OFFSET
1,1
COMMENTS
In general, it seems that for n>2, k>1, A(n, k) = A(n-1, k+1) = A(k, n), except on those two anomalous antidiagonals, first on the thirteenth antidiagonal, where for n=1..13, A(n,14-n) obtains values 5, 7, 11, 27, 43, 107, 235, 363, 619, 1131, 2155, 6251, 10347, and then on the 30th antidiagonal, where for n=1.., A(n,31-n) obtains values 5, 11, 15, 23, 39, 71, 135, 391, 647, 1671, 2695, 4743, 17031, 33415, 49799, 82567, 148103, 410247, etc. The corresponding antidiagonals in A372560 begin as:
233, 933, 14933, 978670933, 64138178286933, 1183140560213014108063589658350933, ..., and:
911, 58325, 933205, 238900565, 15656587449685, 67244531063362552157525, etc. I conjecture that for the former sequence of numbers x, from 933 onward, A372555(x) = 7, and for the latter sequence of numbers y, from 58325 onward, A372555(y) = 9, and that the array A372555(A372560(n, k)) is symmetric apart from its borders, i.e, that for n, k > 1, A372555(A372560(n, k)) = A372555(A372560(k, n)).
EXAMPLE
Array begins:
n\k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
---+----------------------------------------------------------------
1 | 3, 5, 3, 3, 5, 3, 5, 5, 3, 5, 5, 5, 5, 3, 5, 3, 7, 5, 7, 5, 5,
2 | 5, 5, 5, 5, 3, 5, 5, 3, 5, 5, 5, 7, 5, 5, 5, 7, 5, 7, 7, 5, 5,
3 | 5, 5, 5, 3, 5, 5, 3, 5, 5, 5, 11, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7,
4 | 5, 5, 3, 5, 5, 3, 5, 5, 5, 27, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9,
5 | 5, 3, 5, 5, 3, 5, 5, 5, 43, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7,
6 | 3, 5, 5, 3, 5, 5, 5, 107, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7,
7 | 5, 5, 3, 5, 5, 5, 235, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7,
8 | 5, 3, 5, 5, 5, 363, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9,
9 | 3, 5, 5, 5, 619, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7,
10 | 5, 5, 5, 1131, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 1671,
11 | 5, 5, 2155, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 2695, 3,
12 | 5, 6251, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 4743, 3, 5,
13 | 10347, 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 17031, 3, 5, 3,
14 | 5, 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 33415, 3, 5, 3, 5,
15 | 5, 5, 7, 5, 7, 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 49799, 3, 5, 3, 5, 5,
etc.
From column 19 to column 41, the first 11 rows:
n\k|19 20 ........................................................... 40 41
---+-------------------------------------------------------------------------
1 | 7, 5, 5, 5, 7, 7, 5, 5, 5, 7, 7, 5, 3, 3, 3, 5, 5, 5, 5, 3, 3, 3, 1,
2 | 7, 5, 5, 7, 9, 7, 7, 7, 9, 7, 11, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1,
3 | 5, 5, 7, 9, 7, 7, 7, 9, 7, 15, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1,
4 | 5, 7, 9, 7, 7, 7, 9, 7, 23, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1,
5 | 7, 9, 7, 7, 7, 9, 7, 39, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1,
6 | 9, 7, 7, 7, 9, 7, 71, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1,
7 | 7, 7, 7, 9, 7, 135, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1,
8 | 7, 7, 9, 7, 391, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1,
9 | 7, 9, 7, 647, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1,
10 | 9, 7, 1671, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
11 | 7, 2695, 3, 5, 3, 5, 5, 5, 5, 3, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
PROG
(PARI)
\\ Needs also program from A372560.
up_to = 91;
A001045(n) = (2^n - (-1)^n) / 3;
A130249(n) = (#binary(3*n+1)-1);
A372561sq(n, k) = A265745(A372560sq(n, k));
A372561list(up_to) = { my(v = vector(up_to), i=0); for(a=1, oo, for(col=1, a, i++; if(i > up_to, return(v)); v[i] = A372561sq((a-(col-1)), col))); (v); };
v372561 = A372561list(up_to);
A372561(n) = v372561[n];
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Antti Karttunen, May 08 2024
STATUS
approved