A001108 -id:A001108 - 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

A271626 Numbers n such that the sum of the digits of the numbers from 0 to n is a square.

Original entry on oeis.org

0, 1, 8, 17, 19, 27, 46, 62, 91, 99, 145, 152, 304, 359, 472, 513, 571, 684, 720, 799, 913, 1204, 1232, 1413, 1771, 2599, 2907, 3059, 3509, 3769, 3887, 4158, 4507, 4787, 5071, 6209, 7399, 7739, 8059, 8486, 9566, 10709, 11545, 12139, 13284, 13573, 14607, 15417

Offset: 1

Paolo P. Lava, Apr 11 2016

			0 = 0^2 and 1 = 1^2;
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 = 6^2;
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 + 0 + 1 + 1 + 1 + 2 + 1 + 3 + 1 + 4 + 1 + 5 + 1 + 6 + 1 + 7 = 81 = 9^2.

Paolo P. Lava, Table of n, a(n) for n = 1..500

Cf. A001108, A037123, A271629.

with(numtheory): P:=proc(q) local a,b,c,k,n; a:=0; for n from 0 to q do b:=0; c:=n;
for k from 1 to ilog10(n)+1 do b:=b+(c mod 10); c:=trunc(c/10); od; a:=a+b;
if a=trunc(sqrt(a))*trunc(sqrt(a)) then print(n); fi; od; end: P(10^6);

Select[Range[0, 16000], IntegerQ@ Sqrt@ Total@ Map[Total@ IntegerDigits@ # &, Range[0, #]] &] (* Michael De Vlieger, Apr 11 2016 *)

isok(n) = issquare(sum(k=1, n, sumdigits(k))); \\ Michel Marcus, Apr 11 2016

A340560 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k-1} (4sin(aPi/n)^2 + 4sin(bPi/k)^2).

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 49, 49, 1, 1, 288, 1296, 288, 1, 1, 1681, 30625, 30625, 1681, 1, 1, 9800, 707281, 2654208, 707281, 9800, 1, 1, 57121, 16257024, 219069601, 219069601, 16257024, 57121, 1, 1, 332928, 373301041, 17860500000, 62500000000, 17860500000, 373301041, 332928, 1

Offset: 1

Seiichi Manyama, Jan 11 2021

			Square array begins:
  1,    1,      1,         1,           1, ...
  1,    8,     49,       288,        1681, ...
  1,   49,   1296,     30625,      707281, ...
  1,  288,  30625,   2654208,   219069601, ...
  1, 1681, 707281, 219069601, 62500000000, ...

Rows and columns 1..2 give A000012, A001108.
Main diagonal gives A340562.
Cf. A212796, A340475, A340561.

default(realprecision, 120);
{T(n, k) = round(prod(a=1, n-1, prod(b=1, k-1, 4*sin(a*Pi/n)^2+4*sin(b*Pi/k)^2)))}

T(n,k) = T(k,n).

T(n,k) = A212796(n,k)/(n*k).

A076115 Squares (or 0) from A076114.

Original entry on oeis.org

1, 9, 9, 0, 25, 81, 49, 36, 81, 225, 121, 0, 169, 441, 225, 0, 289, 225, 361, 0, 441, 1089, 529, 324, 400, 1521, 729, 0, 841, 2025, 961, 784, 1089, 2601, 1225, 0, 1369, 3249, 1521, 900, 1681, 3969, 1849, 0, 2025, 4761, 2209, 0, 1225, 2025, 2601, 0, 2809, 2025

Offset: 1

Amarnath Murthy, Oct 09 2002

nonn

			a(2) = 4+5 = 9= 3^2. a(8)= 1+2+3+4+5+6+7+8 = 36 = 6^2.

Cf. A001108, A001109, A076114.

More terms from David Wasserman, Apr 02 2005

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

A212614 Least k > 1 such that the product tri(n) * tri(k) is triangular, or zero if no such k exists, where tri(k) is the k-th triangular number.

Original entry on oeis.org

2, 5, 3, 6, 2, 4, 10, 0, 13, 7, 5, 4, 9, 3, 20, 208, 185, 14, 5, 2, 6, 14, 12, 115, 55, 37, 748, 11, 12, 1358, 90, 90, 6, 3, 21, 11, 26, 10, 33, 21, 265, 51, 61, 75, 96, 131, 201, 411, 0, 10, 7, 148, 113, 92, 4, 68, 364, 329, 50, 5083, 43, 329594, 38, 36, 2414

Offset: 1

T. D. Noe, May 31 2012

nonn

That is, tri(k) = k(k+1)/2. It is provable that a(8) and a(49) are zero.

Other terms that are zero are given in sequence A001108. Note that a(71) = 2076978. In general, a Pell equation of the form x^2 = 1 + 2*n(n+1)*y*(y+1) must be solved to find a(n). - T. D. Noe, Jun 03 2012

			For n = 2, tri(n) = 3 and the first k is 5 because tri(5) = 15 and 3*15 = 45 is triangular.

Cf. A188630 (triangular numbers that are tri(x) * tri(y) for some x,y > 1).
Cf. A212615 (similar sequence for pentagonal numbers).
Cf. A000217 (triangular numbers).

kMax = 10^6; TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; Table[t = n*(n+1)/2; k = 2; While[t2 = k*(k+1)/2; k < kMax && ! TriangularQ[t*t2], k++]; If[k == kMax, 0, k], {n, 65}]

A229083 Numbers k such that the distance between the k-th triangular number and the nearest square is at most 1.

Original entry on oeis.org

1, 2, 4, 5, 8, 15, 25, 32, 49, 90, 148, 189, 288, 527, 865, 1104, 1681, 3074, 5044, 6437, 9800, 17919, 29401, 37520, 57121, 104442, 171364, 218685, 332928, 608735, 998785, 1274592, 1940449, 3547970, 5821348, 7428869, 11309768, 20679087, 33929305, 43298624, 65918161

Offset: 1

Ralf Stephan, Sep 13 2013

nonn

The k-th triangular number (A000217) is a square, or a square plus or minus one.

Union of A006451 (k-th triangular number is a square minus one), A072221 (k-th triangular number is a square plus one), and A001108 (k-th triangular number is square). Also, union of A229131 and A001108.

			A000217(4) = 10 and 10 - 3^2 = 1 so 4 is in the sequence.
A000217(5) = 15 and 4^2 - 15 = 1 so 5 is in the sequence.
A000217(8) = 36 = 6^2 so 8 is in sequence.

Cf. A000217, A001108, A006451, A072221, A229118, A229131.

for(n=1,10^8,for(i=-1,1,f=0;if(issquare(n*(n+1)/2+i),f=1;break));if(f,print1(n,",")))

G.f.: (x^7 - 2*x^6 + x^5 - 3*x^4 + x^3 + 2*x^2 + x + 1)/((1-2*x^2+x^4)*(1-2*x^2-x^4)*(1-x)) (conjectured).

A298020 Maximal overhang that can be attained from a stack of blocks of lengths 1,2,...,n (numerators).

Original entry on oeis.org

1, 8, 9, 463, 1057, 4010, 570097, 32903, 828667, 25743541, 431266313, 16610986697, 16089123031, 1971622002613, 2723872673, 159888988030039, 377543463271, 12771918729143, 19566551419628659, 2983570575161357, 1774565709813223

Offset: 1

N. J. A. Sloane, Jan 15 2018

Data supplied by David Treeby.

For the classical problem when there are n blocks all of the same length see A001108/A002805 and A065071.

			The maximal overhangs are 1, 8/3, 9/2, 463/70, 1057/120, 4010/357, 570097/41496, 32903/2016, 828667/43605, 25743541/1180300, 431266313/17472840, 16610986697/601495180, ...

David Treeby, Table of n, a(n) for n = 1..75
David Treeby, Mathematica code
David Treeby, Further Thoughts on a Paradoxical Tower, The American Mathematical Monthly 125.1 (2018): 44-60.

For denominators see A298021.

Cf. A001108/A002805, A065071.

Mathematica
```
(* See link. *)
```

A298021 Maximal overhang that can be attained from a stack of blocks of lengths 1,2,...,n (denominators).

Original entry on oeis.org

1, 3, 2, 70, 120, 357, 41496, 2016, 43605, 1180300, 17472840, 601495180, 525594160, 58565520300, 74066256, 3999303586720, 8733019680, 274477212900, 392291457008100, 56011301214120, 31297881399400, 12003126367152512883720, 5673579586816030200, 1466117956976400, 186018119917880598413465400

Offset: 1

N. J. A. Sloane, Jan 15 2018

Data supplied by David Treeby.

For the classical problem when there are n blocks all of the same length see A001108/A002805 and A065071.

			The maximal overhangs are 1, 8/3, 9/2, 463/70, 1057/120, 4010/357, 570097/41496, 32903/2016, 828667/43605, 25743541/1180300, 431266313/17472840, 16610986697/601495180, ...

David Treeby, Table of n, a(n) for n = 1..75
David Treeby, Mathematica code
David Treeby, Further Thoughts on a Paradoxical Tower, The American Mathematical Monthly 125.1 (2018): 44-60.

For numerators see A298020.

Cf. A001108/A002805, A065071.

Mathematica
```
(* See link. *)
```

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A271626 Numbers n such that the sum of the digits of the numbers from 0 to n is a square.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A340560 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k-1} (4*sin(a*Pi/n)^2 + 4*sin(b*Pi/k)^2).

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A076115 Squares (or 0) from A076114.

Views

Author

Keywords

Examples

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Crossrefs

Extensions

A212614 Least k > 1 such that the product tri(n) * tri(k) is triangular, or zero if no such k exists, where tri(k) is the k-th triangular number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A229083 Numbers k such that the distance between the k-th triangular number and the nearest square is at most 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula

A298020 Maximal overhang that can be attained from a stack of blocks of lengths 1,2,...,n (numerators).

Views

A340560 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Product_{a=1..n-1} Product_{b=1..k-1} (4sin(aPi/n)^2 + 4sin(bPi/k)^2).