cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A156066 Numbers n with property that n^2 is a square arising in A154138.

Original entry on oeis.org

2, 3, 9, 16, 52, 93, 303, 542, 1766, 3159, 10293, 18412, 59992, 107313, 349659, 625466, 2037962, 3645483, 11878113, 21247432, 69230716, 123839109, 403506183, 721787222, 2351806382, 4206884223, 13707332109, 24519518116, 79892186272
Offset: 1

Views

Author

Zak Seidov, Oct 21 2009

Keywords

Comments

Except for the first term, positive values of x (or y) satisfying x^2 - 6xy + y^2 + 23 = 0. - Colin Barker, Feb 08 2014

Links

Crossrefs

Cf. A154138.

Programs

  • GAP
    a:=[2,3,9,16];; for n in [5..30] do a[n]:=6*a[n-2]-a[n-4]; od; a; # Muniru A Asiru, Sep 28 2018
  • Magma
    I:=[2,3,9,16]; [n le 4 select I[n] else 6*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Feb 11 2014
  • Maple
    seq(coeff(series(-x*(x-1)*(x+2)*(2*x+1)/((x^2-2*x-1)*(x^2+2*x-1)),x,n+1), x, n), n = 1..30); # Muniru A Asiru, Sep 28 2018
  • Mathematica
    a[1]=2;a[2]=3;a[3]=9;a[4]=16;a[n_]:=a[n]=6*a[n-2]-a[n-4];A1=Table[a[n],{n,25}]
CoefficientList[Series[-(x - 1) (x + 2) (2 x + 1)/((x^2 - 2 x - 1) (x^2 + 2 x - 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 11 2014 *)
  • PARI
    Vec(-x*(x-1)*(x+2)*(2*x+1)/((x^2-2*x-1)*(x^2+2*x-1)) + O(x^100)) \\ Colin Barker, Feb 08 2014

Formula

a(n) = sqrt((A154138(n)^2 + A154138(n) + 6)/2).
a(1..4) = (2,3,9,16); a(n>4) = 6*a(n-2) - a(n-4).
G.f.: -x*(x-1)*(x+2)*(2*x+1) / ((x^2-2*x-1)*(x^2+2*x-1)). - Colin Barker, Feb 08 2014
a(n) = A006452(n-1) - A006452(n) + A006452(n+1). - Carl Najafi, Sep 27 2018

A154130 Exponents m with decreasing fractional part of (4/3)^m.

Original entry on oeis.org

1, 4, 13, 17, 128, 485, 692, 1738, 12863, 77042, 109705, 289047, 720429, 4475944, 75629223, 182575231
Offset: 1

Views

Author

Hieronymus Fischer, Jan 11 2009

Keywords

Comments

The next term is greater than 3*10^8.

Examples

			a(3)=13, since fract((4/3)^13)=0.0923.., but fract((4/3)^k)>=0.16... for 1<=k<=12; thus fract((4/3)^13)<fract((4/3)^k) for 1<=k<13.

Crossrefs

Cf. A081464, A153669, A153701, A153708, A154138, A154146.
Cf. A002379, A064628.

Formula

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).

Extensions

Extended by Charles R Greathouse IV, Nov 03 2009
a(15)-a(16) from Robert Gerbicz, Nov 21 2010

A175035 Offsets i such that i + n*(n+1)/2 is a perfect square for some positive integer n.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 10, 13, 15, 16, 19, 21, 22, 24, 25, 26, 28, 30, 33, 34, 35, 36, 39, 43, 45, 46, 48, 49, 53, 54, 55, 58, 60, 61, 63, 64, 66, 71, 72, 75, 76, 78, 79, 80, 81, 85, 89, 90, 91, 93, 94, 97, 99, 100, 103, 105, 106, 108, 111, 114, 115, 116, 118, 120
Offset: 1

Views

Author

Ctibor O. Zizka, Nov 10 2009

Keywords

Comments

The ansatz n*(n+1)/2+i=s^2 can be transformed into (2*n+1)^2-2*(2*s)^2 =1-8*i.
A necessary condition for solutions to this Diophantine equation is that D=2 is a quadratic residue of the squarefree part of 8*i-1 (see A057126).
A sufficient condition is then available by a sequence of tests on the continued fractions of a quadratic surd that originates from a solution of this congruence.
See Mollin and Matthews for details. - R. J. Mathar, Nov 16 2009

Links

Crossrefs

Cf. A001108, A006451, A154138 to A154154.
Cf. A230044.

Programs

  • Mathematica
    Take[Rest[Ceiling[Sqrt[#]]^2-#&/@Accumulate[Range[1000]]//Union],70] (* Harvey P. Dale, Sep 07 2019 *)
  • PARI
    is(n)=#bnfisintnorm(bnfinit(z^2-8),-8*n+1) /* Ralf Stephan, Oct 14 2013 */

Extensions

Extended by R. J. Mathar, Nov 26 2009

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

Views

Author

R. J. Mathar, Oct 18 2009

Keywords

Examples

			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.

Links

Crossrefs

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).

Programs

  • Mathematica
    Join[{6, 8}, Select[Range[0, 10^5], ( Ceiling[Sqrt[#*(# + 1)/2]] )^2 - #*(# + 1)/2 == 28 &]] (* G. C. Greubel, Sep 03 2016 *)
  • PARI
    {for (n=0, 10^9, if ( issquare(n*(n+1)\2 + 28), print1(n, ", ") ) );}

Formula

{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

Extensions

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

Views

Author

Ctibor O. Zizka, Nov 09 2009

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • Mathematica
    Ceiling[Sqrt[#]]^2-#&/@Accumulate[Range[0,80]] (* Harvey P. Dale, Aug 25 2013 *)
  • PARI
    a(n) = my(t=n*(n+1)/2); if (issquare(t), 0, (sqrtint(t)+1)^2 - t); \\ Michel Marcus, Nov 06 2022

Formula

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

Extensions

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

Views

Author

Ctibor O. Zizka, Nov 10 2009

Keywords

Comments

Complement of A175035.

Crossrefs

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

Extensions

Extended by R. J. Mathar, Nov 26 2009
Showing 1-6 of 6 results.

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