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A209640 Global ranking function for restricted totally balanced binary strings given in A209641.

%I A209640 #16 Jul 01 2025 15:14:54
%S A209640 0,0,1,0,0,0,0,0,0,0,2,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A209640 0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,5,0,6,0,0,0,7,0,0,0,0,0,0,0,0,0,0,0,
%U A209640 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A209640 Global ranking function for restricted totally balanced binary strings given in A209641.
%C A209640 The given Scheme-program implements a ranking function for the terms of A209641, using Khayyam's triangle A007318.
%e A209640 a(12)=3, as 12 occurs as the 3rd term (zero-based) in A209641.
%e A209640 a(14)=0, as 14 doesn't occur in A209641.
%o A209640 (Scheme) (define (A209640 n) (if (or (zero? n) (not (member_of_A209641? n))) 0 (let* ((w (/ (binwidth n) 2))) (let loop ((rank 0) (row 1) (u (- w 1)) (n (- n (A053644 n))) (i (/ (A053644 n) 2)) (first_0_found? #f)) (cond ((or (zero? row) (zero? u) (zero? n)) (+ (expt 2 (-1+ w)) rank)) ((> i n) (loop rank (- row 1) u n (/ i 2) #t)) (else (loop (+ rank (if first_0_found? (A007318tr (- (+ row u) 1) (- row 1)) (A007318tr (- w 1) (- row 1)))) (+ row 1) (- u 1) (- n i) (/ i 2) first_0_found?)))))))
%o A209640 (define (binwidth n) (let loop ((n n) (i 0)) (if (zero? n) i (loop (floor->exact (/ n 2)) (1+ i)))))
%Y A209640 This is an inverse function for A209641 in the sense that a(A209641(n)) = n for all n. The beginning of sequence coincides with A080300, because A209641 is a subsequence of A014486. Used to compute the permutation A209861.
%K A209640 nonn
%O A209640 0,11
%A A209640 _Antti Karttunen_, Mar 24 2012