A064866 Write numbers 1, then 1 up to 2^2, then 1 up to 3^2, then 1 up to 4^2 and so on.

1, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28

Offset: 1

Author

Floor van Lamoen, Oct 08 2001

Keywords

• easy
• nonn
• tabl

Comments

This is a fractal sequence: if the first instance of each number is deleted, the original sequence is recovered. - Franklin T. Adams-Watters, Dec 14 2013

Subsequences start at indices A000330 + 1. - Ralf Stephan, Dec 17 2013

When sequence fills a triangular array by rows, the main diagonal is A064865:

This triangle begins:

1

1 2

3 4 1

2 3 4 5

6 7 8 9 1

From Antti Karttunen, Feb 17 2014: (Start)

A more natural way of organizing this sequence is as an irregular table consisting of successively larger square matrices:

1;

1, 2;

3, 4;

1, 2, 3;

4, 5, 6;

7, 8, 9;

1, 2, 3, 4;

5, 6, 7, 8;

9,10,11,12;

13,14,15,16;

etc.

(End)

Crossrefs

Cf. A002260, A002262, A002024.

Cf. A074279, A121997, A238013, A237451, A237452.

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

Programs

Mathematica

Table[Range[n^2],{n,10}]//Flatten (* Harvey P. Dale, Mar 05 2018 *)

PARI

A064866_vec(N=9)=concat(vector(N, i, vector(i^2, j, j))) \\ Note: This creates a vector; use A064866_vec()[n] to get the n-th term. - M. F. Hasler, Feb 17 2014

Python

from sympy import integer_nthroot
def A064866(n): return n-(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k-1)*((k<<1)-1)//6 # Chai Wah Wu, Nov 04 2024

Formula

a(n) = A237451(n) + (A237452(n)*A074279(n)) + 1. - M. F. Hasler, Feb 17 2014

For 1 <= n <= 650, a(n) = n - t(t-1)(2t-1)/6, where t = floor((3*n)^(1/3)+1/2). - Mikael Aaltonen, Jan 17 2015

a(n) = n-k(k-1)(2k-1)/6 where k = m+1 if n>m(m+1)(2m+1)/6 and k = m otherwise and m = floor((3n)^(1/3)). - Chai Wah Wu, Nov 05 2024

Extensions

Edited by Ralf Stephan, Dec 17 2013