A296142 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.

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

Offset: 1

Author

Jeremy F. Alm, Dec 05 2017

Keywords

• nonn
• tabf

Comments

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.

Examples

			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;
		

Links

• Jeremy F. Alm, Table of n, a(n) for n = 1..236
• Jeremy F. Alm, First 10 rows

Programs

Mathematica

NestList[Complement[DeleteDuplicates@ Join[# /. s_ /; IntegerQ@ Sqrt@ s :> Sqrt@ s, # /. k_ :> (k + 5)^2], #] &, {2}, 6] // Flatten (* Michael De Vlieger, Dec 06 2017 *)

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