A065221 -id:A065221 - cp's OEIS frontend

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

Floor van Lamoen, Oct 08 2001

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)

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.

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

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

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

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

Edited by Ralf Stephan, Dec 17 2013

A064766 Fill up an infinite triangular array by rows by successively writing numbers 1, then 1 up to 23/2, then 1 up to 34/2, then 1 up to 4*5/2 and so on, at the k-th step writing the numbers from 1 up to the k-th triangular number. The final elements of the rows of this infinite triangle form the sequence.

Original entry on oeis.org

1, 2, 2, 6, 5, 1, 8, 1, 10, 20, 10, 22, 7, 21, 36, 16, 33, 6, 25, 45, 11, 33, 56, 14, 39, 65, 14, 42, 71, 10, 41, 73, 1, 35, 70, 106, 23, 61, 100, 4, 45, 87, 130, 21, 66, 112, 159, 36, 85, 135, 186, 48, 101, 155, 210, 56, 113, 171, 230, 59, 120, 182, 245, 56, 121, 187

Offset: 0

Floor van Lamoen, Oct 18 2001

An alternative, equivalent, definition: Let T(i) = [1,2,3,...,i*(i+1)/2], and let S be the concatenation of T(1), T(2), ... Create a triangle by partitioning S into rows of lengths 1, 2, 3, ... The right-hand elements of the rows of this triangle form the new sequence.

			The first few stages in the construction of the triangular array are:
1
then
..1
.1.2.
3
then
...1
..1.2
.3.1.2
3.4.5.6
then
.....1
....1.2
...3.1.2
..3.4.5.6
.1.2.3.4.5
6.7.8.9.10.
The right-most diagonal forms the sequence: 1,2,2,6,5,...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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

With[{nn=75},TakeList[Flatten[Range[1,#]&/@Accumulate[Range[nn]]],Range[nn]][[;;,-1]]] (* Harvey P. Dale, Oct 31 2024 *)

Edtied by N. J. A. Sloane, Oct 31 2024

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.

Original entry on oeis.org

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

Floor van Lamoen, Oct 08 2001

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

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

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

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.

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 *)

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

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

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

A065228 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the square numbers. The first elements of the rows form a(n).

Original entry on oeis.org

1, 1, 3, 2, 6, 2, 8, 15, 7, 16, 1, 12, 24, 1, 15, 30, 46, 14, 32, 51, 7, 28, 50, 73, 16, 41, 67, 94, 22, 51, 81, 112, 23, 56, 90, 125, 17, 54, 92, 131, 2, 43, 85, 128, 172, 21, 67, 114, 162, 211, 36, 87, 139, 192, 246, 45, 101, 158, 216, 275, 46, 107, 169, 232, 296, 37

Offset: 0

Floor van Lamoen, Oct 22 2001

Alois P. Heinz, Table of n, a(n) for n = 0..10000

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

A065222 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the hexagonal numbers. The final elements of the rows form a(n).

Original entry on oeis.org

1, 2, 5, 3, 8, 14, 6, 14, 23, 5, 16, 28, 41, 10, 25, 41, 58, 10, 29, 49, 70, 1, 24, 48, 73, 99, 6, 34, 63, 93, 124, 3, 36, 70, 105, 141, 178, 26, 65, 105, 146, 188, 231, 44, 89, 135, 182, 230, 3, 53, 104, 156, 209, 263, 318, 49, 106, 164, 223, 283, 344, 28, 91, 155, 220

Offset: 0

Floor van Lamoen, Oct 22 2001

			The array begins
....1
...1.2
..3.4.5
.6.1.2.3
4.5.6.7.8

Cf. A000384.

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

A065223 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the heptagonal numbers (A000566). The final elements of the rows form a(n).

Original entry on oeis.org

1, 2, 5, 2, 7, 13, 2, 10, 19, 29, 6, 18, 31, 45, 5, 21, 38, 56, 75, 14, 35, 57, 80, 104, 17, 43, 70, 98, 127, 9, 40, 72, 105, 139, 174, 21, 58, 96, 135, 175, 216, 23, 66, 110, 155, 201, 248, 10, 59, 109, 160, 212, 265, 319, 32, 88, 145, 203, 262, 322, 383, 42, 105

Offset: 0

Floor van Lamoen, Oct 22 2001

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

A065224 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the octagonal numbers. The final elements of the rows form a(n).

Original entry on oeis.org

1, 2, 5, 1, 6, 12, 19, 6, 15, 25, 36, 8, 21, 35, 50, 1, 18, 36, 55, 75, 96, 22, 45, 69, 94, 120, 14, 42, 71, 101, 132, 164, 21, 55, 90, 126, 163, 201, 15, 55, 96, 138, 181, 225, 270, 36, 83, 131, 180, 230, 281, 333, 45, 99, 154, 210, 267, 325, 384, 36, 97, 159, 222

Offset: 0

Floor van Lamoen, Oct 22 2001

Cf. A000567.

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

A065225 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the 9-gonal (nonagonal) numbers. The final elements of the rows form a(n).

Original entry on oeis.org

1, 2, 5, 9, 5, 11, 18, 2, 11, 21, 32, 44, 11, 25, 40, 56, 73, 16, 35, 55, 76, 98, 10, 34, 59, 85, 112, 140, 15, 45, 76, 108, 141, 175, 6, 42, 79, 117, 156, 196, 237, 18, 61, 105, 150, 196, 243, 291, 15, 65, 116, 168, 221, 275, 330, 386, 47, 105, 164, 224, 285, 347, 410

Offset: 0

Floor van Lamoen, Oct 22 2001

Cf. A001106.

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

A065226 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the decagonal numbers. The final elements of the rows form a(n).

Original entry on oeis.org

1, 2, 5, 9, 4, 10, 17, 25, 7, 17, 28, 40, 1, 15, 30, 46, 63, 81, 15, 35, 56, 78, 101, 125, 24, 50, 77, 105, 134, 164, 20, 52, 85, 119, 154, 190, 227, 33, 72, 112, 153, 195, 238, 282, 30, 76, 123, 171, 220, 270, 321, 3, 56, 110, 165, 221, 278, 336, 395, 4, 65, 127, 190

Offset: 0

Floor van Lamoen, Oct 22 2001

Cf. A001107.

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

A065227 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the triangular numbers. The first elements of the rows form a(n).

Original entry on oeis.org

1, 1, 3, 3, 1, 6, 2, 9, 2, 11, 21, 11, 23, 8, 22, 1, 17, 34, 7, 26, 46, 12, 34, 57, 15, 40, 66, 15, 43, 72, 11, 42, 74, 2, 36, 71, 107, 24, 62, 101, 5, 46, 88, 131, 22, 67, 113, 160, 37, 86, 136, 187, 49, 102, 156, 1, 57, 114, 172, 231, 60, 121, 183, 246, 57, 122, 188, 255

Offset: 0

Floor van Lamoen, Oct 22 2001

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

cp's OEIS Frontend

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.

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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.

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A065228 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the square numbers. The first elements of the rows form a(n).

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A065222 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the hexagonal numbers. The final elements of the rows form a(n).

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A065223 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the heptagonal numbers (A000566). The final elements of the rows form a(n).

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A065224 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the octagonal numbers. The final elements of the rows form a(n).

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A065225 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the 9-gonal (nonagonal) numbers. The final elements of the rows form a(n).

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A065226 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the decagonal numbers. The final elements of the rows form a(n).

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A065227 Fill a triangular array by rows by writing numbers 1 up to b(0), 1 up to b(1), etc., where b(n) are the triangular numbers. The first elements of the rows form a(n).

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