A064865 Fill a triangular array by rows by writing numbers 1, then 1 up to 2^2, then 1 up to 3^2, then 1 up to 4^2 and so on. The final elements of the rows form the sequence.

1, 2, 1, 5, 1, 7, 14, 6, 15, 25, 11, 23, 36, 14, 29, 45, 13, 31, 50, 6, 27, 49, 72, 15, 40, 66, 93, 21, 50, 80, 111, 22, 55, 89, 124, 16, 53, 91, 130, 1, 42, 84, 127, 171, 20, 66, 113, 161, 210, 35, 86, 138, 191, 245, 44, 100, 157, 215, 274, 45, 106, 168, 231, 295, 36

Offset: 1

Author

Floor van Lamoen, Oct 08 2001

Keywords

• nonn
• look
• easy

Comments

Does every number appear at least once? Do some numbers like 1 appear infinitely often? - Robert G. Wilson v, Oct 10 2001

Difference between n-th triangular number and largest square pyramidal number (A000330) less than it. - Franklin T. Adams-Watters, Sep 11 2006

Examples

			The triangle begins:
....1
...1.2
..3.4.1
.2.3.4.5
6.7.8.9.1
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Table: A064866.

Mini-index to these sequences: A064766, A064865, A064866, A065221-A655234 are all of the same type. See A064766 for a detailed explanation.

Cf. A000217, A000330.

Programs

Mathematica

a = {}; Do[a = Append[a, Table[i, {i, 1, n^2} ]], {n, 1, 100} ]; a = Flatten[a]; Do[Print[a[[n(n + 1)/2]]], {n, 1, 100} ]
With[{nn=20},TakeList[Flatten[Table[Range[n^2],{n,nn}]],Range[Floor[ (Sqrt[8*nn^3+12*nn^2+4*nn+3]/Sqrt[3]-1)/2]]]][[All,-1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 04 2020 *)

Python

from sympy import integer_nthroot
def A064865(n): return 1+(k:=(n*(n+1)>>1)-1)-(r:=(m:=integer_nthroot(3*k, 3)[0])-(6*kChai Wah Wu, Nov 05 2024

Formula

a(n) = n(n+1)/2 - max_{p(m) < n(n+1)/2} p(m), where p(m) = m(m+1)(2m+1)/6. - Franklin T. Adams-Watters, Sep 11 2006

Extensions

More terms from Robert G. Wilson v, Oct 10 2001