A154140 -id:A154140 - cp's OEIS frontend

A154153 Numbers k such that 28 plus the k-th triangular number is a perfect square.

6, 8, 47, 57, 278, 336, 1623, 1961, 9462, 11432, 55151, 66633, 321446, 388368, 1873527, 2263577, 10919718, 13193096, 63644783, 76895001, 370948982, 448176912, 2162049111, 2612166473, 12601345686, 15224821928, 73446025007, 88736765097, 428074804358, 517195768656

Offset: 1

R. J. Mathar, Oct 18 2009

nonn

			6, 8, 47, and 57 are terms:
   6* (6+1)/2 + 28 =  7^2,
   8* (8+1)/2 + 28 =  8^2,
  47*(47+1)/2 + 28 = 34^2,
  57*(57+1)/2 + 28 = 41^2.

F. T. Adams-Watters, SeqFan Discussion, Oct 2009
David A. Corneth, Conjectured formula for a(n)

Cf. A000217, A000290.

Cf. 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), this sequence (28), A154154 (30).

Join[{6, 8}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 28 &]] (* G. C. Greubel, Sep 03 2016 *)

{for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 28), print1(n, ", ") ) );}

{k: 28+k*(k+1)/2 in A000290}.
Conjectures: (Start)
a(n) = +a(n-1) +6*a(n-2) -6*a(n-3) -a(n-4) +a(n-5).
G.f.: x*(-6-2*x-3*x^2+2*x^3+7*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).
G.f.: ( 14 + 1/(x-1) + (14+29*x)/(x^2-2*x-1) + (-1-12*x)/(x^2+2*x-1) )/2. (End)
See also the Corneth link - David A. Corneth, Mar 18 2019

a(21)-a(30) from Amiram Eldar, Mar 18 2019

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

A175034 Offsets i such that i + n*(n+1)/2 is never a perfect square for any n>0.

Original entry on oeis.org

2, 5, 7, 11, 12, 14, 17, 18, 20, 23, 27, 29, 31, 32, 37, 38, 40, 41, 42, 44, 47, 50, 51, 52, 56, 57, 59, 62, 65, 67, 68, 69, 70, 73, 74, 77, 82, 83, 84, 86, 87, 88, 92, 95, 96, 98, 101, 102, 104, 107, 109, 110, 112, 113, 117, 119, 122, 125, 126, 127, 128, 131, 132, 135, 137, 139

Offset: 1

Ctibor O. Zizka, Nov 10 2009

Complement of A175035.

Cf. A001108, A006451, A154138, A154139, A154140, A154141

Extended by R. J. Mathar, Nov 26 2009

A154132 Minimal exponents m such that the fractional part of (4/3)^m increases monotonically (when starting with m=1).

Original entry on oeis.org

1, 2, 8, 39, 2495, 3895, 4714, 8592

Offset: 1

Hieronymus Fischer, Jan 11 2009

Recursive definition: a(1)=1, a(n) = least number m>a(n-1) such that the fractional part of (4/3)^m is greater than the fractional part of (4/3)^k for all k, 1<=k

The next such number must be greater than 200000.

			a(4)=39, since fract((4/3)^39)= 0.999186..., but fract((4/3)^k)<0.9887... for 1<=k<=38; thus fract((4/3)^39)>fract((4/3)^k) for 1<=k<39 and 39 is the minimal exponent > 8 with this property.

Cf. A153663, A153671, A091560, A153710, A154140, A154148.

Cf. A002379, A064628.

Recursion: a(1):=1, a(k):=min{ m>1 | fract((4/3)^m) > fract((4/3)^a(k-1))}, where fract(x) = x-floor(x).

Showing 1-4 of 4 results.

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A154153 Numbers k such that 28 plus the k-th triangular number is a perfect 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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A175034 Offsets i such that i + n*(n+1)/2 is never a perfect square for any n>0.

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A154132 Minimal exponents m such that the fractional part of (4/3)^m increases monotonically (when starting with m=1).

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