A124108 -id:A124108 - cp's OEIS frontend

A048678 Binary expansion of nonnegative integers expanded to "Zeckendorffian format" with rewrite rules 0->0, 1->01.

0, 1, 2, 5, 4, 9, 10, 21, 8, 17, 18, 37, 20, 41, 42, 85, 16, 33, 34, 69, 36, 73, 74, 149, 40, 81, 82, 165, 84, 169, 170, 341, 32, 65, 66, 133, 68, 137, 138, 277, 72, 145, 146, 293, 148, 297, 298, 597, 80, 161, 162, 325, 164, 329, 330, 661, 168, 337, 338, 677, 340

Offset: 0

Antti Karttunen

No two adjacent 1-bits. Permutation of A003714.

Replace 1 with 01 in binary. - Ralf Stephan, Oct 07 2003

			11=1011 in binary, thus is rewritten as 100101 = 37 in decimal.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A003714, A005203, A048679, A048680.
MASKTRANS transform of A053644.
Cf. A084471, A088697, A088698.
Cf. A124108.

a048678 0 = 0
a048678 x = 2 * (b + 1) * a048678 x' + b
            where (x', b) = divMod x 2
-- Reinhard Zumkeller, Mar 31 2015

rewrite_0to0_1to01 := proc(n) option remember; if(n < 2) then RETURN(n); else RETURN(((2^(1+(n mod 2))) * rewrite_0to0_1to01(floor(n/2))) + (n mod 2)); fi; end;

f[n_] := FromDigits[ Flatten[IntegerDigits[n, 2] /. {1 -> {0, 1}}], 2]; Table[f@n, {n, 0, 60}] (* Robert G. Wilson v, Dec 11 2009 *)

a(n)=if(n<1,0,(3-(-1)^n)*a(floor(n/2))+(1-(-1)^n)/2)

a(n) = if(n == 0, 0, my(A = -2); sum(i = 0, logint(n, 2), A++; if(bittest(n, i), 1 << (A++)))) \\ Mikhail Kurkov, Mar 14 2024

def a(n):
    return 0 if n==0 else (3 - (-1)**n)*a(n//2) + (1 - (-1)**n)//2
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 30 2017

def A048678(n): return int(bin(n)[2:].replace('1','01'),2) # Chai Wah Wu, Mar 18 2024

a(n) = rewrite_0to0_1to01(n) [ Each 0->1, 1->10 in binary expansion of n ].
a(0)=0; a(n) = (3-(-1)^n)*a(floor(n/2))+(1-(-1)^n)/2. - Benoit Cloitre, Aug 31 2003
a(0)=0, a(2n) = 2a(n), a(2n+1) = 4a(n) + 1. - Ralf Stephan, Oct 07 2003

A322131 In the decimal expansion of n, replace each digit d with 2*d.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 210, 212, 214, 216, 218, 40, 42, 44, 46, 48, 410, 412, 414, 416, 418, 60, 62, 64, 66, 68, 610, 612, 614, 616, 618, 80, 82, 84, 86, 88, 810, 812, 814, 816, 818, 100, 102, 104, 106, 108, 1010, 1012, 1014

Offset: 0

Rémy Sigrist, Nov 27 2018

This is an operation on digit strings: 1066 becomes 201212, for example. 86420 becomes 1612840. The result is always even - see A330336. - N. J. A. Sloane, Dec 17 2019

This sequence is a variant of A124108 in decimal base.

			For n = 109:
- we replace "1" with "2", "0" with "0" and "9" with "18",
- hence a(109) = 2018.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A048385, A061581, A066686, A124108, A330336.

f:= proc(n) option remember;
local m,d;
d:= n mod 10; m:= floor(n/10);
if d >= 5 then 100*procname(m) + 2*d
else 10*procname(m)+2*d
fi
end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Nov 28 2018

a[n_] := FromDigits@Flatten@IntegerDigits[2*IntegerDigits[n]]; Array[a, 60, 0] (* Amiram Eldar, Nov 28 2018 *)

a(n, base=10) = my (d=digits(n, base), v=0); for (i=1, #d, v = v*base^max(1,#digits(2*d[i],base)) + 2*d[i]); v

def A322131(n):
   return int(''.join(str(int(d)*2) for d in str(n))) # Chai Wah Wu, Nov 29 2018

A061581(n+1) = a(A061581(n)).
A066686(a(n), a(k)) = a(A066686(n, k)) for any n > 0 and k > 0.
a(10*n + d) = 10*a(n) + 2*d for any n >= 0 and d = 0..4.
a(10*n + d) = 100*a(n) + 2*d for any n >= 0 and d = 5..9.
G.f. g(x) satisfies g(x) = (2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 16*x^8 + 18*x^9)/(1-x^10) + (10 + 10*x + 10*x^2 + 10*x^3 + 10*x^4 + 100*x^5 + 100*x^6 + 100*x^7 + 100*x^8 + 100*x^9)*g(x^10). - Robert Israel, Nov 28 2018

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A048678 Binary expansion of nonnegative integers expanded to "Zeckendorffian format" with rewrite rules 0->0, 1->01.

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A322131 In the decimal expansion of n, replace each digit d with 2*d.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula