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 A218890 T(n,k) = ((n + k - 1)*(n + k - 2) - (-1 + (-1)^floor((n + k)/2))*n + (1 +(-1)^floor((n + k)/2))*k)/2; n , k > 0, read by antidiagonals. 4
 1, 2, 3, 6, 5, 4, 10, 9, 8, 7, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 28, 27, 26, 25, 24, 23, 22, 36, 35, 34, 33, 32, 31, 30, 29, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 78 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Self-inverse permutation of the natural numbers. a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. In general, let m be natural number. Natural numbers placed in table T(n,k) by antidiagonals. The order of placement - at the beginning m antidiagonals downwards, next m antidiagonals upwards and so on. T(n,k) read by antidiagonals downwards. For m = 1 the result is A056011. This sequence is result for m = 2. A056023 is result for m = 1 and the changed order of placement - at the beginning m antidiagonals upwards, next m antidiagonals downwards and so on. LINKS Boris Putievskiy, Rows n = 1..140 of triangle, flattened Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012. Eric W. Weisstein, MathWorld: Pairing functions FORMULA For the general case. As a table T(n,k) = ((n + k - 1)*(n + k - 2) - (-1 + (-1)^floor((n + k + m - 2)/m))*n + (1 +(-1)^floor((n + k + m - 2)/m))*k)/2. As linear sequence a(n) = ((z - 1)*(z - 2) - (-1 + (-1)^floor((z + m - 2)/m))*i + (1 + (-1)^floor((z + m - 2)/m))*j)/2, where i = n - t*(t + 1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1 + sqrt(8*n - 7))/2), z = i + j. If we change the order of placement - m antidiagonals upwards, m antidiagonals downwards and so on. As a table T(n,k) = ((n + k - 1)*(n + k - 2) - (-1 + (-1)^(floor((n + k + m - 2)/m) + 1))*n + (1 + (-1)^(floor((n + k + m - 2)/m) + 1))*k)/2. As linear sequence a(n) = ((z - 1)*(z - 2) - (-1 + (-1)^(floor((z + m - 2)/m) + 1))*i + (1 + (-1)^(floor((z + m - 2)/m) + 1))*j)/2, where i = n - t*(t + 1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1 + sqrt(8*n - 7))/2), z = i + j. For this sequence. As a table T(n,k) = ((n + k - 1)*(n + k - 2) - (-1 +(-1)^floor((n + k)/2))*n + (1 + (-1)^floor((n + k)/2))*k)/2. As linear sequence a(n) = ((z - 1)*(z - 2) - (-1 + (-1)^floor(z/2))*i + (1 + (-1)^floor(z/2))*j)/2, where i = n - t*(t + 1)/2, j = (t*t + 3*t + 4)/2 - n, t = floor((-1 + sqrt(8*n - 7))/2), z = i + j. EXAMPLE The start of the sequence as table. The direction of the placement denotes by ">" and "v". v...v v...v ..1...2...6..10..11..16..28..36... ..3...5...9..12..17..27..35..38... > 4...8..13..18..26..34..39..48... > 7..14..19..25..33..40..49..63... .15..20..24..32..41..50..62..74... .21..23..31..42..51..61..73..84... >22..30..43..52..60..72..85..98... >29..44..53..59..71..86..99.113... . . . The start of the sequence as triangle array read by rows: 1; 2, 3; 6, 5, 4; 10, 9, 8, 7; 11, 12, 13, 14, 15; 16, 17, 18, 19, 20, 21; 28, 27, 26, 25, 24, 23, 22; 36, 35, 34, 33, 32, 31, 30, 29; ... Row r consists of r consecutive numbers from r*r/2 - r/2 + 1 to r*r/2 + r. If r congruent to 1 or 2 mod 4 rows are increasing. If r congruent to 0 or 3 mod 4 rows are decreasing. MAPLE T:=(n, k)->((n+k-1)*(n+k-2)-(-1+(-1)^floor((n+k)/2))*n+(1+(-1)^floor((n+k)/2))*k)/2: seq(seq(T(k, n-k), k=1..n-1), n=1..13); # Muniru A Asiru, Dec 13 2018 MATHEMATICA T[n_, k_] := ((n+k-1)(n+k-2) - (-1 + (-1)^Floor[(n+k)/2])n + (1 + (-1)^Floor[(n+k)/2]) k)/2; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 06 2018 *) PROG (Python) t=int((math.sqrt(8*n-7) - 1)/ 2) i=n-t*(t+1)/2 j=(t*t+3*t+4)/2-n z=i+j result=((z-1)*(z-2)-(-1+(-1)**int(z/2))*i+(1+(-1)**int(z/2))*j)/2 (Maxima) T(n, k) = ((n + k - 1)*(n + k - 2) - (-1 + (-1)^floor((n + k)/2))*n + (1 +(-1)^floor((n + k)/2))*k)/2\$ create_list(T(k, n - k), n, 1, 12, k, 1, n - 1); /* Franck Maminirina Ramaharo, Dec 13 2018 */ CROSSREFS Cf. A056011, A056023. Sequence in context: A254118 A056895 A254117 * A269373 A269374 A137761 Adjacent sequences: A218887 A218888 A218889 * A218891 A218892 A218893 KEYWORD nonn,tabl AUTHOR Boris Putievskiy, Feb 19 2013 STATUS approved

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Last modified March 27 20:39 EDT 2023. Contains 361575 sequences. (Running on oeis4.)