This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
A000081(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, A014486(5) = 44 (= 101100 in binary = A063171(5)), encodes the following plane tree: .....o .....| .o...o ..\./. ...*.. Matula-Goebel encoding for this tree gives a code number A000040(1) * A000040(A000040(1)) = 2*3 = 6, thus a(5)=6. Likewise, A014486(6) = 50 (= 110010 in binary = A063171(6)) encodes the plane tree: .o .| .o...o ..\./. ...*.. Matula-Goebel encoding for this tree gives a code number A000040(A000040(1)) * A000040(1) = 3*2 = 6, thus a(6) is also 6, which shows these two trees are identical if one ignores their orientation.
mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
binbalQ[n_]:=n==0||With[{dig=IntegerDigits[n,2]},And@@Table[If[k==Length[dig],SameQ,LessEqual][Count[Take[dig,k],0],Count[Take[dig,k],1]],{k,Length[dig]}]];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[mgnum[bint[n]],{n,Select[Range[0,1000],binbalQ]}] (* Gus Wiseman, Nov 22 2022 *)
(define (A127301 n) (*A127301 (A014486->parenthesization (A014486 n)))) ;; A014486->parenthesization given in A014486. (define (*A127301 s) (if (null? s) 1 (fold-left (lambda (m t) (* m (A000040 (*A127301 t)))) 1 s)))
def a005811(n): return bin(n^(n>>1))[2:].count("1")
def ok(n): # This function after Peter Luschny
B=bin(n)[2:] if n!=0 else 0
s=0
for b in B:
s+=1 if b=="1" else -1
if s<0: return 0
return s==0
def A(n): return [0] + [i for i in range(1, n + 1) if ok(i)]
l=A(200)
print([a005811(l[i])//2 for i in range(len(l))]) # Indranil Ghosh, May 21 2017
The 11th term of A063171 is 10110010, corresponding to parenthesization ()(())(), thus its Łukasiewicz word is 3010. The 18th term of A063171 is 11011000, corresponding to parenthesization (()(())), thus its Łukasiewicz word is 1201. I.e., in the latter example there is one list on the top-level, which in turn contains two sublists, of which the first is zero elements long and the second is a sublist containing one empty sublist (the last zero is omitted).
(define (A125985 n) (A080300 (rising-list->binexp (A125985-aux2 (A014486 n))))) (define (A125985-aux2 n) (let loop ((lists (A125985-aux1 n)) (z (list)) (m 1)) (if (null? lists) z (loop (cdr lists) (m-join z (car lists) m) (+ m 1))))) (define (A125985-aux1 n) (if (zero? n) (list) (let ((begin_from (<< 1 (- (- (A000523 n) (A090996 n)) 1)))) (let loop ((s (A090996 n)) (t 0) (nth_list 1) (p begin_from) (b (if (= 0 (A004198bi n begin_from)) 0 1)) (lists (list (list)))) (cond ((< s 1) (cond ((< p 1) (reverse! lists)) (else (loop (- t (- 1 b)) b (+ 1 nth_list) (>> p 1) (if (= 0 (A004198bi n (>> p 1))) 0 1) (cons (list (+ b 1 nth_list)) lists))))) (else (loop (- s (- 1 b)) (+ t b) nth_list (>> p 1) (if (= 0 (A004198bi n (>> p 1))) 0 1) (cons (cons (+ b nth_list) (car lists)) (cdr lists))))))))) (define (A125985-aux2 n) (let loop ((lists (A125985-aux1 n)) (z (list)) (m 1)) (if (null? lists) z (loop (cdr lists) (m-join z (car lists) m) (+ m 1))))) (define (m-join a b m) (let loop ((a a) (b b) (c (list))) (cond ((and (not (pair? a)) (not (pair? b))) (reverse! c)) ((not (pair? a)) (loop a (cdr b) (cons (car b) c))) ((not (pair? b)) (loop (cdr a) b (cons (car a) c))) ((equal? (car a) (car b)) (loop (cdr a) (cdr b) (cons (car a) c))) ((> (car b) m) (loop a (cdr b) (cons (car b) c))) (else (loop (cdr a) b (cons (car a) c)))))) (define (rising-list->binexp rising-list) (let loop ((s 0) (i 0) (h 0) (fs rising-list)) (cond ((null? fs) (+ s (<< (- (<< 1 h) 1) i))) ((> (car fs) h) (loop s (+ i 1) (car fs) (cdr fs))) (else (loop (+ s (<< (- (<< 1 (+ 1 (- h (car fs)))) 1) i)) (+ i 2 (- h (car fs))) (car fs) (cdr fs)))))) (define (>> n i) (if (zero? i) n (>> (floor->exact (/ n 2)) (- i 1)))) (define (<< n i) (if (<= i 0) (>> n (- i)) (<< (+ n n) (- i 1))))
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