login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A220466 a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(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

Stephan Ralf, Some divide-and-conquer sequences with simple ordinary generating functions, The OEIS, Jan 01 2004.

FORMULA

a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p) , p >= 0 and n >= 1. Observe that a(2^p) = A007582(p).

a(n) = ((n+1)/2)*(A060818(n)/A060818(n-1))

a(n) = (-1/64)*(q(n+1)/q(n))/(2*n+1) with q(n) = (-1)^(n+1)*2^(4*n-5)*(2*n)!*A060818(n-1) or q(n) = (1/8)*A220002(n-1)*1/(A098597(2*n-1)/A046161(2*n))*1/(A008991(n-1)/A008992(n-1))

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*n-1)*2^p) := 4^p*(n-1)  + 2^(p-1)*(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*(n1-1)+2^(p-1)*(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 Ralph Stephan's recurrence:

import Data.List (transpose)

a220466 n = a006519_list !! (n-1)

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’s-counting sequence), A000265 (remove 2’s from n), A001316 (Gould’s sequence), A001511 (the ruler function), A003484 (Hurwitz-Radon numbers), A003602 (a fractal sequence), A006519 (highest power of 2 dividing n), A007814 (binary carry sequence), A010060 (Thue-Morse sequence), A014577 (dragon curve), A014707 (dragon curve), A025480 (nim-values), A026741, A035263 (first Feigenbaum symbolic sequence), A037227, A038712, A048460, A048896, A051176, A053381 (smooth nowhere-zero 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

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 21 01:47 EDT 2018. Contains 316405 sequences. (Running on oeis4.)