The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A052945

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A052945

Number of compositions of n when each odd part can be of two kinds.
(history; published version)

#48 by Charles R Greathouse IV at Thu Sep 08 08:44:59 EDT 2022

PROG

(MAGMAMagma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^2)/(1-2*x-2*x^2) )); // G. C. Greubel, Oct 18 2019

Discussion

Thu Sep 08

08:44

OEIS Server: https://oeis.org/edit/global/2944

#47 by Susanna Cuyler at Sun Oct 20 09:37:45 EDT 2019

STATUS

proposed

approved

#46 by Michel Marcus at Sat Oct 19 01:04:30 EDT 2019

STATUS

editing

proposed

#45 by Michel Marcus at Sat Oct 19 01:04:18 EDT 2019

COMMENTS

Numbers of straight-chain fatty acids involving oxo groups (or hydroxy groups), if cis-/trans isomerism is considered while stereoisomerism is neglected. - Stefan Schuster, Apr 19, 2018

STATUS

proposed

editing

#44 by G. C. Greubel at Fri Oct 18 17:15:41 EDT 2019

STATUS

editing

proposed

#43 by G. C. Greubel at Fri Oct 18 17:14:55 EDT 2019

LINKS

G. C. Greubel, <a href="/A052945/b052945.txt">Table of n, a(n) for n = 0..1000</a>

FORMULA

G.f.: (-1 + - x)*(1 + x)/(-1 + - 2*x + - 2*x^2).

Recurrence: a(0)=1, a(1)=2, a(2)=5; for n>2, 2*a(n) + 2*a(n+1) - a(n+2).

a(n) = 2*(a(n-1) + a(n-2)).

a(n) = Sum(1/4*_alpha^(-1-n), __{alpha=RootOf(-1+2*_Zz+2*_Z2z^2)} alpha^(-1-n)/4.

MAPLE

spec := [S, {S=Sequence(Prod(Union(Sequence(Prod(Z, Z)), Sequence(Z)), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);

seq(coeff(series((1-x^2)/(1-2*x-2*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 18 2019

MATHEMATICA

Join[{a=1}, b=1; Table[c=(a+b)*2; a=b; b=c, {n, 0, 20}]/2] (* Vladimir Joseph Stephan Orlovsky, Nov 22 2010 *)

LinearRecurrence[{2, 2, }, {1, 2, 5}, 30] (* G. C. Greubel, Oct 18 2019 *)

PROG

(PARI) Vec((x-1)*(1+x)/(-1+2*x+2*x^2)+O(x^9930)) \\ Charles R Greathouse IV, Nov 20 2011

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x^2)/(1-2*x-2*x^2) )); // G. C. Greubel, Oct 18 2019

(Sage)

def A052945_list(prec):

P.<x> = PowerSeriesRing(ZZ, prec)

return P( (1-x^2)/(1-2*x-2*x^2) ).list()

A052945_list(30) # G. C. Greubel, Oct 18 2019

(GAP) a:=[2, 5];; for n in [3..30] do a[n]:=2*(a[n-1]+a[n-2]); od; Concatenation([1], a); # G. C. Greubel, Oct 18 2019

STATUS

approved

editing

#42 by N. J. A. Sloane at Fri Apr 20 11:50:28 EDT 2018

STATUS

proposed

approved

#41 by Bruno Berselli at Thu Apr 19 09:04:37 EDT 2018

STATUS

editing

proposed

#40 by Bruno Berselli at Thu Apr 19 09:04:15 EDT 2018

FORMULA

First binomial transform of 2, 3, 6, 9, 18, 27, 54, 81, ... starting after 1. (End)

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proposed

editing

#39 by Bruno Berselli at Thu Apr 19 08:57:30 EDT 2018

STATUS

editing

proposed