login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A131708 A024494 prefixed by a 0. 18

%I

%S 0,1,2,3,5,10,21,43,86,171,341,682,1365,2731,5462,10923,21845,43690,

%T 87381,174763,349526,699051,1398101,2796202,5592405,11184811,22369622,

%U 44739243,89478485,178956970,357913941,715827883,1431655766,2863311531,5726623061,11453246122

%N A024494 prefixed by a 0.

%C Sequence is identical to its 3rd differences. a(n)=3a(n-1)-3a(n-2)+2a(n-2), n=3,4.. Also binomial transform of 0, 1, 0. Also A024495 = first differences. Recurrence: a(n+1)-2a(n)= 1, 0, -1, -1, 0, 1, 1 .

%C {A024493, A131708, A024495} is the difference analog of the hyperbolic functions {h_1(x), h_2(x), h_3(x)} of order 3. For the definitions of {h_i(x)} and the difference analog {H_i(n)} see [Erdelyi] and the Shevelev link respectively. - _Vladimir Shevelev_, Aug 01 2017

%D A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

%H Vladimir Shevelev, <a href="https://arxiv.org/abs/1706.01454">Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n</a>, arXiv:1706.01454 [math.CO], 2017.

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2).

%F G.f.: x*(-1+x)/(2*x-1)/(x^2-x+1). - _R. J. Mathar_, Nov 14 2007

%F Recurrences: a(n) = k*a(n - 1) + (6 - 3k)*a(n - 2) + (3k - 7)*a(n - 3) + (6 - 2k)*a(n - 4);k = 0: a(n) = 6a(n - 2) - 7a(n - 3) + 6a(n - 4), k = 1: a(n) = a(n - 1) + 3a(n - 2) - 4a(n - 3) + 4a(n - 4), k = 2: a(n) = 2a(n - 1) - a(n - 3) + 2a(n - 4), cf. A113405, A135350, k = 3: a(n) = 3a(n - 1) - 3a(n - 2) + 2a(n - 3), here and many other sequences, k = 4: a(n) = 4a(n - 1) - 6a(n - 2) + 5a(n - 3) - 2a(n - 4), k = 5: a(n) = 5a(n - 1) - 9a(n - 2) + 8a(n - 3) - 4a(n - 4). For k sum of coefficients = 5 - k. Of the family k=3 gives the best recurrence.

%F a(n+m) = a(n)*A024493(m) + A024493(n)*a(m) + A024495(n)*A024495(m). - _Vladimir Shevelev_, Aug 01 2017

%t LinearRecurrence[{3,-3,2},{0,1,2},40] (* _Harvey P. Dale_, Nov 27 2013 *)

%o (PARI) v=vector(99,i,i);for(i=4,#v,v[i]=3*v[i-1]-3*v[i-2]+2*v[i-3]);v \\ _Charles R Greathouse IV_, Jun 01 2011

%K nonn,easy

%O 0,3

%A _Paul Curtz_, Sep 14 2007, Mar 01 2008

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 8 05:50 EDT 2020. Contains 336290 sequences. (Running on oeis4.)