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.

A221874 Numbers m such that 10*m^2 + 6 is a square.

Original entry on oeis.org

1, 5, 43, 191, 1633, 7253, 62011, 275423, 2354785, 10458821, 89419819, 397159775, 3395598337, 15081612629, 128943316987, 572704120127, 4896450447169, 21747674952197, 185936173675435, 825838944063359, 7060678149219361, 31360132199455445
Offset: 1

Views

Author

Bruno Berselli, Jan 28 2013

Keywords

Comments

The Diophantine equation 10*x^2 + k = y^2, for |k| < 10, has integer solutions with the following k values:
k = 1, the nonnegative x values are in A084070;
k = -1, " A097315;
k = 4, " 2*A084070;
k = -4, " 2*A097315;
k = 6, " this sequence;
k = -6, " A221875;
k = 9, " A075836;
k = -9, " A052454.
a(n+1)/a(n) tends alternately to (sqrt(2)+sqrt(5))^2/3 and (2*sqrt(2)+sqrt(5))^2/3; a(n+2)/a(n) tends to A176398^2.

Links

Crossrefs

Cf. A097315, A084070, A075836, A052454, A129556, A221875.
Subsequence of A031150.

Programs

  • Magma
    m:=22; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)*(1+4*x+x^2)/((1-6*x-x^2)*(1+6*x-x^2))));
  • Maple
    A221874:=proc(q)
local n;
for n from 1 to q do if type(sqrt(10*n^2+6),integer) then print(n);
fi; od; end:
A221874(100000000000000000); # Paolo P. Lava, Feb 11 2013
  • Mathematica
    LinearRecurrence[{0, 38, 0, -1}, {1, 5, 43, 191}, 22]
  • Maxima
    makelist(expand(((-5*(-1)^n+2*sqrt(10))*(3+sqrt(10))^(2*floor(n/2))-(5*(-1)^n+2*sqrt(10))*(3-sqrt(10))^(2*floor(n/2)))/10), n, 1, 22);

Formula

G.f.: x*(1+x)*(1+4*x+x^2)/((1-6*x-x^2)*(1+6*x-x^2)).
a(n) = ((-5*(-1)^n+2*t)*(3+t)^(2*floor(n/2)) - (5*(-1)^n+2*t)*(3-t)^(2*floor(n/2)))/10, where t=sqrt(10).
a(n) = 2*A129556(n) + 1.
a(n)*a(n-3) - a(n-1)*a(n-2) = -36 + 12(-1)^n.

A222390 Nonnegative integers m such that 10*m*(m+1)+1 is a square.

Original entry on oeis.org

0, 3, 15, 132, 588, 5031, 22347, 191064, 848616, 7255419, 32225079, 275514876, 1223704404, 10462309887, 46468542291, 397292260848, 1764580902672, 15086643602355, 67007605759263, 572895164628660, 2544524437949340, 21754929612286743, 96624921036315675
Offset: 1

Views

Author

Bruno Berselli, Feb 18 2013

Keywords

Comments

a(n+1)/a(n) tends alternately to (7+2*sqrt(10))/3 and (13+4*sqrt(10))/3; a(n+2)/a(n) tends to A176398^2.
Subsequence of A014601.

Links

Crossrefs

Cf. nonnegative integers m such that k*m*(m+1)+1 is a square: A001652 (k=2), A001921 (k=3), A001477 (k=4), A053606 (k=5), A105038 (k=6), A105040 (k=7), A053141 (k=8), this sequence (k=10), A105838 (k=11), A061278 (k=12), A104240 (k=13); A105063 (k=17), A222393 (k=18), A101180 (k=19), A077259 (k=20) [incomplete list].
Cf. A221875.

Programs

  • Magma
    m:=22; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(3*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2))));
  • Magma
    I:=[0,3,15,132,588]; [n le 5 select I[n] else Self(n-1) +38*Self(n-2)-38*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // Vincenzo Librandi, Aug 18 2013
  • Mathematica
    LinearRecurrence[{1, 38, -38, -1, 1}, {0, 3, 15, 132, 588}, 23]
CoefficientList[Series[3 x (1 + 4 x + x^2)/((1 - x) (1 - 6 x - x^2) (1 + 6 x - x^2)), {x, 0, 25}], x] (* Vincenzo Librandi, Aug 18 2013 *)
  • Maxima
    makelist(expand(-1/2+((5+(-1)^n*sqrt(10))*(3-sqrt(10))^(2*floor(n/2))+(5-(-1)^n*sqrt(10))*(3+sqrt(10))^(2*floor(n/2)))/20), n, 1, 23);
  • PARI
    x='x+O('x^30); concat([0], Vec(3*x*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2)))) \\ G. C. Greubel, Jul 15 2018

Formula

G.f.: 3*x*(1+4*x+x^2)/((1-x)*(1-6*x-x^2)*(1+6*x-x^2)).
a(n) = a(-n+1) = a(n-1)+38*a(n-2)-38*a(n-3)-a(n-4)+a(n-5).
a(n) = -1/2+((5+t*(-1)^n)*(3-t)^(2*floor(n/2))+(5-t*(-1)^n)*(3+t)^(2*floor(n/2)))/20, where t=sqrt(10).
2*a(n)+1 = A221875(n).

A281647 Solutions x to the negative Pell equation x^2 - 10*y^2 = -6 with x > y > 0.

Original entry on oeis.org

2, 22, 98, 838, 3722, 31822, 141338, 1208398, 5367122, 45887302, 203809298, 1742509078, 7739386202, 66169457662, 293892866378, 2512696882078, 11160189536162, 95416312061302, 423793309507778, 3623307161447398, 16092985571759402, 137590255822939822
Offset: 1

Views

Author

Colin Barker, Jan 26 2017

Keywords

Comments

The corresponding values of y are in A221875.

Examples

			22 is in the sequence because (x, y) = (22, 7) is a solution to x^2 - 10*y^2 = -6.

Links

Crossrefs

Cf. A221875.

Programs

  • Mathematica
    CoefficientList[ Series[(2 (1 + 11x + 11x^2 + x^3))/(1 - 38x^2 + x^4), {x, 0, 21}],  x] (* or *)
LinearRecurrence[{0, 38, 0, -1}, {2, 22, 98, 838}, 22] (* Robert G. Wilson v, Jan 26 2017 *)
  • PARI
    Vec(2*x*(1 + x)*(1 + 10*x + x^2) / ((1 + 6*x - x^2)*(1 - 6*x - x^2)) + O(x^30))

Formula

G.f.: 2*x*(1 + x)*(1 + 10*x + x^2) / ((1 + 6*x - x^2)*(1 - 6*x - x^2)).
a(n) = 38*a(n-2) - a(n-4) for n>4.
a(n) = ((3-r)^n + (-3-r)^n*(-3+r) - 3*(-3+r)^n - r*(-3+r)^n + (3+r)^n)/2, where r=sqrt(10).
Showing 1-3 of 3 results.

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