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User: Aki Halme

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A243907 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.

Original entry on oeis.org

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
Offset: 2

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Author

Aki Halme, Jun 14 2014

Keywords

Comments

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

Examples

			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.

Crossrefs

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).
Cf. A053726, A244418.

Programs

  • 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

Extensions

More terms from Michel Marcus, Jun 29 2014
Edited. Title reformulated. Crossrefs A053726 and A244418 added. - Wolfdieter Lang, Jul 10 2014

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