OFFSET
0,4
COMMENTS
a(n) or a(n+1) gives the number of times certain simple recursive procedures are called to effect a reversal of a sequence of n elements (including both the top-level call and any subsequent recursive calls). See example and program lines.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..300
Antti Karttunen, More information
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).
FORMULA
a(n) = (3/4)*(1+sqrt(2))^(n-1) + 3/4*(1-sqrt(2))^(n-1) - 1/2 + 3*0^n, with n >= 0. - Jaume Oliver Lafont, Sep 10 2009
G.f.: (1 - 2*x - x^2 + 3*x^3)/((1-x)*(1-2*x-x^2)). - Jaume Oliver Lafont, Sep 09 2009
a(n) = 3*a(n-1) - a(n-2) - a(n-3), a(0)=1, a(1)=1, a(2)=1, a(3)=4. - Harvey P. Dale, Nov 20 2011
a(n) = (3*A001333(n-1) - 1)/2. - R. J. Mathar, Mar 04 2013
a(n) = -1/2 - (3/4)*(1+sqrt(2))^n - (3/4)*sqrt(2)*(1-sqrt(2))^n - (3/4)*(1-sqrt(2))^n + (3/4)*(1+sqrt(2))^n*sqrt(2) for n >= 1. - Alexander R. Povolotsky, Mar 05 2013
E.g.f.: 3 + (1/2)*exp(x)*(-1 - 3*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, Oct 13 2019
EXAMPLE
See the Python, Erlang (myrev), PARI (rev) and Forth definitions (REV3) given at Program section.
PARI, Python and Erlang functions are called a(n+1) times for the list of length n, while Forth word REV3 is called a(n) times if there are n elements in the parameter stack.
MAPLE
seq(coeff(series((1 -2*x -x^2 +3*x^3)/((1-x)*(1-2*x-x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 13 2019
MATHEMATICA
Join[{1}, RecurrenceTable[{a[0]==a[1]==1, a[n]==2a[n-1]+a[n-2]+1}, a, {n, 30}]] (* or *) LinearRecurrence[{3, -1, -1}, {1, 1, 1, 4}, 30] (* Harvey P. Dale, Nov 20 2011 *)
Table[If[n==0, 1, (3*LucasL[n-1, 2] -2)/4], {n, 0, 30}] (* G. C. Greubel, Oct 13 2019 *)
PROG
(Haskell)
a033539 n = a033539_list !! n
a033539_list =
1 : 1 : 1 : (map (+ 1) $ zipWith (+) (tail a033539_list)
(map (2 *) $ drop 2 a033539_list))
-- Reinhard Zumkeller, Aug 14 2011
(PARI)
/* needs version >= 2.5 */
/* function demonstrating the reversal of the lists and counting the function calls: */
rev( L )={ cnt++; if( #L>1, my(x, y); x=L[#L]; listpop(L); L=rev(L); y=L[#L]; listpop(L); L=rev(L); listput(L, x); L=rev(L); listput(L, y)); L }
for(n=0, 50, cnt=0; print(n": rev(", L=List(vector(n, i, i)), ")=", rev(L), ", cnt="cnt)) \\ Antti Karttunen, Mar 05 2013, partially based on previous PARI code from Michael Somos, 1999. Edited by M. F. Hasler, Mar 05 2013
(Python)
# function, demonstrating the reversal of the lists:
def myrev(lista):
'''Reverses a list, in cumbersome way.'''
if(len(lista) < 2): return(lista)
else:
tr = myrev(lista[1:])
return([tr[0]]+myrev([lista[0]]+myrev(tr[1:])))
(Erlang)
# definition, demonstrating the reversal of the lists:
myrev([]) -> [];
myrev([A]) -> [A];
myrev([X|Y]) ->
[A|B] = myrev(Y),
[A|myrev([X|myrev(B)])].
(Forth)
# definition, demonstrating how to turn a parameter stack upside down:
: REV3 DEPTH 0= IF ELSE DEPTH 1 = IF ELSE DEPTH 2 = IF SWAP ELSE >R RECURSE R> SWAP >R >R RECURSE R> RECURSE R> THEN THEN THEN ;
-- Antti Karttunen, Mar 04 2013
(PARI) concat([1], vector(30, n, (3*sum(k=0, (n-1)\2, binomial(n-1, 2*k) * 2^k) - 1)/2 )) \\ G. C. Greubel, Oct 13 2019
(Magma) I:=[1, 1, 4]; [1] cat [n le 3 select I[n] else 3*Self(n-1) - Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Oct 13 2019
(Sage) [1]+[(3*lucas_number2(n-1, 2, -1) -2)/4 for n in (1..30)] # G. C. Greubel, Oct 13 2019
(GAP) Concatenation([1], List([1..30], n-> (3*Lucas(2, -1, n-1)[2] -2)/4 )); # G. C. Greubel, Oct 13 2019
(Prolog)
rev([], []).
rev([A], [A]).
rev([A|XB], [B|YA]) :-
rev(XB, [B|Y]), rev(Y, X), rev([A|X], YA). % Lewis Baxter, Jan 04 2021
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved