A085215 - OEIS

%I #13 May 18 2024 14:51:44

%S 0,1,1,2,3,2,3,7,8,3,4,9,26,9,4,5,13,32,27,10,5,6,15,50,33,28,11,6,7,

%T 25,56,51,34,29,30,7,8,27,122,57,52,35,126,31,8,9,31,128,123,58,53,

%U 150,127,32,9,10,33,146,129,124,59,246,151,128,33,10,11,37,152,147,130,125,270

%N Square array A(x,y) = the number whose factorial expansion A007623 is that of x and y concatenated; zero expanded as empty string; read by ascending antidiagonals: A(0,0), A(1,0), A(0,1), A(2,0), A(1,1), A(0,2), ...

%F A(x,y) = A322001(concat(A007623(x), A007623(y))), where A322001 is a left inverse of A007623. - _M. F. Hasler_, Nov 27 2018

%e From _M. F. Hasler_, Nov 27 2018:

%e The array starts:

%e    0   1   2   3   4   5  6 ...

%e    1   3   8   9  10  11 30 ...

%e    2   7  26  27  28  29 ...

%e    3   9  32  33  34  ...

%e    4  13  50  51  ...

%e   (...)  (End)

%e A(4,3) = 51 which has a factorial expansion '2011' (2*24+0*6+1*2+1*1), a concatenation of factorial expansions of 4, '20' and of 3, '11'. Similarly, A(3,4) = 34 which has a factorial expansion '1120' (1*24+1*6+2*2+0*1). See A085217 for the corresponding factorial expansions.

%o (MIT/GNU Scheme) (define (A085215bi x y) (let loop ((x x) (y y) (i 2) (j (1+ (A084558 y)))) (cond ((zero? x) y) (else (loop (floor->exact (/ x i)) (+ (* (A000142 j) (modulo x i)) y) (1+ i) (1+ j))))))

%o (define (A085215 n) (A085215bi (A025581 n) (A002262 n)))

%o (define (A085216 n) (A085215bi (A002262 n) (A025581 n)))

%o (PARI) A085215(x,y)=A322001(eval(Str(A007623(x),A007623(y)))) \\ _M. F. Hasler_, Nov 27 2018

%Y Transpose: A085216. Variant: A085219. Can be used to compute A085201. Cf. A000142, A007623, A084558, A025581, A002262.

%K nonn,tabl

%O 0,4

%A _Antti Karttunen_, Jun 23 2003