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 1-4 of 4 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.

A228209 x-values in the solutions to x^2 - 10*y^2 = 9.

Original entry on oeis.org

3, 7, 13, 57, 253, 487, 2163, 9607, 18493, 82137, 364813, 702247, 3119043, 13853287, 26666893, 118441497, 526060093, 1012639687, 4497657843, 19976430247, 38453641213, 170792556537, 758578289293, 1460225726407, 6485619490563, 28805998562887, 55450123962253
Offset: 1

Views

Author

Colin Barker, Aug 16 2013

Keywords

Links

Crossrefs

Cf. A075836.

Programs

  • Magma
    I:=[3,7,13,57,253,487]; [n le 6 select I[n] else 38*Self(n-3)-Self(n-6): n in [1..30]]; // Vincenzo Librandi, Aug 17 2013
  • Mathematica
    CoefficientList[Series[-(7 x^6 + 13 x^5 + 57 x^4 - 13 x^3 - 7 x^2 - 3 x) / (x^6 - 38 x^3 + 1), {x, 1, 40}], x] (* Vincenzo Librandi, Aug 17 2013 *)
LinearRecurrence[{0,0,38,0,0,-1},{3,7,13,57,253,487},30] (* Harvey P. Dale, Jan 06 2014 *)
  • PARI
    Vec(-x*(7*x^5+13*x^4+57*x^3-13*x^2-7*x-3)/(x^6-38*x^3+1) + O(x^100))

Formula

G.f.: -x*(7*x^5+13*x^4+57*x^3-13*x^2-7*x-3) / (x^6-38*x^3+1).
a(n) = 38*a(n-3)-a(n-6).

Extensions

a(1)=3 prepended by Max Alekseyev, Sep 04 2013

A075873 40*n^2 + 9 is a square.

Original entry on oeis.org

0, 1, 2, 9, 40, 77, 342, 1519, 2924, 12987, 57682, 111035, 493164, 2190397, 4216406, 18727245, 83177404, 160112393, 711142146, 3158550955, 6080054528, 27004674303, 119941758886, 230881959671, 1025466481368, 4554628286713
Offset: 1

Views

Author

Gregory V. Richardson, Oct 16 2002

Keywords

Comments

Lim. n-> Inf. a(n)/a(n-3) = 19 + 6*Sqrt(10). Lim. n-> Inf. a(3*k)/a(3*k-1) = (11 + 2*Sqrt(10))/9. Lim. n-> Inf. a(3*k+1)/a(3*k) = (7 + 2*Sqrt(10))/3. Lim. n-> Inf. a(3*k+2)/a(3*k+1) = (7 + 2*Sqrt(10))/3.

References

  • A. H. Beiler, "The Pellian", ch. 22 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. Dover, New York, New York, pp. 248-268, 1966.
  • L. E. Dickson, History of the Theory of Numbers, Vol. II, Diophantine Analysis. AMS Chelsea Publishing, Providence, Rhode Island, 1999, pp. 341-400.
  • Peter G. L. Dirichlet, Lectures on Number Theory (History of Mathematics Source Series, V. 16); American Mathematical Society, Providence, Rhode Island, 1999, pp. 139-147.

Links

Programs

  • Mathematica
    LinearRecurrence[{0,0,38,0,0,-1},{0,1,2,9,40,77},30] (* Harvey P. Dale, Sep 05 2020 *)
  • PARI
    a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,0,0,38,0,0]^(n-1)*[0;1;2;9;40;77])[1,1] \\ Charles R Greathouse IV, Jul 09 2024

Formula

G.f.: x^2*(x^5+2x^4+9x^3+2x^2+x)/(x^6-38x^3+1).
a(n) = A075836(n)/2.

A359800 a(n) is the least m such that the concatenation of n^2 and m is a square.

Original entry on oeis.org

6, 9, 61, 9, 6, 1, 284, 516, 225, 489, 104, 4, 744, 249, 625, 3201, 444, 9, 201, 689, 4201, 416, 984, 4801, 681, 5201, 316, 996, 5801, 601, 6201, 144, 936, 6801, 449, 7201, 7401, 804, 7801, 225, 8201, 8401, 6, 8801, 9001, 9201, 9401, 324, 9801, 19344, 769, 38025
Offset: 1

Views

Author

Mohammed Yaseen, Jan 13 2023

Keywords

Comments

The only one-digit terms are 1, 4, 6 and 9. Proof: Squares mod 10 are 0, 1, 4, 5, 6 and 9. Concatenation of a square and 0 is 10 times that square, which is not a square. So 0 is ruled out. Squares with last digit 5 have second last digit 2. Since no square ends in 2, 5 is also ruled out.
From Thomas Scheuerle, Jan 14 2023: (Start)
The only term with two digits is a(3) = 61.
Some terms with an odd number of digits appear infinitely often, for example, 516 for n = 8, 1352, 632958674, ... .
If a term has an even number of digits and is of the form 1+2*k*10^(d+1) with 10^d <= 2*k < 10^(d+1), then it appears only once at k = n in this sequence. Are terms with an even number of digits possible which are not of this form? (End)

Examples

			For n=3, 61 is the least number m such that the concatenation of 3^2 and m is a square: 961 = 31^2. So a(3) = 61.
For n=7, 284 is the least number m such that the concatenation of 7^2 and m is a square: 49284 = 222^2. So a(7) = 284.

Links

Crossrefs

Cf. A000290, A071176, A075836, A084070, A221874, A246560.

Programs

  • PARI
    a(n)={my(m=n^2, b=1); while(1, m*=10; my(r=(sqrtint(m+b-1)+1)^2-m); b*=10; if(rAndrew Howroyd, Jan 13 2023
  • Python
    from math import isqrt
def a(n):
    t, k = str(n*n), isqrt(10*n**2)
    while not (s:=str(k*k)).startswith(t) or s[len(t)]=="0": k += 1
    return int(s[len(t):])
print([a(n) for n in range(1, 53)]) # Michael S. Branicky, Jan 15 2023
  • Python
    from math import isqrt
from sympy.ntheory.primetest import is_square
def A359800(n):
    m = 10*n*n
    if is_square(m): return 0
    a = 1
    while (k:=(isqrt(a*(m+1)-1)+1)**2-m*a)>=10*a:
        a *= 10
    return k # Chai Wah Wu, Feb 15 2023

Formula

a(n) = A071176(n^2) = A071176(A000290(n)).
From Thomas Scheuerle, Jan 13 2023: (Start)
a(A084070(n)) = 1.
a(2*A084070(n)) = 4.
a(A221874(n)) = 6.
a(A075836(n)) = 9. (End)
Showing 1-4 of 4 results.

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