A106524 Interleave A038573(n+1) and 2*A038573(n+1).

1, 2, 1, 2, 3, 6, 1, 2, 3, 6, 3, 6, 7, 14, 1, 2, 3, 6, 3, 6, 7, 14, 3, 6, 7, 14, 7, 14, 15, 30, 1, 2, 3, 6, 3, 6, 7, 14, 3, 6, 7, 14, 7, 14, 15, 30, 3, 6, 7, 14, 7, 14, 15, 30, 7, 14, 15, 30, 15, 30, 31, 62, 1, 2, 3, 6, 3, 6, 7, 14, 3, 6, 7, 14, 7, 14, 15, 30, 3, 6, 7, 14, 7, 14, 15, 30, 7, 14, 15

Offset: 0

Author

Paul Barry, May 06 2005

Keywords

• easy
• nonn

Comments

Row sums of number the number triangle (A106522 mod 2).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000034, A001316, A038573, A106522.

Programs

Magma

A106524:= func< n | 2^Multiplicity(Intseq(n+2, 2), 1) - 2^(n mod 2) >;
[A106524(n): n in [0..100]]; // G. C. Greubel, Aug 12 2021

Mathematica

a[n_]:= (2^DigitCount[Floor[(n+2)/2], 2, 1] - 1)*(3 - (-1)^n)/2;
Table[a[n], {n, 0, 100}] (* G. C. Greubel, Aug 11 2021 *)

PARI

a(n) = bitneg(n%2, hammingweight(n+2)); \\ Kevin Ryde, Aug 25 2021

Sage

def A000120(n): return sum(n.digits(2))
def A106524(n): return 2^A000120(n+2) - 2^(n%2)
[A106524(n) for n in (0..100)] # G. C. Greubel, Aug 11 2021

Formula

a(n) = (Sum_{k=0..n+2} binomial(n+2, k)) mod 2 - (3 - (-1)^n)/2.

a(n) = ( (Sum_{k=0..(n/2+1)} binomial(n/2+1, k)) mod 2 - 1 )*(1 + (-1)^n)/2 + ( (Sum_{k=0..(n+1)/2} binomial((n+1)/2, k)) mod 2 - 1)*(1 - (-1)^n)/2.

a(n) = A001316(n+2) - A000034(n).