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.
%I A085215 #27 Aug 16 2025 03:19:20
%S A085215 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 A085215 25,56,51,34,29,30,7,8,27,122,57,52,35,126,31,8,9,31,128,123,58,53,
%U A085215 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
%N 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), ...
%H A085215 Antti Karttunen, <a href="/A085215/b085215.txt">Table of n, a(n) for n = 0..22154; the first 210 antidiagonals of the square array</a>
%H A085215 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A085215 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
%e A085215 From _M. F. Hasler_, Nov 27 2018: (Start)
%e A085215 The array starts:
%e A085215 0 1 2 3 4 5 6 ...
%e A085215 1 3 8 9 10 11 30 ...
%e A085215 2 7 26 27 28 29 ...
%e A085215 3 9 32 33 34 ...
%e A085215 4 13 50 51 ...
%e A085215 (...) (End)
%e A085215 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 A085215 (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 A085215 (define (A085215 n) (A085215bi (A025581 n) (A002262 n)))
%o A085215 (define (A085216 n) (A085215bi (A002262 n) (A025581 n)))
%o A085215 (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
%o A085215 (PARI)
%o A085215 up_to = 78;
%o A085215 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); };
%o A085215 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); };
%o A085215 v085215 = A085215list(up_to);
%o A085215 A085215(n) = v085215[1+n]; \\ _Antti Karttunen_, Mar 24 2025
%Y A085215 Transpose: A085216. Variant: A085219. Can be used to compute A085201. Cf. A000142, A007623, A084558, A025581, A002262.
%K A085215 nonn,tabl
%O A085215 0,4
%A A085215 _Antti Karttunen_, Jun 23 2003
%E A085215 More terms added by _Antti Karttunen_, Mar 24 2025