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A085215
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), ...
4
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, 25, 56, 51, 34, 29, 30, 7, 8, 27, 122, 57, 52, 35, 126, 31, 8, 9, 31, 128, 123, 58, 53, 150, 127, 32, 9, 10, 33, 146, 129, 124, 59, 246, 151, 128, 33, 10, 11, 37, 152, 147, 130, 125, 270, 247, 152, 129, 34, 11
OFFSET
0,4
FORMULA
A(x,y) = A322001(concat(A007623(x), A007623(y))), where A322001 is a left inverse of A007623. - M. F. Hasler, Nov 27 2018. Note: this formula is valid only with x and y such that A322001(A007623(x)) = x and A322001(A007623(y)) = y, i.e., at least for all x,y <= 36287999. See N. J. A. Sloane's Jun 04 2012 comment in A007623. - Antti Karttunen, Feb 23 2025
EXAMPLE
From M. F. Hasler, Nov 27 2018: (Start)
The array starts:
0 1 2 3 4 5 6 ...
1 3 8 9 10 11 30 ...
2 7 26 27 28 29 ...
3 9 32 33 34 ...
4 13 50 51 ...
(...) (End)
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.
PROG
(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))))))
(define (A085215 n) (A085215bi (A025581 n) (A002262 n)))
(define (A085216 n) (A085215bi (A002262 n) (A025581 n)))
(PARI) A085215(x, y)=A322001(eval(Str(A007623(x), A007623(y)))) \\ M. F. Hasler, Nov 27 2018. N.B. This has the same caveat as Hasler's formula. See below for program that is correct for all x, y >= 0. - Antti Karttunen, Mar 24 2025
(PARI)
up_to = 78;
A085215sq(x, y) = { my(i=2, j=2, z=0, f=1); while(y>0, z += (y%i)*f; f *= i; y \= i; i++); while(x>0, z += (x%j)*f; f *= i; x \= j; i++; j++); (z); };
A085215list(up_to) = { my(v = vector(up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A085215sq(a-col, col))); (v); };
v085215 = A085215list(up_to);
A085215(n) = v085215[1+n]; \\ Antti Karttunen, Mar 24 2025
CROSSREFS
Transpose: A085216. Variant: A085219. Can be used to compute A085201. Cf. A000142, A007623, A084558, A025581, A002262.
Sequence in context: A055375 A091533 A055376 * A076731 A347650 A341653
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, Jun 23 2003
EXTENSIONS
More terms added by Antti Karttunen, Mar 24 2025
STATUS
approved