OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Shevelev, On Stephan's conjectures concerning Pascal triangle modulo 2 and their polynomial generalization, arXiv:1011.6083 [math.NT], 2010-2012; J. of Algebra Number Theory: Advances and Appl., 7 (2012), no.1, 11-29.
FORMULA
a(n) = Sum_{k=0..n} (binomial(n, k) mod 2)*8^k.
Conjecture: a(0)=1, a(n+1) = (a(n)*8) XOR a(n), where XOR is the bitwise exclusive-or operator. - Alex Ratushnyak, Apr 22 2012
From Vladimir Shevelev, Dec 26-27 2013: (Start)
Sum_{n>=0} 1/a(n)^r = Product_{k>=0} (1 + 1/(8^(2^k)+1)^r),
Sum_{n>=0} (-1)^A000120(n)/a(n)^r = Product_{k>=0} (1 - 1/(8^(2^k)+1)^r), where r>0 is a real number.
In particular,
Sum_{n>=0} 1/a(n) = Product_{k>=0} (1 + 1/(8^(2^k)+1)) = 1.1284805...;
Sum_{n>=0} (-1)^A000120(n)/a(n) = 7/8.
a(2^n) = 8^(2^n) + 1, n >= 0.
Note that analogs of Stephan's limit formulas (see Shevelev link) reduce to the relations:
a(2^t*n+2^(t-1)) = 63*(8^(2^(t-1)+1))/(8^(2^(t-1))-1) * a(2^t*n+2^(t-1)-2), t >= 2.
In particular, for t=2,3,4, we have the following formulas:
a(4*n+2) = 65 * a(4*n);
a(8*n+4) = 4097/65 * a(8*n+2);
a(16*n+8) = (16777217/266305) * a(16*n+6), etc. (End)
MATHEMATICA
Table[A100311[n], {n, 0, 30}] (* G. C. Greubel, Jan 25 2023 *)
PROG
(Python)
a=1
for i in range(33):
print(a, end=", ")
a = (a*8) ^ a
# Alex Ratushnyak, Apr 22 2012
(Python)
def A100311(n): return sum((bool(~n&n-k)^1)<<3*k for k in range(n+1)) # Chai Wah Wu, May 02 2023
(Magma) [(&+[(Binomial(n, k) mod 2)*8^k: k in [0..n]]): n in [0..40]]; // G. C. Greubel, Jan 25 2023
(SageMath)
def A100311(n): return sum( (binomial(n, k)%2)*8^k for k in range(n+1))
[A100311(n) for n in range(41)] # G. C. Greubel, Jan 25 2023
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Dec 06 2004
STATUS
approved