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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.

%I A243907 #21 Aug 09 2022 14:12:43
%S A243907 23,32,38,41,50,53,59,68,74,77,83,86,88,95,98,104,113,116,122,123,128,
%T A243907 131,137,138,140,143,149,158,163,167,173,176,179,182,185,188,193,194,
%U A243907 200,203,212,213,215,218,221,228,230,233,238,239,242,248,254,257,263
%N 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.
%C A243907 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.
%C A243907 This is the ordered list of integers that appear several times in A144650.
%C A243907 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
%e A243907 23 = 8*2 + 7*1 = 5*3 +4*2.
%e A243907 32 = 11*2 + 10*1 = 5*4 + 4*3.
%e A243907 The first triple solution is 53 = 18*2 + 17*1 = 11*3 + 10*2 = 8*4 + 7*3.
%o A243907 (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
%Y A243907 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.
%Y A243907 Cf. also A144650, with all values organized by rows (but with different offset).
%Y A243907 Cf. A053726, A244418.
%K A243907 nonn
%O A243907 2,1
%A A243907 _Aki Halme_, Jun 14 2014
%E A243907 More terms from _Michel Marcus_, Jun 29 2014
%E A243907 Edited. Title reformulated. Crossrefs A053726 and A244418 added. - _Wolfdieter Lang_, Jul 10 2014