A386652 a(n) = Sum_{k>=0} ( ( binomial(n+k-1, k) * 2^k ) (mod 3^k) ) / 3^k.

2, 5, 9, 13, 16, 18, 21, 25, 28, 31, 41, 44, 44, 52, 57, 55, 57, 62, 69, 75, 76, 75, 84, 85, 92, 92, 99, 95, 103, 105, 119, 114, 118, 125, 127, 135, 125, 152, 137, 145, 141, 151, 167, 155, 162, 178, 172, 175, 185, 186, 195, 209, 205, 208, 201, 211, 219, 228, 210, 224, 228, 238, 232, 242, 235, 242, 254, 242, 254, 267, 278, 266, 258, 280

Offset

1,1

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Example

The terms a(n) equal the sum illustrated below.
a(1) = (1 mod 1) + (2 mod 3)/3 + (4 mod 3^2)/3^2 + (8 mod 3^3)/3^3 + ...
a(2) = (1 mod 1) + (4 mod 3)/3 + (12 mod 3^2)/3^2 + (32 mod 3^3)/3^3 + ...
a(3) = (1 mod 1) + (6 mod 3)/3 + (24 mod 3^2)/3^2 + (80 mod 3^3)/3^3 + ...
a(4) = (1 mod 1) + (8 mod 3)/3 + (40 mod 3^2)/3^2 + (160 mod 3^3)/3^3 + ...
a(5) = (1 mod 1) + (10 mod 3)/3 + (60 mod 3^2)/3^2 + (280 mod 3^3)/3^3 + ...
a(6) = (1 mod 1) + (12 mod 3)/3 + (84 mod 3^2)/3^2 + (448 mod 3^3)/3^3 + ...
...
More explicitly,
a(1) = 0 + 2/3 + 4/3^2 + 8/3^3 + 16/3^4 + 32/3^5 + 64/3^6 + 128/3^7 + ...
a(2) = 0 + 1/3 + 3/3^2 + 5/3^3 + 80/3^4 + 192/3^5 + 448/3^6 + 1024/3^7 + ...
a(3) = 0 + 0/3 + 6/3^2 + 26/3^3 + 78/3^4 + 186/3^5 + 334/3^6 + 234/3^7 + ...
a(4) = 0 + 2/3 + 4/3^2 + 25/3^3 + 74/3^4 + 91/3^5 + 273/3^6 + 51/3^7 + ...
a(5) = 0 + 1/3 + 6/3^2 + 10/3^3 + 67/3^4 + 144/3^5 + 318/3^6 + 687/3^7 + ...
a(6) = 0 + 0/3 + 3/3^2 + 16/3^3 + 72/3^4 + 45/3^5 + 408/3^6 + 774/3^7 + ...
a(7) = 0 + 2/3 + 4/3^2 + 24/3^3 + 39/3^4 + 204/3^5 + 87/3^6 + 948/3^7 + ...
a(8) = 0 + 1/3 + 0/3^2 + 15/3^3 + 15/3^4 + 72/3^5 + 474/3^6 + 1896/3^7 + ...
...

Prog

(PARI) {a(n) = round( sum(k=0, 20*n, ( (binomial(n+k-1, k) * 2^k) % 3^k) / 3^k *1. ) )}
for(n=1, 100, print1(a(n), ", "))

Crossrefs

Cf. A386651, A386653, A386654.

Sequence in context: A360899 A104862 A190686 * A247986 A259252 A114471

Adjacent sequences: A386649 A386650 A386651 * A386653 A386654 A386655

Keyword

nonn

Author

Paul D. Hanna, Jul 27 2025

Status

approved