A327891 Square array A(n,k) (n, k >= 1) read by antidiagonals, where A(n,1) = n-1; A(n,k) is the number of occurrences of A(n,k-1) in the row up to k-1.
0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 1, 4, 3, 3, 2, 1, 1, 5, 1, 1, 2, 2, 1, 1, 6, 4, 4, 3, 1, 2, 1, 1, 7, 1, 1, 1, 3, 1, 2, 1, 1, 8, 5, 5, 3, 2, 3, 1, 2, 1, 1, 9, 1, 1, 2, 2, 1, 3, 1, 2, 1, 1, 10, 6, 6, 4, 3, 4, 1, 3, 1, 2, 1, 1, 11, 1, 1, 1, 3, 2, 4, 1, 3, 1, 2, 1, 1
Offset: 1
Examples
The square array begins: 0, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, ... 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, ... 2, 1, 1, 2, 2, 3, 1, 3, 2, 4, 1, 4, 2, 5, 1, 5, ... 3, 1, 1, 2, 1, 3, 2, 2, 3, 3, 4, 1, 4, 2, 4, 3, ... 4, 1, 1, 2, 1, 3, 1, 4, 2, 2, 3, 2, 4, 3, 3, 4, ... 5, 1, 1, 2, 1, 3, 1, 4, 1, 5, 2, 2, 3, 2, 4, 2, ... 6, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 2, 2, 3, 2, ... 7, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 2, 2, ... 8, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, ... From _M. F. Hasler_, May 08 2025: (Start) The quasi-periodic pattern that starts in row 3, column 3, is: (1 2 2 3), (1 3 2 4), ..., that is, (1 m-1 2 m), starting with m = 3, then m = 4, 5, 6, ... The pattern that starts in row 4, column 8, is: (2 3 3 4 1 4), (2 4 3 5 1 5), ..., that is, (2 m-1 3 m 1 m), starting with m = 4, then m = 5, 6, 7, ... The pattern that starts in row 5, column 15, is (3 m-1 4 m 1 m 2 m), starting with m = 5, then m = 6, 7, 8, ... In row 6, from column 24 on, we have (4 m-1 5 m 1 m 2 m 3 m), with m = 6, 7, 8, ... And so on: In row n, from column n(n-2) on, we have the pattern (n-2 m-1 n-1 m 1 ... n-3 m), starting with m = n. (End)
Crossrefs
The second row is A158416.
Programs
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PARI
M327891=Map(); apply( {A327891(n, k=0)= k|| [n+=(1-k=ceil(sqrt(8*n+1)/2-.5))*k\2, k+=1-n]; if(k==1, n-1, n>k\/2 || n<3, (k\/2)^(n==2==k%2), mapisdefined(M327891,[n,k], &n), n, mapput(M327891,[n,k], n=#[0| j<-[1..k-2], A327891(n, j)==A327891(n, k-1)]+1); n)}, [1..20]) \\ A327891 gives A(n,k) if k is given, otherwise a(n). The "quasi-periodic property" is not used. matrix(9,19,n,k, A327891(n, k)) \\ M. F. Hasler, May 08 2025
Formula
A(n, 1) = n-1; A(2, 2k-1) = k = A(n, 2k) if n = 1 or n > k; A(2, 2k) = 1 = A(n, 2k-1) if n = 1 or n > k. - M. F. Hasler, May 08 2025
Comments