A059134 -id:A059134 - cp's OEIS frontend

A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.

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

Offset: 1

Johannes W. Meijer, Dec 24 2012

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.

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

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).

-- Following Ralf 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

# 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);

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 *)

a(n)=if(n%2,n\2+1,4*a(n/2)-2^valuation(n/2,2)) \\ Ralf Stephan, Dec 17 2013

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

A059127 A hierarchical sequence (W2{3} - see A059126).

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 4, 5, 4, 6, 4, 5, 4, 1, 2, 1, 3, 1, 2, 1, 7, 8, 7, 9, 7, 8, 7, 1, 2, 1, 3, 1, 2, 1, 4, 5, 4, 6, 4, 5, 4, 1, 2, 1, 3, 1, 2, 1, 10, 11, 10, 12, 10, 11, 10, 1, 2, 1, 3, 1, 2, 1, 4, 5, 4, 6, 4, 5, 4, 1, 2, 1, 3, 1, 2, 1, 7, 8, 7, 9, 7, 8, 7, 1, 2, 1, 3, 1, 2, 1, 4, 5, 4, 6, 4, 5, 4, 1, 2, 1

Offset: 0

Jonas Wallgren, Jan 19 2001

Antti Karttunen, Table of n, a(n) for n = 0..14335
J. Wallgren, Hierarchical sequences

Cf. A001511, A059126, A059128, A059129, A059130, A059134, A059135.

Block[{m = 3, i = 7, a}, a = IntegerExponent[2 Range@ i, 2]; Array[IntegerExponent[2 #, 2] /. k_ :> (a + m k - m) &, 15]] // Flatten (* Michael De Vlieger, Nov 02 2018 *)

a(7n+3) = 3*A001511(1+n), a(7n) = a(7n+2) = a(7n+4) = a(7n+6) = a(7n+3)-2, a(7n+1) = a(7n+5) = a(7n+3)-1. - Antti Karttunen, May 28 2017

A059132 A hierarchical sequence (W2{2}c - see A059126).

Original entry on oeis.org

4, 10, 4, 16, 4, 10, 4, 22, 4, 10, 4, 16, 4, 10, 4, 28, 4, 10, 4, 16, 4, 10, 4, 22, 4, 10, 4, 16, 4, 10, 4, 34, 4, 10, 4, 16, 4, 10, 4, 22, 4, 10, 4, 16, 4, 10, 4, 28, 4, 10, 4, 16, 4, 10, 4, 22, 4, 10, 4, 16, 4, 10, 4, 40, 4, 10, 4, 16, 4, 10, 4, 22, 4, 10, 4, 16, 4, 10, 4, 28, 4, 10, 4, 16

Offset: 0

Jonas Wallgren, Jan 19 2001

Antti Karttunen, Table of n, a(n) for n = 0..16383
Jonas Wallgren, Hierarchical sequences

Cf. A007814, A059126, A059134, A059137, A059139.

Array[6 IntegerExponent[2 #, 2] - 2 &, 105] (* Michael De Vlieger, Nov 02 2018 *)

A059132(n) = if(!(n%2), 4, 6+A059132((n-1)/2)); \\ Antti Karttunen, Nov 01 2018

a(n) = 4 + 6*A007814(n+1). a(2n) = 4, a(2n+1) = a(n) + 6. - Ralf Stephan, Oct 09 2003. Typo in the first formula corrected by Antti Karttunen, Nov 01 2018

a(n) = A059126(3*n) + A059126(3*n+1) + A059126(3*n+2). - Sean A. Irvine, Sep 12 2022

A059139 A hierarchical sequence (W2{2}*c - see A059126).

Original entry on oeis.org

4, 7, 4, 10, 4, 7, 4, 13, 4, 7, 4, 10, 4, 7, 4, 16, 4, 7, 4, 10, 4, 7, 4, 13, 4, 7, 4, 10, 4, 7, 4, 19, 4, 7, 4, 10, 4, 7, 4, 13, 4, 7, 4, 10, 4, 7, 4, 16, 4, 7, 4, 10, 4, 7, 4, 13, 4, 7, 4, 10, 4, 7, 4, 22, 4, 7, 4, 10, 4, 7, 4, 13, 4, 7, 4, 10, 4, 7, 4, 16, 4, 7, 4, 10, 4, 7, 4, 13, 4, 7, 4, 10, 4

Offset: 0

Jonas Wallgren, Jan 19 2001

Let the general sequence H(a,b,n) = a*A007814(n+1) + b, then this sequence is H(3,4). Other hierarchical sequences of this type are A059132 (H(6,4)), A059141 (H(7,11)), A059144 (H(9,15)), A059134 (H(21,11)), A059137 (H(36,18)).

Ralf Stephan, Some divide-and-conquer sequences ...
J. Wallgren, Hierarchical sequences

a(n) = 4 + 3*A007814(n). a(2n) = 4, a(2n+1) = a(n) + 3. - Ralf Stephan, Oct 09 2003

cp's OEIS Frontend

A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.

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A059127 A hierarchical sequence (W2{3} - see A059126).

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A059132 A hierarchical sequence (W2{2}c - see A059126).

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A059139 A hierarchical sequence (W2{2}*c - see A059126).

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cp's OEIS Frontend

A220466 a((2*n-1)*2^p) = 4^p*(n-1) + 2^(p-1)*(1+2^p), p >= 0 and n >= 1.

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Haskell

Maple

Mathematica

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A059127 A hierarchical sequence (W2{3} - see A059126).

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Author

Keywords

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Crossrefs

Programs

Mathematica

Formula

A059132 A hierarchical sequence (W2{2}c - see A059126).

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A059139 A hierarchical sequence (W2{2}*c - see A059126).

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Author

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Formula

A220466 a((2n-1)2^p) = 4^p(n-1) + 2^(p-1)(1+2^p), p >= 0 and n >= 1.