

A243907


Numbers that can be expressed as n*m + (n1)*(m1), n = 2, 3, ... , m = n, n+1, n+2, ... in at least two different ways. Ordered increasingly.


1



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

2,1


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 + (n1)*(m1) = (m1)*(2*n1) + n == n (mod (2*n1)), 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


LINKS

Table of n, a(n) for n=2..56.


EXAMPLE

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.


PROG

(PARI) lista(nn=200) = {v = []; vres = []; for (n=2, nn, for (m=2, n, new = n*m + (n1)*(m1); if (! vecsearch(v, new), v = vecsort(concat(v, n*m + (n1)*(m1))), if (! vecsearch(vres, new), vres = vecsort(concat(vres, new))); ); ); ); for (i=1, min(60, #vres), print1(vres[i], ", ")); } \\ Michel Marcus, Jun 29 2014


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 StarsandStripes, 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.
Sequence in context: A087563 A093477 A070664 * A096083 A112687 A107282
Adjacent sequences: A243904 A243905 A243906 * A243908 A243909 A243910


KEYWORD

nonn,changed


AUTHOR

Aki Halme, Jun 14 2014


EXTENSIONS

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


STATUS

approved



