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 A210530 T(n,k) = (k + 3*n - 2 - (k+n-2)*(-1)^(k+n))/2 n, k > 0, read by antidiagonals. 9
 1, 2, 3, 1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row T(n,k) for odd n is even numbers sandwiched between n's starts from n and 2*n. Row T(n,k) for even n is odd numbers sandwiched between n's starts from 2*n-1 and n. Antidiagonal T(1,k), T(2,k-1), ..., T(k,1) for odd k is 1,2,3,...,k. Antidiagonal T(1,k), T(2,k-1), ..., T(k,1) for even k is k+1, k+2, ..., 2*k+1. The main diagonal is A000027. Diagonal, located above the main diagonal T(1,k), T(2,k+1), T(3,k+2), ... for odd k is A000027. Diagonal, located above the main diagonal T(1,k), T(2,k+1), T(3,k+2), ... for even k is k, k+3, k+6, ..., A016789, A016777, A008585. Diagonal, located below the main diagonal T(n,1), T(n+1,2), T(n+2,3), ... for odd n is n,n+1, n+2, ... A000027. Diagonal, located below the main diagonal T(n,1), T(n+1,2), T(n+2,3), ... for even n is 2*n-1, 2*n+2, 2*n+5, ... A008585, A016777, A016789. The table contains: A124625 as row 1, A114753 as column 1, A109043 as column 2, A066104 as column 4. 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. FORMULA As table T(n,k) = (k + 3*n - 2 - (k+n-2)*(-1)^(k+n))/2. As linear sequence a(n) = A000027(n) - A204164(n)*(2*A204164(n)-3) - 1. a(n) = n - v*(2*v-3) - 1, where t = floor((-1 + sqrt(8*n-7))/2) and v = floor((t+2)/2). G.f. of the table: (y*(- 1 + 3*y^2) + x^2*(2 + 5*y - 2*y^2 - 7*y^3) + x^3*(4 + y - 6*y^2 - y^3) + x*(y + 2*y^2 - y^3))/((- 1 + x)^2*(1 + x)^2*(-1 + y)^2*(1 + y)^2). - Stefano Spezia, Nov 17 2018 EXAMPLE The start of the sequence as table:    1   2   1   4   1   6   1   8   1  10    3   2   5   2   7   2   9   2  11   2    3   6   3   8   3  10   3  12   3  14    7   4   9   4  11   4  13   4  15   4    5  10   5  12   5  14   5  16   5  18   11   6  13   6  15   6  17   6  19   6    7  14   7  16   7  18   7  20   7  22   15   8  17   8  19   8  21   8  23   8    9  18   9  20   9  22   9  24   9  26   19  10  21  10  23  10  25  10  27  10   ... The start of the sequence as triangle array read by rows:    1;    2,  3;    1,  2,  3;    4,  5,  6,  7;    1,  2,  3,  4,  5;    6,  7,  8,  9, 10, 11;    1,  2,  3,  4,  5,  6,  7;    8,  9, 10, 11, 12, 13, 14, 15;    1,  2,  3,  4,  5,  6,  7,  8,  9;   10, 11, 12, 13, 14, 15, 16, 17, 18, 19;   ... Row number r contains r numbers. If r is  odd: 1,2,3,...,r. If r is even: r, r+1, r+3, ..., 2*r-1. The start of the sequence as array read by rows, the length of row r is 4*r-1. First 2*r-1 numbers are from the row number 2*r-1 of triangle array, located above. Last 2*r numbers are from the row number 2*r of triangle array, located above.   1,2,3;   1,2,3,4,5,6,7;   1,2,3,4,5,6,7,8,9,10,11;   1,2,3,4,5,6,7,8,9,10,11,12,13,14,15;   1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19;   ... Row number r contains 4*r-1 numbers: 1,2,3,...,4*r-1. MATHEMATICA T[n_, k_] := (k+3n-2-(k+n-2)(-1)^(k+n))/2; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Nov 17 2018 *) PROG (PARI) T(n, k) = (k+3*n-2-(k+n-2)*(-1)^(k+n))/2; \\ Andrew Howroyd, Jan 11 2018 (Python) t=int((math.sqrt(8*n-7)-1)/2) v=int((t+2)/2) result=n-v*(2*v-3)-1 CROSSREFS Cf. A124625, A114753, A109043, A066104, A000027, A016789, A016777, A008585, A204164. Sequence in context: A025481 A124171 A276146 * A076645 A011448 A174981 Adjacent sequences:  A210527 A210528 A210529 * A210531 A210532 A210533 KEYWORD nonn,tabl AUTHOR Boris Putievskiy, Jan 28 2013 STATUS approved

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Last modified July 29 05:59 EDT 2021. Contains 346340 sequences. (Running on oeis4.)