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 A156567 Positive numbers y such that y^2 is of the form x^2+(x+23)^2 with integer x. 7
 17, 23, 37, 65, 115, 205, 373, 667, 1193, 2173, 3887, 6953, 12665, 22655, 40525, 73817, 132043, 236197, 430237, 769603, 1376657, 2507605, 4485575, 8023745, 14615393, 26143847, 46765813, 85184753, 152377507, 272571133, 496493125 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS (-8, a(1)) and(A118337(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+23)^2 = y^2. lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2). lim_{n -> infinity} a(n)/a(n-1) = (27+10*sqrt(2))/23 for n mod 3 = {0, 2}. lim_{n -> infinity} a(n)/a(n-1) = (627+238*sqrt(2))/23^2 for n mod 3 = 1. For the generic case x^2+(x+p)^2=y^2 with p=m^2-2 a prime number in A028871, m>=2, 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+2, a(3)=3m^2-10m+8, a(4)=3p, a(5)=3m^2+10m+8, a(6)=20m^2-58m+42.Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(1)=p, b(2)=m^2+2m+2, b(3)=5m^2-14m+10, b(4)=5p, b(5)=5m^2+14m+10, b(6)=29m^2-82m+58. [From Mohamed Bouhamida, Sep 09 2009] LINKS FORMULA a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=17, a(2)=23, a(3)=37, a(4)=65, a(5)=115, a(6)=205. G.f.: x*(1-x)*(17+40*x+77*x^2+40*x^3+17*x^4)/(1-6*x^3+x^6). EXAMPLE (-8, a(1)) = (-8, 17) is a solution: (-8)^2+(-8+23)^2 = 64+225 = 289 = 17^2. (A118337(1), a(2)) = (0, 23) is a solution: 0^2+(0+23)^2 = 529 = 23^2. (A118337(3), a(4)) = (33, 65) is a solution: 33^2+(33+23)^2 = 1089+3136 = 4225 = 65^2. PROG (PARI) {forstep(n=-8, 360000000, [1, 3], if(issquare(2*n*(n+23)+529, &k), print1(k, ", ")))} CROSSREFS Cf. A118337, A156035 (decimal expansion of 3+2*sqrt(2)), A156571 (decimal expansion of (27+10*sqrt(2))/23), A157472 (decimal expansion of (627+238*sqrt(2))/23^2). A156570 (first trisection), A156568 (second trisection), A156569 (third trisection). Sequence in context: A103805 A333327 A241528 * A231332 A243137 A256397 Adjacent sequences:  A156564 A156565 A156566 * A156568 A156569 A156570 KEYWORD nonn AUTHOR Klaus Brockhaus, Feb 11 2009 , Feb 16 2009 EXTENSIONS G.f. corrected, third and fourth comment edited, cross-reference added by Klaus Brockhaus, Sep 18 2009 STATUS approved

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Last modified July 24 18:34 EDT 2021. Contains 346273 sequences. (Running on oeis4.)