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A243907
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Numbers that can be expressed as n*m + (n-1)*(m-1), n = 2, 3, ... , m = n, n+1, n+2, ... in at least two different ways. Ordered increasingly.
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1
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23, 32, 38, 41, 50, 53, 59, 68, 74, 77, 83, 86, 88, 95, 98, 104, 113, 116, 122, 123, 128, 131, 137, 138, 140, 143, 149, 158, 163, 167, 173, 176, 179, 182, 185, 188, 193, 194, 200, 203, 212, 213, 215, 218, 221, 228, 230, 233, 238, 239, 242, 248, 254, 257, 263
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refs;
listen;
history;
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internal format)
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OFFSET
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2,1
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COMMENTS
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This sequence was inspired by the flag of the United States. The 50 stars are placed in a rectangular grid with outside dimensions six stars wide by five stars high, but they could also be placed in a grid 17 stars wide by two stars high. This sequence lists, up to 200 stars, all numbers of stars that could be placed in a rectangular field in more than one arrangement.
This is the ordered list of integers that appear several times in A144650.
R(n,m) = n*m + (n-1)*(m-1) = (m-1)*(2*n-1) + n == n (mod (2*n-1)), and also with n interchanged with m. See A244418 for the table a(n,m) = R(n,m) for n >= m >= 1. - Wolfdieter Lang, Jul 10 2014
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LINKS
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EXAMPLE
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23 = 8*2 + 7*1 = 5*3 +4*2.
32 = 11*2 + 10*1 = 5*4 + 4*3.
The first triple solution is 53 = 18*2 + 17*1 = 11*3 + 10*2 = 8*4 + 7*3.
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PROG
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(PARI) lista(nn=200) = {v = []; vres = []; for (n=2, nn, for (m=2, n, new = n*m + (n-1)*(m-1); if (! vecsearch(v, new), v = vecsort(concat(v, n*m + (n-1)*(m-1))), if (! vecsearch(vres, new), vres = vecsort(concat(vres, new))); ); ); ); for (i=1, min(60, #vres), print1(vres[i], ", ")); } \\ Michel Marcus, Jun 29 2014
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CROSSREFS
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The sequence A186041 lists all possible solutions, including single ones, and has four additional terms at the start. The sequence A140646 also refers to the Stars-and-Stripes, but gives the history, not the geometry of the current arrangement.
Cf. also A144650, with all values organized by rows (but with different offset).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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