A026147 a(n) = position of n-th 1 in A001285 or A010059 (Thue-Morse sequence).

1, 4, 6, 7, 10, 11, 13, 16, 18, 19, 21, 24, 25, 28, 30, 31, 34, 35, 37, 40, 41, 44, 46, 47, 49, 52, 54, 55, 58, 59, 61, 64, 66, 67, 69, 72, 73, 76, 78, 79, 81, 84, 86, 87, 90, 91, 93, 96, 97, 100, 102, 103, 106, 107, 109, 112, 114, 115, 117, 120, 121, 124, 126, 127, 130

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Barbeau notes that if we let A = the first 2^k terms of this sequence and B = the first 2^k terms of A181155, then the two sets A and B have the same sum of powers for first up to the k-th power. I note it holds for 0th power also. - Michael Somos, Jun 09 2013

Examples

			Let k=2. Then A = {1,4,6,7} and B = {2,3,5,8} have the property that 1^0+4^0+6^0+7^0 = 2^0+3^0+5^0+8^0 = 4, 1^1+4^1+6^1+7^1 = 2^1+3^1+5^1+8^1 = 18, and 1^2+4^2+6^2+7^2 = 2^2+3^2+5^2+8^2 = 102. - _Michael Somos_, Jun 09 2013
		

References

• Edward J. Barbeau, Power Play, MAA, 1997. See p. 104.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001969, A181155.

Programs

Mathematica

a[ n_] := If[ n < 1, 0, 2 n + Mod[ Total[ IntegerDigits[ n - 1, 2]], 2] - 1] (* Michael Somos, Jun 09 2013 *)

PARI

a(n)=2*n+hammingweight(n-1)%2-1 \\ Charles R Greathouse IV, Mar 22 2013

PARI

{a(n) = if( n<1, 0, 2*n + subst( Pol( binary( n-1)), x, 1)%2 - 1)} /* Michael Somos, Jun 09 2013 */

Python

def A026147(n): return 1+((m:=n-1).bit_count()&1)+(m<<1) # Chai Wah Wu, Mar 03 2023

Formula

a(n) = 1+A001969(n).

a(n) = Sum_{k=0..2n} mod(-2 + Sum_{j=0..k} floor(C(k, j)/2), 3). - Paul Barry, Dec 24 2004

a(n) + A010059(n+1) = 2n + 2 for n >= 0. - Clark Kimberling, Oct 06 2014