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.
The first 15 rows of this irregular triangular table:
0,
1,
2, 2,
3, 3, 3,
3, 4, 4, 3,
4, 4, 5, 4, 4,
4, 5, 5, 5, 5, 4,
4, 5, 6, 5, 6, 5, 4,
4, 5, 6, 6, 6, 6, 5, 4,
5, 5, 6, 6, 7, 6, 6, 5, 5,
5, 6, 6, 6, 7, 7, 6, 6, 6, 5,
4, 6, 7, 6, 7, 7, 7, 6, 7, 6, 4,
5, 5, 7, 7, 7, 7, 7, 7, 7, 7, 5, 5,
5, 6, 6, 7, 8, 7, 7, 7, 8, 7, 6, 6, 5,
6, 6, 7, 6, 8, 8, 7, 7, 8, 8, 6, 7, 6, 6
etc.
E.g., we have
a(1) = T(0,0) = a(0) + a(0) + 1 = 1,
a(2) = T(1,0) = a(1) + a(0) + 1 = 2,
a(3) = T(0,1) = a(0) + a(1) + 1 = 2,
a(4) = T(2,0) = a(2) + a(0) + 1 = 3, etc.
up_to = 105; A002260(n) = (n-binomial((sqrtint(8*n)+1)\2, 2)); \\ From A002260 A004736(n) = (1-n+(n=sqrtint(8*n)\/2)*(n+1)\2); \\ From A004736 A071673list(up_to) = { my(v=vector(1+up_to)); v[1] = 0; for(n=1,up_to,v[1+n] = 1 + v[A004736(n)] + v[A002260(n)]); (v); }; v071673 = A071673list(up_to); A071673(n) = v071673[1+n]; \\ Antti Karttunen, Aug 17 2021
(define (A071673 n) (cond ((zero? n) n) (else (+ 1 (A071673 (A025581 (-1+ n))) (A071673 (A002262 (-1+ n)))))))
;; Functions below show the essential idea: (define A072634 (lexrank->arithrank-bijection packA054238)) (define (lexrank->arithrank-bijection packfun) (lambda (n) (rank-bintree (binexp->parenthesization (A014486 n)) packfun))) (define (rank-bintree bt packfun) (cond ((not (pair? bt)) 0) (else (1+ (packfun (rank-bintree (car bt) packfun) (rank-bintree (cdr bt) packfun)))))) (define (packA054238 x y) (+ (A000695 x) (* 2 (A000695 y))))
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