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A049777
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Triangular array read by rows: T(m,n) = n + n+1 + ... + m = (m+n)(m-n+1)/2.
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15
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1, 3, 2, 6, 5, 3, 10, 9, 7, 4, 15, 14, 12, 9, 5, 21, 20, 18, 15, 11, 6, 28, 27, 25, 22, 18, 13, 7, 36, 35, 33, 30, 26, 21, 15, 8, 45, 44, 42, 39, 35, 30, 24, 17, 9, 55, 54, 52, 49, 45, 40, 34, 27, 19, 10, 66, 65, 63, 60, 56, 51, 45, 38, 30, 21, 11, 78, 77, 75, 72, 68, 63, 57, 50
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OFFSET
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1,2
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COMMENTS
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No primes and all composite numbers (except 2^x) are generated after the first two columns of the square array for this sequence. In other words, no primes and all composites except 2^x are generated when m-n >= 2. - Bob Selcoe, Jun 18 2013
Diagonal sums in the square array equal partial sums of squares (A000330). - Bob Selcoe, Feb 14 2014
The following apply to the triangle as a square array read by rows unless otherwise specified (see Table link);
Conjecture: There is at least one prime in interval [T(n,k), T(n,k+1)]. Since T(n,k+1)/T(n,k) decreases to (k+1)/k as n increases, this is true for k=1 ("Bertrand's Postulate", first proved by P. Chebyshev), k=2 (proved by El Bachraoui) and k=3 (proved by Loo).
Starting with T(1,1), The falling diagonal of the first 2 numbers in each column (read by column) are the generalized pentagonal numbers (A001318). That is, the coefficients of T(1,1), T(2,1), T(2,2), T(3,2), T(3,3), T(4,3), T(4,4) etc. are the generalized pentagonal numbers. These are A000326 and A005449 (Pentagonal and Second pentagonal numbers: n*(3*n+1)/2, respectively), interweaved.
Let D(n,k) denote falling diagonals starting with T(n,k):
Treating n as constant: pentagonal numbers of the form n*k + 3*k*(k-1)/2 are D(n,1); sequences A000326, 005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542 are formed by n = 1 through 12, respectively.
Treating k as constant: D(1,k) are (3*n^2 + (4k-5)*n + (k-1)*(k-2))/2. When k = 2(mod3), D(1,k), is same as D(k+1,1) omitting the first (k-2)/3 numbers in the sequences. So D(1,2) is same as D(3,1); D(1,5) is same as D(6,1) omitting the 6; D(1,8) is same as D(9,1) omitting the 9 and 21; etc.
(End)
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LINKS
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FORMULA
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Read as a square array: T(n,k) = k*(k+2n-1)/2. - Bob Selcoe, Oct 27 2014
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EXAMPLE
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Rows: {1}; {3,2}; {6,5,3}; ...
Triangle begins:
1;
3, 2;
6, 5, 3;
10, 9, 7, 4;
15, 14, 12, 9, 5;
21, 20, 18, 15, 11, 6;
28, 27, 25, 22, 18, 13, 7;
36, 35, 33, 30, 26, 21, 15, 8;
45, 44, 42, 39, 35, 30, 24, 17, 9;
55, 54, 52, 49, 45, 40, 34, 27, 19, 10; ...
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MATHEMATICA
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Flatten[Table[(n+k) (n-k+1)/2, {n, 15}, {k, n}]] (* Harvey P. Dale, Feb 27 2012 *)
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PROG
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(PARI) {T(n, k) = if( k<1 || n<k, 0, (n + k) * (n - k + 1) / 2 )} /* Michael Somos, Oct 06 2007 */
(Magma) /* As triangle */ [[(m+n)*(m-n+1) div 2: n in [1..m]]: m in [1.. 15]]; // Vincenzo Librandi, Oct 27 2014
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CROSSREFS
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Cf. A001318 (generalized pentagonal numbers).
Cf. A000326, 005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542 (pentagonal numbers of form n*k + 3*k*(k-1)/2).
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KEYWORD
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AUTHOR
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STATUS
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approved
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