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A230417 Lower triangular region of A230415, a triangular table read by rows: T(n, k) tells in how many digit positions the factorial base representations (A007623) of n and k differ, where (n, k) = (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), ..., n >= 0 and (0 <= k <= n).

%I A230417 #27 Dec 11 2021 04:41:36
%S A230417 0,1,0,1,2,0,2,1,1,0,1,2,1,2,0,2,1,2,1,1,0,1,2,2,3,2,3,0,2,1,3,2,3,2,
%T A230417 1,0,2,3,1,2,2,3,1,2,0,3,2,2,1,3,2,2,1,1,0,2,3,2,3,1,2,1,2,1,2,0,3,2,
%U A230417 3,2,2,1,2,1,2,1,1,0,1,2,2,3,2,3,1,2,2,3,2,3,0,2,1,3,2,3,2,2,1,3,2,3,2,1,0,2,3,1,2,2,3,2,3,1,2,2,3,1,2,0
%N A230417 Lower triangular region of A230415, a triangular table read by rows: T(n, k) tells in how many digit positions the factorial base representations (A007623) of n and k differ, where (n, k) = (0,0), (1,0), (1,1), (2,0), (2,1), (2,2), ..., n >= 0 and (0 <= k <= n).
%H A230417 Antti Karttunen, <a href="/A230417/b230417.txt">Rows n = 0..120 of triangle, flattened</a>
%F A230417 a(n) = A230415bi(A003056(n),A002262(n)). [As a sequence, this is obtained by taking a subsection from array A230415.]
%F A230417 T(n,0) = A060130(n) [the leftmost column].
%F A230417 For n >= 1, T(n,n-1) = A055881(n) [the last nonzero column].
%F A230417 Each entry T(n,k) <= A231714(n,k).
%e A230417 This triangular table begins:
%e A230417   0;
%e A230417   1, 0;
%e A230417   1, 2, 0;
%e A230417   2, 1, 1, 0;
%e A230417   1, 2, 1, 2, 0;
%e A230417   2, 1, 2, 1, 1, 0;
%e A230417   1, 2, 2, 3, 2, 3, 0;
%e A230417   ...
%e A230417 Please see A230415 for examples showing how the terms are computed.
%o A230417 (Scheme)
%o A230417 (define (A230417 n) (A230415bi (A003056 n) (A002262 n)))
%o A230417 (define (A230415bi x y) (let loop ((x x) (y y) (i 2) (d 0)) (cond ((and (zero? x) (zero? y)) d) (else (loop (floor->exact (/ x i)) (floor->exact (/ y i)) (+ i 1) (+ d (if (= (modulo x i) (modulo y i)) 0 1)))))))
%Y A230417 This is a lower, or equivalently, an upper triangular subregion of symmetric square array A230415.
%Y A230417 Cf. A231714, A060130, A055881.
%K A230417 nonn,base,tabl
%O A230417 0,5
%A A230417 _Antti Karttunen_, Nov 10 2013