

A220466


a((2*n1)*2^p) = 4^p*(n1) + 2^(p1)*(1+2^p), p >= 0 and n >= 1.


40



1, 3, 2, 10, 3, 7, 4, 36, 5, 11, 6, 26, 7, 15, 8, 136, 9, 19, 10, 42, 11, 23, 12, 100, 13, 27, 14, 58, 15, 31, 16, 528, 17, 35, 18, 74, 19, 39, 20, 164, 21, 43, 22, 90, 23, 47, 24, 392, 25, 51, 26, 106, 27, 55, 28, 228, 29, 59, 30, 122, 31, 63, 32, 2080, 33, 67, 34, 138, 35
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

The a(n) appeared in the analysis of A220002, a sequence related to the Catalan numbers.
The first Maple program makes use of a program by Peter Luschny for the calculation of the a(n) values. The second Maple program shows that this sequence has a beautiful internal structure, see the first formula, while the third Maple program makes optimal use of this internal structure for the fast calculation of a(n) values for large n.
The cross references lead to sequences that have the same internal structure as this sequence.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ralf Stephan, Some divideandconquer sequences with simple ordinary generating functions, The OEIS, Jan 01 2004.


FORMULA

a((2*n1)*2^p) = 4^p*(n1) + 2^(p1)*(1+2^p), p >= 0 and n >= 1. Observe that a(2^p) = A007582(p).
a(n) = ((n+1)/2)*(A060818(n)/A060818(n1))
a(n) = (1/64)*(q(n+1)/q(n))/(2*n+1) with q(n) = (1)^(n+1)*2^(4*n5)*(2*n)!*A060818(n1) or q(n) = (1/8)*A220002(n1)*1/(A098597(2*n1)/A046161(2*n))*1/(A008991(n1)/A008992(n1))
Recurrence: a(2n) = 4a(n)  2^A007814(n), a(2n+1) = n+1.  Ralf Stephan, Dec 17 2013


MAPLE

# First Maple program
a := n > 2^padic[ordp](n, 2)*(n+1)/2 : seq(a(n), n=1..69); # Peter Luschny, Dec 24 2012
# Second Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n1)*2^p) := 4^p*(n1) + 2^(p1)*(1+2^p) od: od: seq(a(n), n=1..nmax);
# Third Maple program
nmax:=69: for p from 0 to ceil(simplify(log[2](nmax))) do n:=2^p: n1:=1: while n <= nmax do a(n) := 4^p*(n11)+2^(p1)*(1+2^p): n:=n+2^(p+1): n1:= n1+1: od: od: seq(a(n), n=1..nmax);


MATHEMATICA

A220466 = Module[{n, p}, p = IntegerExponent[#, 2]; n = (#/2^p + 1)/2; 4^p*(n  1) + 2^(p  1)*(1 + 2^p)] &; Array[A220466, 50] (* JungHwan Min, Aug 22 2016 *)


PROG

(PARI) a(n)=if(n%2, n\2+1, 4*a(n/2)2^valuation(n/2, 2)) \\ Ralf Stephan, Dec 17 2013
(Haskell)  Following Ralf Stephan's recurrence:
import Data.List (transpose)
a220466 n = a006519_list !! (n1)
a220466_list = 1 : concat
(transpose [zipWith () (map (* 4) a220466_list) a006519_list, [2..]])
 Reinhard Zumkeller, Aug 31 2014


CROSSREFS

Cf. A000027 (the natural numbers), A000120 (1'scounting sequence), A000265 (remove 2's from n), A001316 (Gould's sequence), A001511 (the ruler function), A003484 (HurwitzRadon numbers), A003602 (a fractal sequence), A006519 (highest power of 2 dividing n), A007814 (binary carry sequence), A010060 (ThueMorse sequence), A014577 (dragon curve), A014707 (dragon curve), A025480 (nimvalues), A026741, A035263 (first Feigenbaum symbolic sequence), A037227, A038712, A048460, A048896, A051176, A053381 (smooth nowherezero vector fields), A055975 (Gray code related), A059134, A060789, A060819, A065916, A082392, A085296, A086799, A088837, A089265, A090739, A091512, A091519, A096268, A100892, A103391, A105321 (a fractal sequence), A109168 (a continued fraction), A117973, A129760, A151930, A153733, A160467, A162728, A181988, A182241, A191488 (a companion to Gould's sequence), A193365, A220466 (this sequence).
Sequence in context: A135515 A114486 A176743 * A090780 A184174 A277821
Adjacent sequences: A220463 A220464 A220465 * A220467 A220468 A220469


KEYWORD

nonn,easy,look


AUTHOR

Johannes W. Meijer, Dec 24 2012


STATUS

approved



