2..6....

....1.-1..2.-1..2..-2...3..-2..3..-3..4....

......-2..3.-3..3..-4...5..-5..5..-6..7....

..........5.-6..6..-7...9.-10.10.-11.13...

...........-11.12.-13..16.-19.20.-21.24.-27

...............23.-25..29.-35.39.-41.45.-51

The left edge is A130668.

I discovered the array 1 1 -2 1 -3 2 in studying the singular points of planar polynomial differential systems (inspired by the reference).

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Paul Curtz, Stabilite locale des systemes quadratiques, Ann. sc. Ecole Norm. Sup. vol 13 no 3 (1980) pp 293-302.

Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Formula

a(2n) = A004526(n).

a(2n+1) = A000027(n+1) .

G.f.: x*(1+x^2+x^3)/((1-x)^2*(1+x)^2*(1+x^2)). - R. J. Mathar, Feb 25 2009

From G. C. Greubel, Sep 17 2024: (Start)

a(n) = (1/8)*(3*n + 1 - (-1)^n*(n + 3) + i^n*(1 + (-1)^n)).

E.g.f.: (1/4)*( cos(x) - (1-2*x)*cosh(x) + (2+x)*sinh(x) ). (End)

Mathematica

a[n_?OddQ] := (n+1)/2; a[n_?EvenQ] := Floor[n^2/16] - Floor[(n-2)^2/16]; Table[a[n], {n, 0, 51}] (* Jean-François Alcover, Aug 13 2012 *)
LinearRecurrence[{0, 1, 0, 1, 0, -1}, {0, 1, 0, 2, 1, 3}, 61] (* G. C. Greubel, Sep 17 2024 *)

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 61);
[0] cat Coefficients(R!( x*(1+x^2+x^3)/((1-x^2)*(1-x^4)) )); // G. C. Greubel, Sep 17 2024
(SageMath)
def A124072_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x*(1+x^2+x^3)/((1-x^2)*(1-x^4)) ).list()
A124072_list(60) # G. C. Greubel, Sep 17 2024

Crossrefs

Cf. A000027, A004526, A129819, A130668.

Sequence in context: A115121 A323523 A371092 * A189357 A100053 A029194

Adjacent sequences: A124069 A124070 A124071 * A124073 A124074 A124075

Keyword

nonn,easy

Author

Paul Curtz, Jun 26 2007

Extensions

Partially edited by R. J. Mathar, Jul 07 2008

Status

approved