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

Author

Paul Curtz, Jun 26 2007

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000027, A004526, A129819, A130668.

Programs

Magma

R:=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

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 *)

SageMath

def A124072_list(prec):
    P. = 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

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)

Extensions

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