A133197 Triangular numbers such that moving the first digit to the end produces a square number.

1, 10, 136, 406, 111628, 400960, 624403, 40423536, 1119043156276, 4291092052416, 600441627703203, 93344240136333376, 4950849307261614030, 9159508712581260256, 91853946457361410960, 94418158421136440556, 108128255436355107240, 111546878242671354528

Offset

1,2

Comments

The square roots are: 1, 1, 19, 8, 341, 98, 494, 2058, 1091069, 1706142, 2101494 - Robert G. Wilson v, Oct 14 2007

a(17) > 10^20. - Donovan Johnson, Mar 14 2011

Links

Giovanni Resta, Table of n, a(n) for n = 1..87 (terms < 10^63)

Example

136 is a triangular number, 361 is a square number - hence 136 belongs to this sequence.

Mathematica

Select[Table[n(n + 1)/2, {n, 1000000}], IntegerQ[Sqrt[ FromDigits[ Append[Drop[IntegerDigits[ # ], 1], First[IntegerDigits[ # ]]]]]] &]
lst = {}; Do[ t = n (n + 1)/2; id = IntegerDigits@ t; If[MemberQ[{1, 4, 5, 6, 9}, id[[1]]] && IntegerQ@ Sqrt@ FromDigits@ RotateLeft@ id, AppendTo[lst, t]; Print@t], {n, 44000000}]; lst - Robert G. Wilson v, Oct 14 2007
Select[Accumulate[Range[10^6]], IntegerQ[Sqrt[FromDigits[RotateLeft[IntegerDigits[#]]]]]&] (* The program generates the first 8 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* Harvey P. Dale, Dec 29 2023 *)

Crossrefs

Cf. A133198.

Sequence in context: A095653 A024135 A050408 * A287473 A240917 A240654

Adjacent sequences: A133194 A133195 A133196 * A133198 A133199 A133200

Keyword

base,nonn

Author

Tanya Khovanova, Oct 09 2007

Extensions

More terms from Robert G. Wilson v, Oct 14 2007

a(12) from Donovan Johnson, May 13 2010

a(13)-a(16) from Donovan Johnson, Mar 14 2011

a(17)-a(18) from Giovanni Resta, May 11 2016

Status

approved