|
| |
|
|
A129836
|
|
Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+97)^2 = y^2.
|
|
15
| |
|
|
0, 15, 228, 291, 368, 1575, 1940, 2387, 9416, 11543, 14148, 55115, 67512, 82695, 321468, 393723, 482216, 1873887, 2295020, 2810795, 10922048, 13376591, 16382748, 63658595, 77964720, 95485887, 371029716, 454411923, 556532768
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| Also values x of Pythagorean triples (x, x+97, y).
Corresponding values y of solutions (x, y) are in A157469.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436, the x values are given by the sequence defined by a(n) = 6*a(n-3)-a(n-6)+2p with a(1)=0, a(2)=2m+1, a(3)=6m^2-10m+4, a(4)=3p, a(5)=6m^2+10m+4, a(6)=40m^2-58m+21 (cf. A118673).
Pairs (p, m) are (7, 2), (17, 3), (31, 4), (71, 6), (97, 7), (127, 8), (199, 10), (241, 11), (337, 13), (449, 15), (577, 17), (647, 18), (881, 21), (967, 22), ...
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (99+14*sqrt(2))/97 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (19491+12070*sqrt(2))/97^2 for n mod 3 = 0.
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=2m^2+2m+1, b(3)=10m^2-14m+5, b(4)=5p, b(5)=10m^2+14m+5, b(6)=58m^2-82m+29. [From Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 09 2009]
|
|
|
REFERENCES
| Mohammad K. Azarian, Diophantine Pair, Problem B-881, Fibonacci Quarterly, Vol. 37, No. 3, August 1999, pp. 277-278. Solution appeared in Vol. 38, No. 2, May 2000, pp. 183-184.
|
|
|
FORMULA
| a(n) = 6*a(n-3)-a(n-6)+194 for n > 6; a(1)=0, a(2)=15, a(3)=228, a(4)=291, a(5)=368, a(6)=1575.
G.f.: x*(15+213*x+63*x^2-13*x^3-71*x^4-13*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 97*A001652(k) for k >= 0.
|
|
|
MATHEMATICA
| ClearAll[a]; Evaluate[Array[a, 6]] = {0, 15, 228, 291, 368, 1575}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 194; Table[a[n], {n, 1, 29}] (* From Jean-François Alcover, Dec 27 2011, after given formula *)
|
|
|
PROG
| (PARI) {forstep(n=0, 600000000, [3, 1], if(issquare(2*n^2+194*n+9409), print1(n, ", ")))}
|
|
|
CROSSREFS
| Cf. A157469, A066436 (primes of the form 2*n^2-1), A001652, A118673, A118674, A156035 (decimal expansion of 3+2*sqrt(2)), A157470 (decimal expansion of (99+14*sqrt(2))/97), A157471 (decimal expansion of (19491+12070*sqrt(2))/97^2).
Sequence in context: A067222 A154597 A041422 * A075262 A097185 A178299
Adjacent sequences: A129833 A129834 A129835 * A129837 A129838 A129839
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Mohamed Bouhamida (bhmd95(AT)yahoo.fr), May 21 2007
|
|
|
EXTENSIONS
| Edited and two terms added by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Mar 12 2009
|
| |
|
|
