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.

A129975 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2.

Original entry on oeis.org

0, 132, 2295, 2859, 3535, 15792, 19060, 22984, 94363, 113407, 136275, 552292, 663288, 796572, 3221295, 3868227, 4645063, 18777384, 22547980, 27075712, 109444915, 131421559, 157811115, 637894012, 765983280, 919792884, 3717921063
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 13 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+953, y).
Corresponding values y of solutions (x, y) are in A160212.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (969+124*sqrt(2))/953 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1947891+1218490*sqrt(2))/953^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A160212, A001652, A129974, A156035 (decimal expansion of 3+2*sqrt(2)), A160213 (decimal expansion of (969+124*sqrt(2))/953), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,132,2295,2859,3535,15792,19060},30] (* Harvey P. Dale, Apr 12 2013 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1906*n+908209), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1906 for n > 6; a(1)=0, a(2)=132, a(3)=2295, a(4)=2859, a(5)=3535, a(6)=15792.
G.f.: x*(132+2163*x+564*x^2-116*x^3-721*x^4-116*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 953*A001652(k) for k >= 0.
a(1)=0, a(2)=132, a(3)=2295, a(4)=2859, a(5)=3535, a(6)=15792, a(7)=19060, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Apr 12 2013

Extensions

Edited and two terms added by Klaus Brockhaus, May 18 2009

A160213 Decimal expansion of (969+124*sqrt(2))/953.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, May 18 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129975.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160212.

Examples

			(969+124*sqrt(2))/953 = 1.20080008576522957612...

Links

Crossrefs

Cf. A129975, A160212, A002193 (decimal expansion of sqrt(2)), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).

Programs

  • Magma
    (969 +124*Sqrt(2))/953; // G. C. Greubel, Apr 08 2018
  • Mathematica
    RealDigits[(969 +124*Sqrt[2])/953, 10, 100][[1]] (* G. C. Greubel, Apr 08 2018 *)
  • PARI
    (969 +124*sqrt(2))/953 \\ G. C. Greubel, Apr 08 2018

Formula

Equals (31+2*sqrt(2))/(31-2*sqrt(2)).

A160212 Positive numbers y such that y^2 is of the form x^2+(x+953)^2 with integer x.

Original entry on oeis.org

845, 953, 1093, 3977, 4765, 5713, 23017, 27637, 33185, 134125, 161057, 193397, 781733, 938705, 1127197, 4556273, 5471173, 6569785, 26555905, 31888333, 38291513, 154779157, 185858825, 223179293, 902119037, 1083264617, 1300784245
Offset: 1

Views

Author

Klaus Brockhaus, May 18 2009

Keywords

Comments

(-116, a(1)) and (A129975(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (969+124*sqrt(2))/953 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1947891+1218490*sqrt(2))/953^2 for n mod 3 = 1.

Examples

			(-116, a(1)) = (-116, 845) is a solution: (-116)^2+(-116+953)^2 = 13456+700569 = 714025 = 845^2.
(A129975(1), a(2)) = (0, 953) is a solution: 0^2+(0+953)^2 = 908209 = 953^2.
(A129975(3), a(4)) = (2295, 3977) is a solution: 2295^2+(2295+953)^2 = 5267025+10549504 = 15816529 = 3977^2.

Links

Crossrefs

Cf. A129975, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160213 (decimal expansion of (969+124*sqrt(2))/953), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).

Programs

  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1},{845,953,1093,3977,4765,5713},30] (* Harvey P. Dale, Feb 18 2024 *)
  • PARI
    {forstep(n=-116, 10000000, [3, 1], if(issquare(2*n^2+1906*n+908209, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=845, a(2)=953, a(3)=1093, a(4)=3977, a(5)=4765, a(6)=5713.
G.f.: (1-x)*(845+1798*x+2891*x^2+1798*x^3+845*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 953*A001653(k) for k >= 1.
Showing 1-3 of 3 results.

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