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 A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2-n^2, m >= n+1. 0
 1, 3, 4, 5, 8, 9, 7, 12, 15, 16, 9, 16, 21, 24, 25, 11, 20, 27, 32, 35, 36, 13, 24, 33, 40, 45, 48, 49, 15, 28, 39, 48, 55, 60, 63, 64, 17, 32, 45, 56, 65, 72, 77, 80, 81, 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 121, 23, 44, 63, 80 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers (2n+1) and (2n+3) for n>0, at least one will be the product of 2 primes (not necessarily distinct). Example: n=3 for consecutive odd integers a(7)=7 and a(11)=9 and of the 3 sequence entries a(8)=12, a(9)=15 and a(10)=16 between them, one is the product of 2 primes a(9)=15=3*5. - Michael Hiebl, Jul 15 2007 A024352 gives distinct values in the array, minus the first row (1, 4, 9, 16, etc.). a(n) gives all solutions to the equation x^2 + xy = n, with y mod 2 = 0, x > 0, y >= 0. - Andrew S. Plewe, Oct 19 2007 Alternatively, triangular sequence of coefficients of Dynkin diagram weights for the Cartan groups C_n: t(n,m)=m*(2*n - m). Row sums are A002412. - Roger L. Bagula, Aug 05 2008 REFERENCES R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139. LINKS EXAMPLE Array begins: 1 4 9 16 25 36 49 64 81 100 ... 3 8 15 24 35 48 63 80 99 120 ... 5 12 21 32 45 60 77 96 117 140 ... 7 16 27 40 55 72 91 112 135 160 ... 9 20 33 48 65 84 105 128 153 180 ... ... Triangle begins: {1}, {3, 4}, {5, 8, 9}, {7, 12, 15, 16}, {9, 16, 21, 24, 25}, {11, 20, 27, 32, 35, 36}, {13, 24, 33, 40, 45, 48, 49}, {15, 28, 39, 48, 55, 60, 63, 64}, {17, 32, 45, 56, 65, 72, 77, 80, 81}, {19, 36, 51, 64, 75, 84, 91, 96, 99, 100} MATHEMATICA t[n_, m_] := (n^2 - m^2); Flatten[ Table[ t[i, j], {i, 12}, {j, i - 1, 0, -1}]] (* Robert G. Wilson v, Jul 11 2005 *) (* to view table *) Table[t[i, j], {j, 0, 6}, {i, j + 1, 10}] // TableForm Clear[T, n, m, a] T[n_, m_] = m*(2*n - m ); a = Table[Table[T[n, m], {m, 1, n}], {n, 1, 10}]; Flatten[a] (* Roger L. Bagula, Aug 05 2008 *) CROSSREFS Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850. Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567. Diagonals give A033428, A045944, A067725. Sequence in context: A215497 A092997 A021747 * A286053 A112594 A188003 Adjacent sequences:  A105017 A105018 A105019 * A105021 A105022 A105023 KEYWORD nonn,tabl,easy AUTHOR Andrew S. Plewe and Franklin T. Adams-Watters, Jul 11 2005 EXTENSIONS More terms from Robert G. Wilson v, Jul 11 2005 STATUS approved

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Last modified October 18 00:15 EDT 2019. Contains 328135 sequences. (Running on oeis4.)