A159550 -id:A159550 - cp's OEIS frontend

A129993 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+199)^2 = y^2.

0, 21, 504, 597, 704, 3441, 3980, 4601, 20540, 23681, 27300, 120197, 138504, 159597, 701040, 807741, 930680, 4086441, 4708340, 5424881, 23818004, 27442697, 31619004, 138821981, 159948240, 184289541, 809114280, 932247141, 1074118640

Offset: 1

Mohamed Bouhamida, Jun 14 2007

Also values x of Pythagorean triples (x, x+199, y).
Corresponding values y of solutions (x, y) are in A159548.
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) = (201+20*sqrt(2))/199 for n mod 3 = {1, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 0.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A159548, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159549 (decimal expansion of (201+20*sqrt(2))/199), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).

I:=[0,21,504,597,704,3441,3980]; [n le 7 select I[n] else Self(n-1) + 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // G. C. Greubel, Mar 31 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,21,504,597,704,3441,3980},30] (* Harvey P. Dale, Jun 03 2012 *)

{forstep(n=0, 500000000, [1, 3], if(issquare(2*n^2+398*n+39601), print1(n, ",")))};

a(n) = 6*a(n-3) - a(n-6) + 398 for n > 6; a(1)=0, a(2)=21, a(3)=504, a(4)=597, a(5)=704, a(6)=3441.
G.f.: x*(21+483*x+93*x^2-19*x^3-161*x^4-19*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 199*A001652(k) for k >= 0.
a(1)=0, a(2)=21, a(3)=504, a(4)=597, a(5)=704, a(6)=3441, a(7)=3980, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Jun 03 2012

Edited and two terms added by Klaus Brockhaus, Apr 14 2009

A159549 Decimal expansion of (201+20*sqrt(2))/199.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 14 2009

lim_{n -> infinity} b(n)/b(n-1) = (201+20*sqrt(2))/199 for n mod 3 = {1, 2}, b = A129993.

lim_{n -> infinity} b(n)/b(n-1) = (201+20*sqrt(2))/199 for n mod 3 = {0, 2}, b = A159548.

			(201+20*sqrt(2))/199 = 1.15218226757518543204...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A129993, A159548, A002193 (decimal expansion of sqrt(2)), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).

(201 + 20*Sqrt(2))/199 // G. C. Greubel, Mar 30 2018

with(MmaTranslator[Mma]): Digits:=100:
RealDigits(evalf((201+20*sqrt(2))/199))[1]; # Muniru A Asiru, Mar 31 2018

RealDigits[(201+20*Sqrt[2])/199, 10, 100][[1]] (* G. C. Greubel, Mar 30 2018 *)

(201+20*sqrt(2))/199 \\ G. C. Greubel, Mar 30 2018

(201+20*sqrt(2))/199 = (20+sqrt(2))/(20-sqrt(2)).

A159548 Positive numbers y such that y^2 is of the form x^2+(x+199)^2 with integer x.

Original entry on oeis.org

181, 199, 221, 865, 995, 1145, 5009, 5771, 6649, 29189, 33631, 38749, 170125, 196015, 225845, 991561, 1142459, 1316321, 5779241, 6658739, 7672081, 33683885, 38809975, 44716165, 196324069, 226201111, 260624909, 1144260529, 1318396691

Offset: 1

Klaus Brockhaus, Apr 14 2009

(-19,a(1)) and (A129993(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+199)^2 = y^2.

			(-19, a(1)) = (-19, 181) is a solution: (-19)^2+(-19+199)^2 = 361+32400 = 32761 = 181^2.
(A129993(1), a(2)) = (0, 199) is a solution: 0^2+(0+199)^2 = 39601 = 199^2.
(A129993(3), a(4)) = (504, 865) is a solution: 504^2+(504+199)^2 = 254016+494209 = 748225 = 865^2.

Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A129993, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159549 (decimal expansion of (201+20*sqrt(2))/199), A159550 (decimal expansion of (91443+58282*sqrt(2))/199^2).

LinearRecurrence[{0,0,6,0,0,-1},{181,199,221,865,995,1145},30] (* Harvey P. Dale, Aug 09 2025 *)

{forstep(n=-20, 50000000, [1, 3], if(issquare(2*n^2+398*n+39601, &k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=181, a(2)=199, a(3)=221, a(4)=865, a(5)=995, a(6)=1145.
G.f.: x*(1-x)*(181+380*x+601*x^2+380*x^3+181*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 199*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) = (201+20*sqrt(2))/199 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = (91443+58282*sqrt(2))/199^2 for n mod 3 = 1.

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A129993 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x+199)^2 = y^2.

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A159549 Decimal expansion of (201+20*sqrt(2))/199.

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A159548 Positive numbers y such that y^2 is of the form x^2+(x+199)^2 with integer x.

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