How to optimize this query?

how_to_optimize.sql

CREATE TABLE clients (
  first_name text,
  last_name text
);

INSERT INTO clients (first_name, last_name) 
  (SELECT md5(random()::text) AS first_name, md5(random()::text) AS last_name
   FROM (SELECT * FROM generate_series(1,1000000) AS id) AS x);

# trigram index
CREATE INDEX clients_search_idx
ON clients
USING gin (first_name gin_trgm_ops);

# btree index
CREATE INDEX btree_index_on_client_first_name
ON clients USING BTREE (first_name);

\timing

SELECT count(*) FROM clients WHERE first_name ILIKE '%abc%';
# Time: 42,086 ms

# but

SELECT count(*) FROM clients WHERE first_name ILIKE '%ab%';
# Time: 2661,724 ms

# ############################################################################

# 3 chars

EXPLAIN SELECT count(*) FROM clients WHERE first_name ILIKE '%abc%';
                                         QUERY PLAN                                          
---------------------------------------------------------------------------------------------
 Aggregate  (cost=12561.04..12561.05 rows=1 width=0)
   ->  Bitmap Heap Scan on clients  (cost=122.28..12535.79 rows=10101 width=0)
         Recheck Cond: (first_name ~~* '%abc%'::text)
         ->  Bitmap Index Scan on clients_search_idx  (cost=0.00..119.76 rows=10101 width=0)
               Index Cond: (first_name ~~* '%abc%'::text)

# 2 chars

EXPLAIN SELECT count(*) FROM clients WHERE first_name ILIKE '%ab%';
                              QUERY PLAN                              
----------------------------------------------------------------------
 Aggregate  (cost=25123.78..25123.79 rows=1 width=0)
   ->  Seq Scan on clients  (cost=0.00..24846.00 rows=111111 width=0)
         Filter: (first_name ~~* '%ab%'::text)

# Why doesn't it use btree index?