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A244418
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Triangle read by rows T(n,m) = n*m +(n-1)*(m-1), for n >= m >= 1.
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4
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1, 2, 5, 3, 8, 13, 4, 11, 18, 25, 5, 14, 23, 32, 41, 6, 17, 28, 39, 50, 61, 7, 20, 33, 46, 59, 72, 85, 8, 23, 38, 53, 68, 83, 98, 113, 9, 26, 43, 60, 77, 94, 111, 128, 145, 10, 29, 48, 67, 86, 105, 124, 143, 162, 181, 11, 32, 53, 74, 95, 116, 137, 158, 179, 200, 221
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OFFSET
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1,2
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COMMENTS
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This table is motivated by an entry of Aki Halme (A243907); see also A053726. a(n,m) is the number of stars on an array similar to the one appearing on the flag of the United States with n columns of m stars interchanged with (n-1) columns of (m-1) stars, for n>=2 and m = 2, ..., n.
The column sequences of the rectangular array R(n,m) = n*m + (n-1)*(m-1) = (2*n-1)*(m-1) + n for n >= 1 and m >= 1 (just symmetrize the given triangular array) are congruent n (mod (2*n-1)), n >= 1. With the odd modulus M = 2*n-1 and for M = d*L, that is d|n and L = (2*n-1)/d one can derive an identity for R(n,m) = d*(L*(m-1) + x) + (n - x*d) = d*k + (d+1)/2 (new modulus d) with k = L*(m-1) + x and n - x*d = (d+1)/2, that is x = ((2*n-1) - d)/(2*d) = (L-1)/2 which is a positive integer because L is odd. Then k = (2*L*m - (L+1))/2, also an integer. Thus for each divisor d of n the identity R(n, m) = R((d+1)/2, k+1) = R((d+1)/2, ((2*m-1)*(2*n-1)/d + 1)/2) holds.
The preceding identity shows that for odd composite moduli M = 2*n - 1 (with nontrivial divisors d of M) the sequence R(n,m), m >= 1 is a subsequence of the one for each modulus d. For example, for M = 15 = 3*5, n = 8, 15*(m-1) + 8 = 3*(5*m-3) + 2 = 5*(3*m-2) + 3 for m >= 1.
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LINKS
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FORMULA
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T(n,m) = n*m + (n-1)*(m-1) = (2*n-1)*(m-1) + n, for n>=m, else 0.
G.f. for column m: G(m, x) = x^m*((2*m^2 - 2*m + 1) - 2*(m - 1)^2*x)/(1 - x)^2.
G.f. for triangle: sum(n >= 1, sum(m = 1..n, T(n,m)*x^m*y^n ) ) = (x*y+1)*(2*x*y^2-x*y-1)*x*y/((-1+y)^2*(x*y-1)^3). - Robert Israel, Jan 11 2015
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EXAMPLE
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The triangle T(n,m) begins:
n\m 1 2 3 4 5 6 7 8 9 10 ...
1: 1
2: 2 5
3: 3 8 13
4: 4 11 18 25
5: 5 14 23 32 41
6: 6 17 28 39 50 61
7: 7 20 33 46 59 72 85
8: 8 23 38 53 68 83 98 113
9: 9 26 43 60 77 94 111 128 145
10: 10 29 48 67 86 105 124 143 162 181 ...
For more rows see the link.
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PROG
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(PARI) tabl(nn) = {for (n=1, nn, for (m=1, n, print1(n*m + (n-1)*(m-1), ", "); ); print(); ); } \\ Michel Marcus, Jan 11 2015
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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