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A296142
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Triangle read by rows: first row is 2; given row k, define the elements of row k+1 as the (sorted) elements derived from row k by two recursion rules: (i.) if x is in row k, then (x+5)^2 is in row k+1; (ii.) if x^2 is in row k, then x is in row k+1.
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4
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2, 49, 7, 2916, 54, 144, 8532241, 12, 2921, 3481, 22201, 72799221804516, 59, 149, 289, 8532246, 8561476, 12152196, 493106436, 5299726695343845803536039441, 17, 2926, 3486, 4096, 22206, 23716, 86436, 72799221804521, 72799307127001, 73298956913361, 147675989144401, 243153962155686481
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OFFSET
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1,1
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COMMENTS
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Inspired by problem A1 on the 2017 William Lowell Putnam Mathematical Competition.
Every positive integer greater than one will appear, except the multiples of 5.
For k > 2, the number of terms in row k is at least the sum of the number of terms in rows k-1 and k-2.
3 shows up for the first time no later than row 104.
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LINKS
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EXAMPLE
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First few rows are
2;
49;
7, 2916;
54, 144, 8532241;
12, 2921, 3481, 22201, 72799221804516;
59, 149, 289, 8532246, 8561476, 12152196, 493106436, 299726695343845803536039441;
17, 2926, 3486, 4096, 22206, 23716, 86436, 72799221804521, 72799307127001, 73298956913361, 147675989144401, 243153962155686481, 28087103045340200593045577229687766576786428563667986916;
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MATHEMATICA
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NestList[Complement[DeleteDuplicates@ Join[# /. s_ /; IntegerQ@ Sqrt@ s :> Sqrt@ s, # /. k_ :> (k + 5)^2], #] &, {2}, 6] // Flatten (* Michael De Vlieger, Dec 06 2017 *)
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PROG
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(PARI) lista(nn) = my(row = [2], all = row); for (n=1, nn, row = concat(apply(x->(x+5)^2, row), apply(x->sqrtint(x), select(issquare, row))); all = concat(all, vecsort(row, , 8)); ); all; \\ Michel Marcus, Oct 16 2023
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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