OFFSET
1,3
COMMENTS
What is the limit of the average value of a(n)/n as n increases? It appears to be near Pi/4.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..6500
EXAMPLE
Term a(n) equals the sum of the residues of the binomial coefficients in (1 + x)^(2*n) modulo 2^n, divided by 2^n, as illustrated below.
a(1) = (1 + 0 + 1)/2 = 1;
a(2) = (1 + 0 + 2 + 0 + 1)/2^2 = 1;
a(3) = (1 + 6 + 7 + 4 + 7 + 6 + 1)/2^3 = 4;
a(4) = (1 + 8 + 12 + 8 + 6 + 8 + 12 + 8 + 1)/2^4 = 4;
a(5) = (1 + 10 + 13 + 24 + 18 + 28 + 18 + 24 + 13 + 10 + 1)/2^5 = 5;
a(6) = (1 + 12 + 2 + 28 + 47 + 24 + 28 + 24 + 47 + 28 + 2 + 12 + 1)/2^6 = 4;
a(7) = (1 + 14 + 91 + 108 + 105 + 82 + 59 + 104 + 59 + 82 + 105 + 108 + 91 + 14 + 1)/2^7 = 8;
a(8) = (1 + 16 + 120 + 48 + 28 + 16 + 72 + 176 + 70 + 176 + 72 + 16 + 28 + 48 + 120 + 16 + 1)/2^8 = 4;
a(9) = (1 + 18 + 153 + 304 + 500 + 376 + 132 + 80 + 238 + 492 + 238 + 80 + 132 + 376 + 500 + 304 + 153 + 18 + 1)/2^9 = 8;
a(10) = (1 + 20 + 190 + 116 + 749 + 144 + 872 + 720 + 18 + 24 + 436 + 24 + 18 + 720 + 872 + 144 + 749 + 116 + 190 + 20 + 1)/2^10 = 6;
...
SPECIFIC VALUES.
Sum_{n>=1} a(n)*(4/5)^n = 17.30840243003270887738767198463123109202082549...
Sum_{n>=1} a(n)*(3/4)^n = 10.57892430627269874867780443159552606330691496...
Sum_{n>=1} a(n)*(2/3)^n = 5.417748705049056891272117551452109394174599147...
Sum_{n>=1} a(n)/2^n = 1.828467161197177620900317695715194643663356606030162443659...
Sum_{n>=1} a(n)/3^n = 0.6728993248236073006477750390537792815286492439817...
Sum_{n>=1} a(n)/4^n = 0.3970731138083449706711724063387465668154560090032...
Sum_{n>=1} a(n)/5^n = 0.2803736063617441421256259884859153810306621545786...
PROG
(PARI) {a(n) = sum(k=0, 2*n, binomial(2*n, k) % (2^n) )/2^n}
for(n=1, 80, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 08 2024
STATUS
approved