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A286234 Square array A(n,k) = P(A000010(k), floor((n+k-1)/k)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.

%I A286234 #35 Jun 12 2025 10:17:01
%S A286234 1,1,2,3,1,4,3,3,2,7,10,3,3,2,11,3,10,3,5,4,16,21,3,10,3,5,4,22,10,21,
%T A286234 3,10,5,5,7,29,21,10,21,3,10,5,8,7,37,10,21,10,21,3,14,5,8,11,46,55,
%U A286234 10,21,10,21,3,14,5,8,11,56,10,55,10,21,10,21,5,14,8,12,16,67,78,10,55,10,21,10,21,5,14,8,12,16,79,21,78,10,55,10,21,10,27,5,14,8,12,22,92
%N A286234 Square array A(n,k) = P(A000010(k), floor((n+k-1)/k)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), etc. Here P is a two-argument form of sequence A000027 used as a pairing function N x N -> N.
%C A286234 Transpose of A286235.
%H A286234 Antti Karttunen, <a href="/A286234/b286234.txt">Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array</a>
%H A286234 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>
%e A286234 The top left 12 X 12 corner of the array:
%e A286234    1,  1,  3, 3, 10, 3, 21, 10, 21, 10, 55, 10
%e A286234    2,  1,  3, 3, 10, 3, 21, 10, 21, 10, 55, 10
%e A286234    4,  2,  3, 3, 10, 3, 21, 10, 21, 10, 55, 10
%e A286234    7,  2,  5, 3, 10, 3, 21, 10, 21, 10, 55, 10
%e A286234   11,  4,  5, 5, 10, 3, 21, 10, 21, 10, 55, 10
%e A286234   16,  4,  5, 5, 14, 3, 21, 10, 21, 10, 55, 10
%e A286234   22,  7,  8, 5, 14, 5, 21, 10, 21, 10, 55, 10
%e A286234   29,  7,  8, 5, 14, 5, 27, 10, 21, 10, 55, 10
%e A286234   37, 11,  8, 8, 14, 5, 27, 14, 21, 10, 55, 10
%e A286234   46, 11, 12, 8, 14, 5, 27, 14, 27, 10, 55, 10
%e A286234   56, 16, 12, 8, 19, 5, 27, 14, 27, 14, 55, 10
%e A286234   67, 16, 12, 8, 19, 5, 27, 14, 27, 14, 65, 10
%e A286234 The first fifteen rows when viewed as a triangle:
%e A286234    1
%e A286234    1  2
%e A286234    3  1  4
%e A286234    3  3  2  7
%e A286234   10  3  3  2 11
%e A286234    3 10  3  5  4 16
%e A286234   21  3 10  3  5  4 22
%e A286234   10 21  3 10  5  5  7 29
%e A286234   21 10 21  3 10  5  8  7 37
%e A286234   10 21 10 21  3 14  5  8 11 46
%e A286234   55 10 21 10 21  3 14  5  8 11 56
%e A286234   10 55 10 21 10 21  5 14  8 12 16 67
%e A286234   78 10 55 10 21 10 21  5 14  8 12 16 79
%e A286234   21 78 10 55 10 21 10 27  5 14  8 12 22 92
%e A286234   36 21 78 10 55 10 21 10 27  5 19  8 17 22 106
%t A286234 Map[(2 + (#1 + #2)^2 - #1 - 3 #2)/2 & @@ # & /@ Reverse@ # &, Table[{EulerPhi@ k, Floor[n/k]}, {n, 14}, {k, n}]] // Flatten (* _Michael De Vlieger_, May 06 2017 *)
%o A286234 (Scheme)
%o A286234 (define (A286234 n) (A286234bi (A002260 n) (A004736 n)))
%o A286234 (define (A286234bi row col) (let ((a (A000010 col)) (b (quotient (+ row col -1) col))) (* (/ 1 2) (+ (expt (+ a b) 2) (- a) (- (* 3 b)) 2))))
%o A286234 (Python)
%o A286234 from sympy import totient
%o A286234 def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2
%o A286234 def t(n, k): return T(totient(k), n//k)
%o A286234 for n in range(1, 21): print([t(n, k) for k in range(1, n + 1)][::-1]) # _Indranil Ghosh_, May 11 2017
%Y A286234 Transpose: A286235.
%Y A286234 Cf. A000010, A000027, A286156, A286236, A286244.
%K A286234 nonn,tabl
%O A286234 1,3
%A A286234 _Antti Karttunen_, May 05 2017