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A157246 Positive numbers y such that y^2 is of the form x^2+(x+343)^2 with integer x. 1
245, 265, 287, 343, 427, 493, 637, 833, 1097, 1295, 1715, 2275, 2693, 3577, 4753, 6317, 7483, 9947, 13223, 15665, 20825, 27685, 36805, 43603, 57967, 77063, 91297, 121373, 161357, 214513, 254135, 337855, 449155, 532117, 707413, 940457, 1250273 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
(-147, a(1)), (-96, a(2)), (-63, a(3)) and (A118611(n), a(n+3)) are solutions (x, y) to the Diophantine equation x^2+(x+343)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-7) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2)) / ((9+4*sqrt(2))/7)^2 for n mod 7 = {0, 1, 2, 4, 5}.
lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 7 = {3, 6}.
LINKS
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,6,0,0,0,0,0,0,-1).
FORMULA
a(n) = 6*a(n-7)-a(n-14) for n > 14; a(1)=245, a(2)=265, a(3)=287, a(4)=343, a(5)=427, a(6)=493, a(7)=637, a(8)=833, a(9)=1097, a(10)=1295, a(11)=1715, a(12)=2275, a(13)=2693, a(14)=3577.
G.f.: x * (1-x) * (245 +510*x +797*x^2 +1140*x^3 +1567*x^4 +2060*x^5 +2697*x^6 +2060*x^7 +1567*x^8 +1140*x^9 +797*x^10 +510*x^11 +245*x^12) / (1 -6*x^7 +x^14).
a(7*k-3) = 343*A001653(k) for k >= 1.
EXAMPLE
(-147, a(1)) = (-147, 245) is a solution: (-147)^2+(-147+343)^2 = 21609+38416 = 60025 = 245^2.
(A118611(1), a(4)) = (0, 343) is a solution: 0^2+(0+343)^2 = 117649 = 343^2.
(A118611(3), a(6)) = (132, 493) is a solution: 132^2+(132+343)^2 = 17424+225625 = 243049 = 493^2.
MATHEMATICA
Sqrt[#]&/@Select[Table[x^2+(x+343)^2, {x, -150, 885000}], IntegerQ[ Sqrt[#]]&] (* or *) LinearRecurrence[{0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, -1}, {245, 265, 287, 343, 427, 493, 637, 833, 1097, 1295, 1715, 2275, 2693, 3577}, 40](* Harvey P. Dale, Dec 30 2011 *)
PROG
(PARI) {forstep(n=-148, 1000000, [1, 3], if(issquare(n^2+(n+343)^2, &k), print1(k, ", ")))}
CROSSREFS
Cf. A118611, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).
Sequence in context: A013684 A233821 A216233 * A186460 A171994 A129210
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, Feb 25 2009
EXTENSIONS
G.f. adapted to the offset by Bruno Berselli, Apr 01 2011
STATUS
approved

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Last modified May 21 05:34 EDT 2024. Contains 372728 sequences. (Running on oeis4.)