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Square array A(n,k) read by antidiagonals: A(n,k) = T(lcm(n,k), gcd(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.
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%I #12 Feb 16 2025 08:33:44

%S 1,3,3,6,5,6,10,21,21,10,15,14,13,14,15,21,55,78,78,55,21,28,27,120,

%T 25,120,27,28,36,105,34,210,210,34,105,36,45,44,231,90,41,90,231,44,

%U 45,55,171,300,406,465,465,406,300,171,55,66,65,64,63,630,61,630,63,64,65,66,78,253,465,666,820,903,903,820,666,465,253,78,91,90,561,230,1035,324,85,324,1035,230,561,90,91

%N Square array A(n,k) read by antidiagonals: A(n,k) = T(lcm(n,k), gcd(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table.

%C The array is read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

%H Antti Karttunen, <a href="/A286102/b286102.txt">Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array</a>

%H MathWorld, <a href="https://mathworld.wolfram.com/PairingFunction.html">Pairing Function</a>

%F A(n,k) = T(lcm(n,k), gcd(n,k)), where T(n,k) is sequence A000027 considered as a two-dimensional table, that is, as a pairing function from N x N to N.

%F A(n,k) = A(k,n), or equivalently, a(A038722(n)) = a(n). [Array is symmetric.]

%e The top left 12 X 12 corner of the array:

%e 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78

%e 3, 5, 21, 14, 55, 27, 105, 44, 171, 65, 253, 90

%e 6, 21, 13, 78, 120, 34, 231, 300, 64, 465, 561, 103

%e 10, 14, 78, 25, 210, 90, 406, 63, 666, 230, 990, 117

%e 15, 55, 120, 210, 41, 465, 630, 820, 1035, 101, 1540, 1830

%e 21, 27, 34, 90, 465, 61, 903, 324, 208, 495, 2211, 148

%e 28, 105, 231, 406, 630, 903, 85, 1596, 2016, 2485, 3003, 3570

%e 36, 44, 300, 63, 820, 324, 1596, 113, 2628, 860, 3916, 375

%e 45, 171, 64, 666, 1035, 208, 2016, 2628, 145, 4095, 4950, 739

%e 55, 65, 465, 230, 101, 495, 2485, 860, 4095, 181, 6105, 1890

%e 66, 253, 561, 990, 1540, 2211, 3003, 3916, 4950, 6105, 221, 8778

%e 78, 90, 103, 117, 1830, 148, 3570, 375, 739, 1890, 8778, 265

%o (Scheme)

%o (define (A286102 n) (A286102bi (A002260 n) (A004736 n)))

%o (define (A286102bi row col) (A000027bi (lcm row col) (gcd row col)))

%o (define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

%Y Cf. A000027, A003989, A003990, A003991, A038722, A285722, A285732, A286098, A286099, A286101, A285724.

%Y Cf. A000217 (row 1 and column 1), A001844 (main diagonal).

%K nonn,tabl,changed

%O 1,2

%A _Antti Karttunen_, May 03 2017