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 1-5 of 5 results.

A236331 Positive integers n such that x^2 - 18xy + y^2 + n = 0 has integer solutions.

Original entry on oeis.org

64, 256, 320, 576, 704, 1024, 1216, 1280, 1600, 1856, 1984, 2304, 2624, 2816, 2880, 3136, 3520, 3776, 3904, 4096, 4544, 4864, 5056, 5120, 5184, 5696, 6080, 6336, 6400, 6464, 6976, 7424, 7744, 7936, 8000, 8384, 8896, 9216, 9280, 9536, 9664, 9920, 10496, 10816
Offset: 1

Views

Author

Colin Barker, Feb 16 2014

Keywords

Examples

			64 is in the sequence because x^2 - 18xy + y^2 + 64 = 0 has integer solutions, for example (x, y) = (1, 13).

Crossrefs

Cf. A001519 (n = 64), A052995 (n = 256), A055819 (n = 256), A005248 (n = 320), A237132 (n = 704), A237133 (n = 1216).
Cf. A031363, A084917, A237351, A237599, A237606, A237609, A237610, A236330.

A237133 Values of x in the solutions to x^2 - 3xy + y^2 + 19 = 0, where 0 < x < y.

Original entry on oeis.org

4, 5, 7, 11, 17, 28, 44, 73, 115, 191, 301, 500, 788, 1309, 2063, 3427, 5401, 8972, 14140, 23489, 37019, 61495, 96917, 160996, 253732, 421493, 664279, 1103483, 1739105, 2888956, 4553036, 7563385, 11920003, 19801199, 31206973, 51840212, 81700916, 135719437
Offset: 1

Views

Author

Colin Barker, Feb 04 2014

Keywords

Comments

The corresponding values of y are given by a(n+2).
Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 1216 = 0.

Examples

			11 is in the sequence because (x, y) = (11, 28) is a solution to x^2 - 3xy + y^2 + 19 = 0.

Links

Crossrefs

Cf. A001519, A005248, A055819, A237132, A218735.

Programs

  • Mathematica
    LinearRecurrence[{0,3,0,-1},{4,5,7,11},40] (* Harvey P. Dale, Dec 15 2014 *)
  • PARI
    Vec(-x*(x-1)*(4*x^2+9*x+4)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

Formula

a(n) = 3*a(n-2)-a(n-4).
G.f.: -x*(x-1)*(4*x^2+9*x+4) / ((x^2-x-1)*(x^2+x-1)).
a(n) = (1/2) * (F(n+4) + (-1)^n*F(n-5)), n>4, with F the Fibonacci numbers (A000045). - Ralf Stephan, Feb 05 2014

A218735 Values of x in the solutions to x^2 - 3xy + y^2 + 29 = 0, where 0 < x < y.

Original entry on oeis.org

5, 6, 9, 13, 22, 33, 57, 86, 149, 225, 390, 589, 1021, 1542, 2673, 4037, 6998, 10569, 18321, 27670, 47965, 72441, 125574, 189653, 328757, 496518, 860697, 1299901, 2253334, 3403185, 5899305, 8909654, 15444581, 23325777, 40434438, 61067677, 105858733
Offset: 1

Views

Author

Colin Barker, Feb 05 2014

Keywords

Comments

The corresponding values of y are given by a(n+2).
Positive values of x (or y) satisfying x^2 - 18xy + y^2 + 1856 = 0.

Examples

			13 is in the sequence because (x, y) = (13, 33) is a solution to x^2 - 3xy + y^2 + 29 = 0.

Links

Crossrefs

Cf. A001519, A005248, A055819, A237132, A237133.

Programs

  • Mathematica
    LinearRecurrence[{0,3,0,-1},{5,6,9,13},40] (* Harvey P. Dale, Nov 30 2024 *)
  • PARI
    Vec(-x*(x-1)*(5*x^2+11*x+5)/((x^2-x-1)*(x^2+x-1)) + O(x^100))

Formula

a(n) = 3*a(n-2)-a(n-4).
G.f.: -x*(x-1)*(5*x^2+11*x+5) / ((x^2-x-1)*(x^2+x-1)).

A356716 a(n) is the integer w such that (c(n)^2, -d(n)^2, -w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 11^3, where c(n) = F(n+2) + (-1)^n * F(n-3), d(n) = F(n+1) + (-1)^n * F(n-4) and F(n) is the n-th Fibonacci number (A000045).

Original entry on oeis.org

5, 19, 31, 101, 179, 655, 1189, 4451, 8111, 30469, 55555, 208799, 380741, 1431091, 2609599, 9808805, 17886419, 67230511, 122595301, 460804739, 840280655, 3158402629, 5759369251, 21648013631, 39475304069, 148377692755, 270567759199, 1016995835621, 1854499010291
Offset: 1

Views

Author

XU Pingya, Aug 24 2022

Keywords

Comments

Conjecture:
(i) For all k > 2, 2*x^3 + 2*y^3 + z^3 = A089270(k)^3 have primitive solutions form (c(n)^2, -d(n)^2, -w(n)) with d(n) = 3*d(n-2) - d(n-4), c(n) = d(n+2) - d(n) and w(n) = 8*w(n-2) - 8*w(n-4) + w(n-6).
(ii) This sequence is a subsequence of A089270.
From XU Pingya, Jun 07 2024: (Start)
Several positive examples of conjecture:
When A089270(4,5,6,7) = {19,29,31,41}, d(n) can be taken as:
(1/2) * (F(n+3) + (-1)^n * F(n-6));
((1-(-1)^n)/2) * (F(n+3) + F(n-4)) + ((1+(-1)^n)/2) * (F(n+3) - F(n-4));
((1-(-1)^n)/2) * (2*F(n-1) + 3*F(n-3)) + ((1+(-1)^n)/2) * (3*F(n-2) + 2*F(n-4));
and
((1-(-1)^n)/2) * (2*F(n+1) + F(n-5)) + ((1+(-1)^n)/2) * (F(n+2) + 2*F(n-4)).
When A089270(17) = 121, d(n) can be taken as d(1,2,3,4) = {-3,0,7,11}. (End)
From XU Pingya, Jul 17 2024: (Start)
Furthermore, we observe that if (x, y) (y < x/2) is the solution of the Diophantine equation x^2 + x * y - y^2 = A089270(k). Let
d(2*n-1) = x * F(2*n-2) - y * F(2*n-3), c(2*n-1) = d(2*n+1) - d(2*n-1);
d(2*n) = x * F(2*n-2) + y * F(2*n-1), c(2*n) = d(2*n+2) - d(2*n).
Then such c(n) and d(n) satisfy the conjecture. (End)

Examples

			For n=3, 2 * ((F(5) - F(0))^2)^3 + 2 * (-(F(4) - F(-1))^2)^3 + (-31)^3 = 2 * 25^3 - 2 * 4^3 - 31^3 = 1331, a(3) = 31.

Links

Crossrefs

Cf. A000045, A081016, A089270, A206351, A228208, A237132.
Cf. also A337928, A354336, A356717.

Programs

  • Mathematica
    Table[(-1331+2*((Fibonacci[n+2]+(-1)^n*Fibonacci[n-3]))^6-2*(Fibonacci[n+1]+(-1)^n*Fibonacci[n-4])^6)^(1/3), {n,28}]

Formula

a(n) = (-1331 + 2 * A237132(n)^6 - 2 * A228208(n-1)^6)^(1/3).
a(n) = ((1-(-1)^n)/2) * (-1 + 6 * Sum_{k=0..n-1} Fibonacci(4*k-1) + 14 * Sum_{k=0..n-2} Fibonacci(4*k+1)) + ((1+(-1)^n)/2) * (-1 + 6 * Sum_{k=0..n-1} Fibonacci(4*k-1) + 14 * Sum_{k=0..n-1} Fibonacci(4*k+1)).
a(n) = ((1-(-1)^n)/2) * (-1 + 6*A206351(n) + 14*A081016(n-2)) + ((1+(-1)^n)/2) * (-1 + 6*A206351(n) + 14*A081016(n-1)).
From Stefano Spezia, Aug 25 2022: (Start)
G.f.: x*(5 + 14*x - 23*x^2 - 28*x^3 - x^4)/((1 - x)*(1 - 3*x + x^2)*(1 + 3*x + x^2)).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)
From XU Pingya, Jul 17 2024: (Start)
a(2*n-1) = (F(2*n) + F(2*n-2) + F(2*n-5))^2 + (F(2*n) + F(2*n-2) + F(2*n-5)) * (F(2*n-2) + F(2*n-4) + F(2*n-7)) - (F(2*n-2) + F(2*n-4) + F(2*n-7))^2;
a(2*n) = (F(2*n+2) + F(2*n-3))^2 + (F(2*n+2) + F(2*n-3)) * (F(2*n) + F(2*n-5)) - (F(2*n) + F(2*n-5))^2. (End)

A356717 a(n) is the integer w such that (c(n)^2, -d(n)^2, w) is a primitive solution to the Diophantine equation 2*x^3 + 2*y^3 + z^3 = 11^3, where c(n) = F(n+2) + (-1)^n * F(n-3), d(n) = F(n+3) + (-1)^n * F(n-2) and F(n) is the n-th Fibonacci number (A000045).

Original entry on oeis.org

1, 29, 59, 241, 445, 1691, 3089, 11629, 21211, 79745, 145421, 546619, 996769, 3746621, 6831995, 25679761, 46827229, 176011739, 320958641, 1206402445, 2199883291, 8268805409, 15078224429, 56675235451, 103347687745, 388457842781, 708355589819, 2662529664049
Offset: 1

Views

Author

XU Pingya, Aug 24 2022

Keywords

Examples

			For n=3, 2 * ((F(5) - F(0))^2)^3 + 2 * (-(F(6) - F(1))^2)^3 + 59^3 = 2 * 25^3 - 2 * 49^3 + 59^3 = 1331, a(3) = 59.

Links

Crossrefs

Cf. A000045, A081016, A081018, A089270, A228208, A237132.
Cf. also A337929, A354337, A356716.

Programs

  • Mathematica
    Table[(1331-2*((Fibonacci[n+2]+(-1)^n*Fibonacci[n-3]))^6+2*(Fibonacci[n+3]+(-1)^n*Fibonacci[n-2])^6)^(1/3), {n,28}]

Formula

a(n) = (1331 - 2 * A237132(n)^6 + 2 * A228208(n+1)^6)^(1/3).
a(n) = ((1-(-1)^n)/2) * (-5 + 14 * Sum_{k=1..n-1} Fibonacci(4*k-1) + 6 * Sum_{k=0..n-1} Fibonacci(4*k+1)) + ((1+(-1)^n)/2) * (-5 + 14 * Sum_{k=1..n} Fibonacci(4*k-1) + 6 * Sum_{k=0..n-1} Fibonacci(4*k+1)).
a(n) = ((1-(-1)^n)/2) * (-5 + 14 * A081018(n-1) + 6 * A081016(n-1)) + ((1+(-1)^n)/2) * (-5 + 14 * A081018(n) + 6 * A081016(n-1)).
From Stefano Spezia, Aug 25 2022: (Start)
G.f.: x*(1 + 28*x + 23*x^2 - 14*x^3 - 5*x^4)/((1 - x)*(1 - 3*x + x^2)*(1 + 3*x + x^2)).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - a(n-4) + a(n-5) for n > 5. (End)
Showing 1-5 of 5 results.

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