OFFSET
1,1
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..500
EXAMPLE
The terms a(n) equal the sum as illustrated below.
a(1) = (1 mod 1) + (3 mod 4)/4 + (9 mod 4^2)/4^2 + (27 mod 4^3)/4^3 + ...
a(2) = (1 mod 1) + (6 mod 4)/4 + (27 mod 4^2)/4^2 + (108 mod 4^3)/4^3 + ...
a(3) = (1 mod 1) + (9 mod 4)/4 + (54 mod 4^2)/4^2 + (270 mod 4^3)/4^3 + ...
a(4) = (1 mod 1) + (12 mod 4)/4 + (90 mod 4^2)/4^2 + (540 mod 4^3)/4^3 + ...
a(5) = (1 mod 1) + (15 mod 4)/4 + (135 mod 4^2)/4^2 + (945 mod 4^3)/4^3 + ...
a(6) = (1 mod 1) + (18 mod 4)/4 + (189 mod 4^2)/4^2 + (1512 mod 4^3)/4^3 + ...
...
More explicitly,
a(1) = 0 + 3/4 + 9/4^2 + 27/4^3 + 81/4^4 + 243/4^5 + 729/4^6 + ...
a(2) = 0 + 2/4 + 11/4^2 + 44/4^3 + 149/4^4 + 434/4^5 + 1007/4^6 + ...
a(3) = 0 + 1/4 + 6/4^2 + 14/4^3 + 191/4^4 + 1007/4^5 + 4028/4^6 + ...
a(4) = 0 + 0/4 + 10/4^2 + 28/4^3 + 19/4^4 + 296/4^5 + 3892/4^6 + ...
a(5) = 0 + 3/4 + 7/4^2 + 49/4^3 + 38/4^4 + 922/4^5 + 1538/4^6 + ...
a(6) = 0 + 2/4 + 13/4^2 + 40/4^3 + 222/4^4 + 820/4^5 + 926/4^6 + ...
a(7) = 0 + 1/4 + 12/4^2 + 28/4^3 + 114/4^4 + 650/4^5 + 1852/4^6 + ...
a(8) = 0 + 0/4 + 4/4^2 + 40/4^3 + 106/4^4 + 968/4^5 + 1684/4^6 + ...
...
PROG
(PARI) {a(n) = round( sum(k=0, 25*n, ( (binomial(n+k-1, k) * 3^k) % 4^k) / 4^k *1. ) )}
for(n=1, 100, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 27 2025
STATUS
approved
