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

1,2

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

Links

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

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

Example

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

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 *)

Prog

(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*k<m*(m+1)*((m<<1)+1)))*(r+1)*((r<<1)+1)//6 # Chai Wah Wu, Nov 05 2024

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.

Sequence in context: A263454 A036073 A124227 * A178472 A337667 A331888

Adjacent sequences: A064862 A064863 A064864 * A064866 A064867 A064868

Keyword

nonn,look,easy

Author

Floor van Lamoen, Oct 08 2001

Extensions

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

Status

approved