A347595 a(0) = 1; for n>0, a(n) is the smallest positive integer that has not previously occurred such that a(n-1)^2 + n^2 + a(n) is a square.

1, 2, 8, 27, 39, 54, 73, 98, 133, 186, 273, 426, 709, 1250, 2305, 4386, 8517, 16746, 33169, 65978, 131557, 262674, 524865, 1049202, 2097829, 4195034, 8389393, 16778058, 33555333, 67109826, 134218753, 268436546, 536872069, 1073743050, 2147484945, 4294968666, 8589936037, 17179870706

Offset: 0

Author

Scott R. Shannon, Sep 08 2021

Keywords

• nonn

Comments

This sequence uses the same rules as A347594 except here all numbers must be unique. Up to 10^5 terms all terms are larger than the previous term; it is unknown if this holds for all terms as n->infinity.

Examples

			a(1) = 2 as a(0)^2 + 1^2 = 1 + 1 = 2, and 2 + 2 = 4 = 2^2 is the next smallest square.
a(2) = 8 as a(1)^2 + 2^2 = 4 + 4 = 8, and 8 + 8 = 16 = 4^2. Note that although 8 + 1 = 9 = 3^2, 1 cannot be chosen as a(0) = 1.
a(3) = 27 as a(2)^2 + 3^2 = 64 + 9 = 73 and 73 + 27 = 100 = 10^2.  Note that although 73 + 8 = 81 = 9^2, 8 cannot be chosen as a(2) = 8.
a(4) = 39 as a(3)^2 + 4^2 = 729 + 16 = 745, and 745 + 39 = 784 = 28^2 is the next smallest square.
		

Crossrefs

Cf. A347594, A000290, A103605, A009000, A005408, A347754.