A006451 -id:A006451 - cp's OEIS frontend

A120744 Least k>0 such that a centered polygonal number nk(k+1)/2+1 is a perfect square; or -1 if no such number exists.

2, -1, 1, 3, 2, 7, 15, 1, 16, 8, 14, 4, 5, 15, 1, 2, 5, -1, 6, 3, 2, 39, 6, 1, 21, 7, 110, 3, 15, 7, 15, -1, 2, 8, 1, 4, 989, 8, 14, 2, 45, 15, 9, 4, 5, 335, 9, 1, 29, -1, 30, 15, 10, 415, 6, 2, 10, 32, 54, 3, 77, 55, 1, 5, 2, 7, 47750, 11, 15, 23, 47, -1, 48, 24, 16, 12, 5, 8, 2639, 1, 6720, 704, 38, 4, 2, 39, 505, 3, 13, 56, 9, 20, 13, 1631, 41

Offset: 1

Alexander Adamchuk, Apr 26 2007

sign

			a(5) = 2 because A129556(2) = 2>1 and A129556(1) = 0<1.

Max Alekseyev, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Centered Polygonal Number

Cf. A001542, A006451, A001921, A129444, A001570, A001652, A129556, A053606, A105038, A105040, A053141, A061278.

a(n) = -1 for n in A166259.

a(n) = 1 for n = k^2-1.

Edited and b-file provided by Max Alekseyev, Jan 20 2010

A154142 Indices k such that 9 plus the k-th triangular number is a perfect square.

Original entry on oeis.org

0, 10, 13, 63, 80, 370, 469, 2159, 2736, 12586, 15949, 73359, 92960, 427570, 541813, 2492063, 3157920, 14524810, 18405709, 84656799, 107276336, 493415986, 625252309, 2875839119, 3644237520, 16761618730, 21240172813, 97693873263, 123796799360, 569401620850

Offset: 1

R. J. Mathar, Oct 18 2009

nonn

			0*(0+1)/2+9 = 3^2. 10*(10+1)/2+9 = 8^2. 13*(13+1)/2+9 = 10^2. 63*(63+1)/2+9 = 45^2.

Robert Israel, Table of n, a(n) for n = 1..2351
F. T. Adams-Watters, SeqFan Discussion, Oct 2009

Cf. A000217, A000290, A006451, A001109, A001541.

[n: n in [0..2*10^7] | IsSquare(9 + n*(n+1)/2)]; // Vincenzo Librandi, Sep 03 2016

[0] cat [n: n in [0..2*10^7] | (Ceiling(Sqrt(n*(n+ 1)/2)))^2-n*(n+1)/2 eq 9]; // Vincenzo Librandi, Sep 03 2016

seq(seq((8*orthopoly[U](k+j,3) - (8 - (-1)^j)*orthopoly[T](k+j,3)-1)/2, j=0..1),k=0..20); # Robert Israel, Jul 07 2015

Join[{0}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 9 &]] (* G. C. Greubel, Sep 03 2016 *)
Select[Range[0, 2 10^7], IntegerQ[Sqrt[9 + # (# + 1) / 2]] &] (* Vincenzo Librandi, Sep 03 2016 *)
(Sqrt[8#+1]-1)/2&/@Select[Accumulate[Range[0,5*10^6]],IntegerQ[Sqrt[#+9]]&] (* The program generates the first 17 terms of the sequence. *) (* Harvey P. Dale, Oct 21 2024 *)

{k: 9+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^2*(10 +3*x -10*x^2 -x^3)/((1-x) * (x^2-2*x-1) * (x^2+2*x-1))
G.f.: ( 2 + (-5+4*x)/(x^2+2*x-1) + (6+17*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End)
a(1..4) = (0,10,13,63); a(n) = 6*a(n-2) - a(n-4) + 2, for n > 4. - Ctibor O. Zizka, Nov 10 2009
From Robert Israel, Jul 07 2015: (Start)
These conjectures follow from the theory of Pell-like equations.
a(2*k+1) = (8 * A001109(k) -7 * A001541(k) - 1)/2.
a(2*k)   = (8 * A001109(k) -9 * A001541(k) - 1)/2. (End)

a(16)-a(24) from Donovan Johnson, Nov 01 2010

a(25)-a(30) from Lars Blomberg, Jul 07 2015

A154144 Indices k such that 13 plus the k-th triangular number is a perfect square.

Original entry on oeis.org

2, 8, 23, 53, 138, 312, 807, 1821, 4706, 10616, 27431, 61877, 159882, 360648, 931863, 2102013, 5431298, 12251432, 31655927, 71406581, 184504266, 416188056, 1075369671, 2425721757, 6267713762, 14138142488, 36530912903, 82403133173, 212917763658, 480280656552

Offset: 1

R. J. Mathar, Oct 18 2009

nonn

			2*(2+1)/2+13 = 4^2. 8*(8+1)/2+13 = 7^2. 23*(23+1)/2+13 = 17^2. 53*(53+1)/2+13 = 38^2.

F. T. Adams-Watters, SeqFan Discussion, Oct 2009

Cf. A000217, A000290, A006451.

With[{nn=25000},Transpose[Select[Thread[{Range[nn],Accumulate[ Range[nn]]}], IntegerQ[Sqrt[#[[2]]+13]]&]][[1]]] (* Harvey P. Dale, Jan 13 2012 *)
Join[{2, 8}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 13 &]] (* G. C. Greubel, Sep 03 2016 *)

{k: 13+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*(2 +6*x +3*x^2 -6*x^3 -3*x^4)/((1-x) * (x^2-2*x-1) * (x^2+2*x-1))
G.f.: ( 6 + (-3-2*x)/(x^2+2*x-1) + 1/(x-1) + (8+19*x)/(x^2-2*x-1) )/2 . (End)
a(1..4) = (2,8,23,53); a(n) = 6*a(n-2) - a(n-4) + 2, for n>2. - Ctibor O. Zizka, Nov 10 2009

a(16)-a(24) from Donovan Johnson, Nov 01 2010

a(25)-a(30) from Lars Blomberg, Jul 07 2015

A154145 Indices k such that 15 plus the k-th triangular number is a perfect square.

Original entry on oeis.org

1, 4, 6, 11, 20, 33, 43, 70, 121, 196, 254, 411, 708, 1145, 1483, 2398, 4129, 6676, 8646, 13979, 24068, 38913, 50395, 81478, 140281, 226804, 293726, 474891, 817620, 1321913, 1711963, 2767870, 4765441, 7704676, 9978054, 16132331, 27775028

Offset: 1

R. J. Mathar, Oct 18 2009

nonn

a(1..4)=(1,4,6,11); a(n>4)=6*a(n-2)-a(n-4)+2. [From Ctibor O. Zizka, Nov 13 2009]

			1*(1+1)/2+15 = 4^2. 4*(4+1)/2+15 = 5^2. 6*(6+1)/2+15 = 6^2. 11*(11+1)/2+15 = 9^2.

F. T. Adams-Watters, SeqFan Discussion, Oct 2009

Cf. A000217, A000290, A006451.

Flatten[Position[Accumulate[Range[28000000]],?(IntegerQ[Sqrt[#+15]]&)]] (* This program will take a long time to run. *) (* _Harvey P. Dale, Jun 09 2014 *)
Join[{1, 4, 6}, Select[Range[0, 1000], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 15 &]] (* G. C. Greubel, Sep 03 2016 *)

{k: 15+k*(k+1)/2 in A000290}.
Conjectures: (Start)
a(n)= +a(n-1) +6*a(n-4) -6*a(n-5) -a(n-8) +a(n-9).
G.f.: x*(-1-3*x-2*x^2-5*x^3-3*x^4+5*x^5+2*x^6+3*x^7+2*x^8)/((x-1) * (x^4+2*x^2-1) * (x^4-2*x^2-1)).
G.f.: ( 4 + (7+4*x+16*x^2+11*x^3)/(x^4-2*x^2-1) + 1/(x-1) + (-4-7*x-3*x^2-2*x^3)/(x^4+2*x^2-1) )/2. (End)
a(1..4) = (1,4,6,11); a(n) = 6*a(n-2) - a(n-4) + 2, for n>4. - Ctibor O. Zizka, Nov 13 2009

a(32)-a(37) from Donovan Johnson, Nov 01 2010

A154148 Numbers k such that 21 plus the k-th triangular number is a perfect square.

Original entry on oeis.org

5, 7, 40, 50, 237, 295, 1384, 1722, 8069, 10039, 47032, 58514, 274125, 341047, 1597720, 1987770, 9312197, 11585575, 54275464, 67525682

Offset: 1

R. J. Mathar, Oct 18 2009

			5, 7, 40, and 50 are terms:
   5* (5+1)/2 + 21 =  6^2,
   7* (7+1)/2 + 21 =  7^2,
  40*(40+1)/2 + 21 = 29^2,
  50*(50+1)/2 + 21 = 36^2.

F. T. Adams-Watters, SeqFan Discussion, Oct 2009

Cf. A000217, A000290, A006451.

Select[Range[68*10^6],IntegerQ[Sqrt[21+(#(#+1))/2]]&] (* Harvey P. Dale, Mar 07 2017 *)

{for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 21), print1(n, ", ") ) );}

{k: 21+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*(-5-2*x-3*x^2+2*x^3+6*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).
G.f.: ( 12 + (12+25*x)/(x^2-2*x-1) + 1/(x-1) + (-1-10*x)/(x^2+2*x-1) )/2. (End)

A154150 Numbers k such that 24 plus the k-th triangular number is a perfect square.

Original entry on oeis.org

1, 15, 24, 94, 145, 551, 848, 3214, 4945, 18735, 28824, 109198, 168001, 636455, 979184, 3709534, 5707105, 21620751, 33263448, 126014974

Offset: 1

R. J. Mathar, Oct 18 2009

			1, 5, 24, and 94 are terms:
   1* (1+1)/2 + 24 =  5^2,
  15*(15+1)/2 + 24 = 12^2,
  24*(24+1)/2 + 24 = 18^2,
  94*(94+1)/2 + 24 = 67^2.

F. T. Adams-Watters, SeqFan Discussion, Oct 2009

Cf. A000217, A000290, A006451.

Select[Range[12602*10^4],IntegerQ[Sqrt[24+(#(#+1))/2]]&] (* Harvey P. Dale, Jul 07 2019 *)

{for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 24), print1(n, ", ") ) );}

{k: 24+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*(-1-14*x-3*x^2+14*x^3+2*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).
G.f.: ( 4 + 1/(x-1) + (10+27*x)/(x^2-2*x-1) + (-7+4*x)/(x^2+2*x-1) )/2. (End)

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

Original entry on oeis.org

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

A154154 Numbers k such that 30 plus the k-th triangular number is a perfect square.

Original entry on oeis.org

3, 13, 34, 84, 203, 493, 1186, 2876, 6915, 16765, 40306, 97716, 234923, 569533, 1369234, 3319484, 7980483, 19347373, 46513666

Offset: 1

R. J. Mathar, Oct 18 2009

			3, 13, 34, and 84 are terms:
   3* (3+1)/2 + 30 =  6^2,
  13*(13+1)/2 + 30 = 11^2,
  34*(34+1)/2 + 30 = 25^2,
  84*(84+1)/2 + 30 = 60^2.

F. T. Adams-Watters, SeqFan Discussion, Oct 2009

Cf. A000217, A000290, A006451.

Position[Accumulate[Range[8*10^6]],?(IntegerQ[Sqrt[#+30]]&)]//Flatten (* _Harvey P. Dale, May 30 2016 *)
Join[{3, 13}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 30 &]] (* G. C. Greubel, Sep 03 2016 *)

{for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 30), print1(n, ", ") ) );}

{k: 30+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*(-3-10*x-3*x^2+10*x^3+4*x^4)/((x-1) * (x^2-2*x-1) * (x^2+2*x-1)).
G.f.: ( 8 + (-5-2*x)/(x^2+2*x-1) + (12+29*x)/(x^2-2*x-1) + 1/(x-1) )/2. (End)

A217278 Sequences A124174 and A006454 interlaced.

Original entry on oeis.org

0, 0, 1, 3, 10, 15, 45, 120, 351, 528, 1540, 4095, 11935, 17955, 52326, 139128, 405450, 609960, 1777555, 4726275, 13773376, 20720703, 60384555, 160554240, 467889345, 703893960, 2051297326, 5454117903, 15894464365, 23911673955, 69683724540, 185279454480

Offset: 0

Raphie Frank, Sep 29 2012

a(2n) and 2*a(2n) + 1 are triangular.

a(2n + 1) is triangular and a(2n + 1)/2 is the harmonic mean of consecutive triangular numbers (therefore, a(2n + 1) + 1 is square).

			a(18) = 35*(52326 - 1540) + 45 = 1777555,
a(19) = 35*(139128 - 4095) + 120 = 4726275.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,0,34,0,-34,0,-1,0,1).

Cf. (sqrt(8a(2n) + 1) - 1)/2 = A216134(n) = A216162(2n + 1).
Cf. sqrt(a(2n+1) + 1) = A006452(n + 1) = A216162(2n + 2).
Cf. (sqrt(8a(2n+1) + 1) - 1)/2 = A006451(n).

LinearRecurrence[{0,1,0,34,0,-34,0,-1,0,1},{0,0,1,3,10,15,45,120,351,528},40] (* Harvey P. Dale, Aug 04 2019 *)

concat([0,0], Vec(-x^2*(3*x^5+x^4+12*x^3+9*x^2+3*x+1)/((x-1)*(x+1)*(x^2-2*x-1)*(x^2+2*x-1)*(x^4+6*x^2+1)) + O(x^100))) \\ Colin Barker, Jun 23 2015

a(n) = 35*(a(n-4) - a(n-8)) + a(n-12).
lim n --> infinity a(2n)/a(2n - 1) = (3 + sqrt(8))/2.
From Raphie Frank, Dec 21 2015: (Start)
a(2*n + 1) = 1/64*(((4 + sqrt(2)) * (1 - (-1)^(n+1) * sqrt(2))^(2*floor((n+1)/2)) + (4 - sqrt(2)) * (1+(-1)^(n+1) * sqrt(2))^(2*floor((n+1)/2))))^2 - 1.
a(2*n + 2) = 1/2*(3*(a(2*n + 1)) + sqrt((a(2*n + 1)) + 1) * sqrt(8*(a(2*n + 1)) + 1) + 1).
(End)

A226071 Numbers k such that triangular(k)+1 is a square and triangular(k)+2 is a prime.

Original entry on oeis.org

0, 2, 5, 189, 3074, 218685

Offset: 1

Alex Ratushnyak, May 25 2013

Subsequence of A006451 that contains the terms k such that triangular(k)+2 is prime.

a(7) is too large to include here (see b-file). - Max Alekseyev, Jan 30 2014

Max Alekseyev, Table of n, a(n) for n = 1..12

Corresponding primes: A226069.
Roots of corresponding squares: A226070.
Cf. A226071-A226074.

C
```
// see A226069.
```

Select[Range[0,220000],IntegerQ[Sqrt[(#(#+1))/2+1]]&&PrimeQ[(#(#+1))/2+2]&] (* Harvey P. Dale, Mar 01 2023 *)

Terms a(7)-a(12) from Max Alekseyev, Jan 30 2014

cp's OEIS Frontend

A120744 Least k>0 such that a centered polygonal number nk(k+1)/2+1 is a perfect square; or -1 if no such number exists.

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A154142 Indices k such that 9 plus the k-th triangular number is a perfect square.

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A154144 Indices k such that 13 plus the k-th triangular number is a perfect square.

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A154145 Indices k such that 15 plus the k-th triangular number is a perfect square.

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A154148 Numbers k such that 21 plus the k-th triangular number is a perfect square.

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A154150 Numbers k such that 24 plus the k-th triangular number is a perfect square.

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

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Mathematica

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