A193715 Positions of triangular numbers (A000217) in the union of squares and triangular numbers (A005214).

1, 2, 4, 6, 7, 9, 11, 12, 13, 15, 17, 18, 20, 22, 23, 25, 27, 29, 30, 32, 34, 35, 37, 39, 41, 42, 44, 46, 47, 49, 51, 52, 54, 56, 58, 59, 61, 63, 64, 66, 68, 70, 71, 73, 75, 76, 78, 80, 81, 82, 84, 86, 87, 89, 91, 92, 94, 96, 98, 99, 101, 103, 104, 106, 108

Offset: 1

Author

Reinhard Zumkeller, Aug 03 2011

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A005214, A010054, A193714.

Programs

Haskell

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a193715 n = a193715_list !! (n-1)
a193715_list =
   map ((+ 1) . fromJust . (`elemIndex` a005214_list)) $ tail a000217_list

Mathematica

With[{max = 2500}, Position[Union[Join[Accumulate[Range[Floor[(Sqrt[8*max + 1] - 1)/2]]], Range[Floor[Sqrt[max]]]^2]], ?(IntegerQ[Sqrt[8*# + 1]] &)] // Flatten] (* _Amiram Eldar, Apr 04 2025 *)

Formula

A010054(A005214(a(n))) = 1.

a(n) ~ c * n, where c = 1 + sqrt(2)/2 = 1.707106... . - Amiram Eldar, Apr 04 2025