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Square number array read by ascending antidiagonals: T(1,k) = 2*k + 1, and T(n,k) = (2*n^(k+1)-n-1)/(n-1) otherwise.
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%I #11 Oct 14 2023 14:04:39

%S 1,1,1,1,3,1,1,5,5,1,1,7,13,7,1,1,9,25,29,9,1,1,11,41,79,61,11,1,1,13,

%T 61,169,241,125,13,1,1,15,85,311,681,727,253,15,1,1,17,113,517,1561,

%U 2729,2185,509,17,1,1,19,145,799,3109,7811,10921,6559,1021,19,1

%N Square number array read by ascending antidiagonals: T(1,k) = 2*k + 1, and T(n,k) = (2*n^(k+1)-n-1)/(n-1) otherwise.

%F T(0,k) = A000012(k) = 1;

%F T(1,k) = A005408(k) = 2k+1;

%F T(2,k) = A036563(k+2);

%F T(3,k) = A058481(k+1);

%F T(4,k) = A083584(k);

%F T(5,k) = A137410(k);

%F T(6,k) = A233325(k);

%F T(7,k) = A233326(k);

%F T(8,k) = A233328(k);

%F T(9,k) = A211866(k+1);

%F T(10,k) = A165402(k+1);

%F T(n,0) = A000012(n) = 1;

%F T(n,1) = A005408(n) = 2*n+1;

%F T(n,2) = A001844(n) = 2*n^2 + 2*n + 1.

%e Square array begins:

%e 1..1...1.....1......1.......1........1........1...

%e 1..3...5.....7......9......11.......13.......15...

%e 1..5..13....29.....61.....125......253......509...

%e 1..7..25....79....241.....727.....2185.....6559...

%e 1..9..41...169....681....2729....10921....43689...

%e 1.11..61...311...1561....7811....39061...195311...

%e 1.13..85...517...3109...18661...111973...671845...

%e 1.15.113...799...5601...39215...274513..1921599...

%e 1.17.145..1169...9361...74897...599185..4793489...

%e 1.19.181..1639..14761..132859..1195741.10761679...

%e 1.21.221..2221..22221..222221..2222221.22222221...

%p T:= proc(n, k); if n=1 then 2*k+1 else (2*n^(k+1)-n-1)/(n-1) fi end:

%p seq(seq(T(n-k, k), k=0..n), n=0..10); # _Georg Fischer_, Oct 14 2023

%Y Cf. A238303.

%K easy,nonn,tabl

%O 0,5

%A _Philippe Deléham_, Feb 24 2014

%E Definition amended by _Georg Fischer_, Oct 14 2023