A285732 - OEIS

%I #20 Mar 28 2021 19:35:02

%S -1,1,1,2,-2,3,4,3,2,6,7,5,-3,5,10,11,8,6,4,9,15,16,12,9,-4,8,14,21,

%T 22,17,13,10,7,13,20,28,29,23,18,14,-5,12,19,27,36,37,30,24,19,15,11,

%U 18,26,35,45,46,38,31,25,20,-6,17,25,34,44,55,56,47,39,32,26,21,16,24,33,43,54,66,67,57,48,40,33,27,-7,23,32,42,53,65,78

%N Square array A(n,k) read by antidiagonals, A(n,n) = -n, otherwise, if n > k, A(n,k) = T(n-k,k), else A(n,k) = T(n,k-n), 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="/A285732/b285732.txt">Table of n, a(n) for n = 1..10585; the first 145 antidiagonals of the array</a>

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

%F If n = k, A(n,k) = -n, if n > k, A(n,k) = T(n-k,k), otherwise [when n < k], A(n,k) = T(n,k-n), 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) = A285722(n,k) - A286100(n,k).

%e The top left 14 X 14 corner of the array:

%e   -1,  1,  2,  4,  7, 11, 16, 22, 29,  37,  46,  56,  67,  79

%e    1, -2,  3,  5,  8, 12, 17, 23, 30,  38,  47,  57,  68,  80

%e    3,  2, -3,  6,  9, 13, 18, 24, 31,  39,  48,  58,  69,  81

%e    6,  5,  4, -4, 10, 14, 19, 25, 32,  40,  49,  59,  70,  82

%e   10,  9,  8,  7, -5, 15, 20, 26, 33,  41,  50,  60,  71,  83

%e   15, 14, 13, 12, 11, -6, 21, 27, 34,  42,  51,  61,  72,  84

%e   21, 20, 19, 18, 17, 16, -7, 28, 35,  43,  52,  62,  73,  85

%e   28, 27, 26, 25, 24, 23, 22, -8, 36,  44,  53,  63,  74,  86

%e   36, 35, 34, 33, 32, 31, 30, 29, -9,  45,  54,  64,  75,  87

%e   45, 44, 43, 42, 41, 40, 39, 38, 37, -10,  55,  65,  76,  88

%e   55, 54, 53, 52, 51, 50, 49, 48, 47,  46, -11,  66,  77,  89

%e   66, 65, 64, 63, 62, 61, 60, 59, 58,  57,  56, -12,  78,  90

%e   78, 77, 76, 75, 74, 73, 72, 71, 70,  69,  68,  67, -13,  91

%e   91, 90, 89, 88, 87, 86, 85, 84, 83,  82,  81,  80,  79, -14

%o (Scheme)

%o (define (A285732 n) (A285732bi (A002260 n) (A004736 n)))

%o (define (A285732bi row col) (cond ((= row col) (- row)) ((> row col) (A000027bi (- row col) col)) (else (A000027bi row (- col row)))))

%o (Python)

%o def T(n, m): return ((n + m)**2 - n - 3*m + 2)//2

%o def A(n, k): return -n if n == k else T(n - k, k) if n>k else T(n, k - n)

%o for n in range(1, 21): print([A(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, May 03 2017

%Y Transpose: A285733.

%Y Cf. A000124 (row 1, after -1), A000217 (column 1, after -1).

%Y Cf. also A000027, A003989, A072030, A285721, A285722, A286100.

%K sign,tabl

%O 1,4

%A _Antti Karttunen_, May 03 2017