A pattern sequence approach to Stern’s sequence

Suppose that w ∈ 1 { 0 , 1 } ∗ and let a w ( n ) be the number of occurrences of the word w in the binary expansion of n . Let { s ( n ) } n ⩾ 0 denote the Stern sequence, defined by s ( 0 ) = 0 , s ( 1 ) = 1 , and for n ⩾ 1 , s ( 2 n ) = s ( n ) , and s ( 2 n + 1 ) = s ( n ) + s ( n + 1 ) . In this note, we show that s ( n ) = a 1 ( n ) + ∑ w ∈ 1 { 0 , 1 } ∗ s ( [ w ¯ ] 2 ) a w 1 ( n ) where w ¯ denotes the complement of w (obtained by sending 0 ↦ 1 and 1 ↦ 0 ) and [ w ] 2 denotes the integer specified by the word w ∈ { 0 , 1 } ∗ interpreted in base 2 .