A080819 -id:A080819 - cp's OEIS frontend

A080820 Least m such that m^2 >= n*(n+1)/2.

1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 9, 9, 10, 11, 11, 12, 13, 14, 14, 15, 16, 16, 17, 18, 19, 19, 20, 21, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 40, 41, 42, 43, 43, 44, 45, 45, 46, 47, 48, 48, 49, 50, 50, 51, 52, 52

Offset: 1

Amarnath Murthy, Mar 21 2003

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A080817, A080818, A080819.

seq(ceil(sqrt(n*(n+1)/2)),n=1..80); # C. Ronaldo

lst = {}; m = 1; Do[While[n*(n + 1)/2 > m^2, m++]; AppendTo[lst, m], {n, 69}]; lst (* Arkadiusz Wesolowski, Jul 18 2012 *)
Ceiling[Sqrt[#]]&/@Accumulate[Range[80]] (* Harvey P. Dale, Feb 10 2016 *)

a(n) = A080819(n)^(1/2).

a(n) = ceiling(sqrt(n*(n+1)/2)) - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004

Name changed by Arkadiusz Wesolowski, Jul 18 2012

A080817 Leading diagonal of triangle in A080818.

Original entry on oeis.org

1, 3, 6, 10, 6, 10, 15, 8, 13, 19, 26, 15, 22, 30, 16, 24, 33, 43, 25, 35, 46, 25, 36, 48, 61, 36, 49, 63, 35, 49, 64, 33, 48, 64, 81, 46, 63, 81, 43, 61, 80, 100, 58, 78, 99, 54, 75, 97, 49, 71, 94, 118, 66, 90, 115, 60, 85, 111, 138, 79, 106, 134, 72, 100, 129, 159, 93, 123

Offset: 1

Amarnath Murthy, Mar 21 2003

nonn

Cf. A080818, A080819, A080820.

From David Wasserman, May 13 2004: (Start)
a(n) = (ceiling(sqrt(n(n+1)/2)))^2 - n(n-1)/2.
a(n) = A048761(A000217(n)) - A000217(n-1). (End)

More terms from David Wasserman, May 13 2004

A080818 Triangle read by rows in which the n-th row contains n distinct numbers whose sum is the smallest square.

Original entry on oeis.org

1, 1, 3, 1, 2, 6, 1, 2, 3, 10, 1, 2, 3, 4, 6, 1, 2, 3, 4, 5, 10, 1, 2, 3, 4, 5, 6, 15, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 19, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 26, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 15, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 22, 1, 2, 3, 4, 5, 6, 7, 8

Offset: 1

Amarnath Murthy, Mar 21 2003

If n(n+1)/2 is a square triangular number then the n-th row contains first n natural numbers (e.g., the 8th row).

			1; 1,3; 1,2,6; 1,2,3,10; 1,2,3,4,6; 1,2,3,4,5,10; 1,2,3,4,5,6,15; 1,2,3,4,5,6,...

Cf. A080817, A080819, A080820.

seq(op([seq(i,i=1..n-1),(ceil(sqrt(n*(n+1)/2)))^2 - n*(n-1)/2]),n=1..15); # C. Ronaldo

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004

A175032 a(n) is the difference between the n-th triangular number and the next perfect square.

Original entry on oeis.org

0, 0, 1, 3, 6, 1, 4, 8, 0, 4, 9, 15, 3, 9, 16, 1, 8, 16, 25, 6, 15, 25, 3, 13, 24, 36, 10, 22, 35, 6, 19, 33, 1, 15, 30, 46, 10, 26, 43, 4, 21, 39, 58, 15, 34, 54, 8, 28, 49, 0, 21, 43, 66, 13, 36, 60, 4, 28, 53, 79, 19, 45, 72, 9, 36, 64, 93, 26, 55, 85, 15, 45, 76, 3, 34, 66, 99, 22

Offset: 0

Ctibor O. Zizka, Nov 09 2009

All terms are from {0} U A175035. No terms are from A175034.

The sequence consists of ascending runs of length 3 or 4. The first run starts at n = 1 and thereafter the k-th run starts at n = A214858(k - 1). - John Tyler Rascoe, Nov 05 2022

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A001109, A214858.
Cf. A000217, A080819.
Cf. A175034, A175035.
Cf. sequences where a(m)=k: A001108 (0), A006451 (1), A154138 (3), A154139 (4), A154140 (6), A154141 (8), A154142 (9), A154143 (10), A154144 (13), A154145 (15), A154146 (16), A154147 (19), A154148 (21), A154149 (22), A154150(24), A154151 (25), A154151 (26), A154153(28), A154154 (30).

Ceiling[Sqrt[#]]^2-#&/@Accumulate[Range[0,80]] (* Harvey P. Dale, Aug 25 2013 *)

a(n) = my(t=n*(n+1)/2); if (issquare(t), 0, (sqrtint(t)+1)^2 - t); \\ Michel Marcus, Nov 06 2022

a(n) = (ceiling(sqrt(n*(n+1)/2)))^2 - n*(n+1)/2. - Ctibor O. Zizka, Nov 09 2009

a(n) = A080819(n) - A000217(n). - R. J. Mathar, Aug 24 2010

Erroneous formula variant deleted and offset set to zero by R. J. Mathar, Aug 24 2010

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A080820 Least m such that m^2 >= n*(n+1)/2.

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A080817 Leading diagonal of triangle in A080818.

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A080818 Triangle read by rows in which the n-th row contains n distinct numbers whose sum is the smallest square.

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A175032 a(n) is the difference between the n-th triangular number and the next perfect square.

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