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.

A130013 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.

Original entry on oeis.org

0, 37, 1768, 1941, 2128, 11937, 12940, 14025, 71148, 76993, 83316, 416245, 450312, 487165, 2427616, 2626173, 2840968, 14150745, 15308020, 16559937, 82478148, 89223241, 96519948, 480719437, 520032720, 562561045, 2801839768, 3030974373
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+647, y).
Corresponding values y of solutions (x, y) are in A159641.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (649+36*sqrt(2))/647 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1084467+707402*sqrt(2))/647^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159641, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159642 (decimal expansion of (649+36*sqrt(2))/647), A159643 (decimal expansion of (1084467+707402*sqrt(2))/647^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,37,1768,1941,2128,11937,12940},40] (* Harvey P. Dale, Jan 27 2025 *)
  • PARI
    {forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1294*n+418609), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1294 for n > 6; a(1)=0, a(2)=37, a(3)=1768, a(4)=1941, a(5)=2128, a(6)=11937.
G.f.: x*(37+1731*x+173*x^2-35*x^3-577*x^4-35*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 647*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 21 2009

A159643 Decimal expansion of (1084467 + 707402*sqrt(2))/647^2.

Original entry on oeis.org

4, 9, 8, 0, 5, 0, 5, 6, 8, 0, 5, 9, 8, 9, 6, 5, 1, 0, 5, 1, 7, 5, 5, 0, 3, 1, 7, 0, 9, 3, 8, 4, 8, 6, 3, 6, 8, 3, 6, 8, 4, 3, 6, 9, 5, 1, 8, 7, 0, 1, 3, 1, 3, 6, 5, 6, 9, 2, 8, 7, 7, 1, 2, 4, 0, 3, 2, 4, 9, 8, 4, 3, 3, 4, 5, 2, 3, 4, 2, 0, 2, 6, 8, 0, 0, 2, 8, 8, 1, 9, 8, 6, 7, 3, 3, 2, 5, 9, 5, 2, 9, 4, 3, 9, 0
Offset: 1

Views

Author

Klaus Brockhaus, Apr 21 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A130013.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A159641.

Examples

			(1084467 + 707402*sqrt(2))/647^2 = 4.98050568059896510517...

Links

Crossrefs

Cf. A130013, A159641, A002193 (decimal expansion of sqrt(2)), A159642 (decimal expansion of (649+36*sqrt(2))/647).

Programs

  • Magma
    (1084467+707402*Sqrt(2))/647^2; // G. C. Greubel, May 10 2018
  • Mathematica
    RealDigits[(1084467+707402*Sqrt[2])/647^2, 10, 100][[1]] (* G. C. Greubel, May 10 2018 *)
  • PARI
    (1084467+707402*sqrt(2))/647^2 \\ G. C. Greubel, May 10 2018

Formula

Equals (1226 + 577*sqrt(2))/(1226 - 577*sqrt(2)).
Equals (3 + 2*sqrt(2))*(36 - sqrt(2))^2/(36 + sqrt(2))^2.

A159641 Positive numbers y such that y^2 is of the form x^2+(x+647)^2 with integer x.

Original entry on oeis.org

613, 647, 685, 2993, 3235, 3497, 17345, 18763, 20297, 101077, 109343, 118285, 589117, 637295, 689413, 3433625, 3714427, 4018193, 20012633, 21649267, 23419745, 116642173, 126181175, 136500277, 679840405, 735437783, 795581917
Offset: 1

Views

Author

Klaus Brockhaus, Apr 21 2009

Keywords

Comments

(-35,a(1)) and (A130013(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+647)^2 = y^2.

Examples

			(-35, a(1)) = (-35, 613) is a solution: (-35)^2+(-35+647)^2 = 1225+374544 = 375769 = 613^2.
(A130013(1), a(2)) = (0, 647) is a solution: 0^2+(0+647)^2 = 418609 = 647^2.
(A130013(3), a(4)) = (1768, 2993) is a solution: 1768^2+(1768+647)^2 = 3125824+5832225 = 8958049 = 2993^2.

Links

Crossrefs

Cf. A130013, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159642 (decimal expansion of (649+36*sqrt(2))/647), A159643 (decimal expansion of (1084467+707402*sqrt(2))/647^2).

Programs

  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1},{613,647,685,2993,3235,3497},30] (* Harvey P. Dale, Jun 22 2022 *)
  • PARI
    {forstep(n=-36, 10000000, [1, 3], if(issquare(2*n^2+1294*n+418609, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=613, a(2)=647, a(3)=685, a(4)=2993, a(5)=3235, a(6)=3497.
G.f.: (1-x)*(613+1260*x+1945*x^2+1260*x^3+613*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 647*A001653(k) for k >= 1.
Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (649+36*sqrt(2))/647 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = (1084467+707402*sqrt(2))/647^2 for n mod 3 = 1.
Showing 1-3 of 3 results.

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