A286102 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.

1, 3, 3, 6, 5, 6, 10, 21, 21, 10, 15, 14, 13, 14, 15, 21, 55, 78, 78, 55, 21, 28, 27, 120, 25, 120, 27, 28, 36, 105, 34, 210, 210, 34, 105, 36, 45, 44, 231, 90, 41, 90, 231, 44, 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

Offset: 1

Author

Antti Karttunen, May 03 2017

Keywords

• nonn
• tabl

Comments

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.

Examples

			The top left 12 X 12 corner of the array:
   1,   3,   6,  10,   15,   21,   28,   36,   45,   55,   66,   78
   3,   5,  21,  14,   55,   27,  105,   44,  171,   65,  253,   90
   6,  21,  13,  78,  120,   34,  231,  300,   64,  465,  561,  103
  10,  14,  78,  25,  210,   90,  406,   63,  666,  230,  990,  117
  15,  55, 120, 210,   41,  465,  630,  820, 1035,  101, 1540, 1830
  21,  27,  34,  90,  465,   61,  903,  324,  208,  495, 2211,  148
  28, 105, 231, 406,  630,  903,   85, 1596, 2016, 2485, 3003, 3570
  36,  44, 300,  63,  820,  324, 1596,  113, 2628,  860, 3916,  375
  45, 171,  64, 666, 1035,  208, 2016, 2628,  145, 4095, 4950,  739
  55,  65, 465, 230,  101,  495, 2485,  860, 4095,  181, 6105, 1890
  66, 253, 561, 990, 1540, 2211, 3003, 3916, 4950, 6105,  221, 8778
  78,  90, 103, 117, 1830,  148, 3570,  375,  739, 1890, 8778,  265
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of array
• MathWorld, Pairing Function

Crossrefs

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

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

Programs

Scheme

(define (A286102 n) (A286102bi (A002260 n) (A004736 n)))
(define (A286102bi row col) (A000027bi (lcm row col) (gcd row col)))
(define (A000027bi row col) (* (/ 1 2) (+ (expt (+ row col) 2) (- row) (- (* 3 col)) 2)))

Formula

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.

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