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 1-3 of 3 results.

A220082 Numbers k such that 10*k-1 is a square.

Original entry on oeis.org

1, 5, 17, 29, 53, 73, 109, 137, 185, 221, 281, 325, 397, 449, 533, 593, 689, 757, 865, 941, 1061, 1145, 1277, 1369, 1513, 1613, 1769, 1877, 2045, 2161, 2341, 2465, 2657, 2789, 2993, 3133, 3349, 3497, 3725, 3881, 4121, 4285, 4537, 4709, 4973, 5153, 5429, 5617, 5905
Offset: 1

Views

Author

Bruno Berselli, Dec 05 2012

Keywords

Comments

Equivalently, numbers of the form m*(10*m+6)+1, where m=0,-1,1,-2,2,-3,3,...

Links

Crossrefs

Cf. A085787, A132356 (numbers n such that 10*n+1 is a square).
Cf. numbers n such that k*n-1 is a square: A002522 (k=1), A001844 (k=2), A062317 (k=5).

Programs

  • Magma
    [n: n in [1..6000] | IsSquare(10*n-1)]; /* or (see the first comment): */ [1] cat [m*(10*m+6)+1: m in [-n,n], n in [1..24]];
  • Magma
    I:=[1,5,17,29,53]; [n le 5 select I[n] else Self(n-1) +2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..60]]; // Vincenzo Librandi, Aug 18 2013
  • Maple
    A220082:=proc(q)
local n;
for n from 1 to q do if type(sqrt(10*n-1), integer) then print(n);
fi; od; end:
A220082(1000); # Paolo P. Lava, Feb 19 2013
  • Mathematica
    Select[Range[0, 6000], IntegerQ[Sqrt[10 # - 1]] &]
CoefficientList[Series[(1 + 4 x + 10 x^2 + 4 x^3 + x^4) / ((1 + x)^2 (1 - x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2013 *)
LinearRecurrence[{1,2,-2,-1,1},{1,5,17,29,53},50] (* Harvey P. Dale, Nov 19 2023 *)

Formula

G.f.: x*(1+4*x+10*x^2+4*x^3+x^4)/((1+x)^2*(1-x)^3).
a(n) = a(-n+1) = (10*n*(n-1)-(2*n-1)*(-1)^n+3)/4.
For the definition: 10*a(n)-1 = ((10*n-(-1)^n-5)/2)^2.
a(n) = A212570(n)-A212570(n-1) = 4*A085787(n-1)+1 = A132356(n-1)-(2*n-1)*(-1)^n.

A274544 Values of k such that 2*k-1 and 5*k-1 are both perfect squares.

Original entry on oeis.org

1, 13, 925, 18241, 1333345, 26303005, 1922682061, 37928914465, 2772506198113, 54693468355021, 3997952014996381, 78867943439025313, 5765044033118582785, 113727519745606145821, 8313189497804981379085, 163995004605220623248065, 11987613490790750030057281
Offset: 1

Views

Author

Colin Barker, Jun 27 2016

Keywords

Comments

Intersection of A001844 and A062317.

Examples

			13 is in the sequence because 2*13-1 = 25 = 5^2, and 5*13-1 = 64 = 8^2.

Links

Crossrefs

Cf. A001844, A062317, A274545.

Programs

  • Mathematica
    Rest@ CoefficientList[Series[x (1 + 12 x - 530 x^2 + 12 x^3 + x^4)/((1 - x) (1 - 38 x + x^2) (1 + 38 x + x^2)), {x, 0, 17}], x] (* Michael De Vlieger, Jun 27 2016 *)
  • PARI
    Vec(x*(1+12*x-530*x^2+12*x^3+x^4)/((1-x)*(1-38*x+x^2)*(1+38*x+x^2))+ O(x^20))
  • PARI
    isok(n) = issquare(2*n-1) && issquare(5*n-1); \\ Michel Marcus, Jun 28 2016

Formula

a(n) = a(n-1) + 1442*a(n-2) - 1442*a(n-3) - a(n-4) + a(n-5) for n>5.
G.f.: x*(1 + 12*x - 530*x^2 + 12*x^3 + x^4) / ((1 - x)*(1 - 38*x + x^2)*(1 + 38*x + x^2)).

A274545 Values of k such that 5*k-1 and 10*k-1 are both perfect squares.

Original entry on oeis.org

1, 29, 33293, 1130977, 1305146305, 44336554445, 51164345409437, 1738081606216033, 2005744667435597089, 68136275082544365341, 78629202401645931667661, 2671078254047822603875969, 3082421990543579145800043553, 104711609647046466634601365517
Offset: 1

Views

Author

Colin Barker, Jun 27 2016

Keywords

Comments

Intersection of A062317 and A220082.

Examples

			29 is in the sequence because 5*29-1 = 144 = 12^2, and 10*29-1 = 289 = 17^2.

Links

Crossrefs

Cf. A062317, A220082, A274544.

Programs

  • Mathematica
    Rest@ CoefficientList[Series[x (1 + 28 x- 5938 x^2 + 28 x^3 + x^4) / ((1 - x) (1 - 198 x + x^2) (1 + 198 x + x^2)), {x, 0, 17}], x] (* Michael De Vlieger, Jun 27 2016 *)
  • PARI
    Vec(x*(1+x)*(1-6*x+x^2)/((1-x)*(1-34*x+x^2)*(1+x+x^2)) + O(x^20))
  • PARI
    isok(n) = issquare(5*n-1) && issquare(10*n-1); \\ Michel Marcus, Jun 28 2016

Formula

a(n) = a(n-1) + 39202*a(n-2) - 39202*a(n-3) - a(n-4) + a(n-5) for n>5.
G.f.: x*(1 + 28*x - 5938*x^2 + 28*x^3 + x^4)/((1 - x)*(1 - 198*x + x^2)*(1 + 198*x + x^2)).
Showing 1-3 of 3 results.

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