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 A159758 Positive numbers y such that y^2 is of the form x^2+(x+79)^2 with integer x. 4
 65, 79, 101, 289, 395, 541, 1669, 2291, 3145, 9725, 13351, 18329, 56681, 77815, 106829, 330361, 453539, 622645, 1925485, 2643419, 3629041, 11222549, 15406975, 21151601, 65409809, 89798431, 123280565, 381236305, 523383611, 718531789 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS (-16, a(1)) and (A118676(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+79)^2 = y^2. Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2). Lim_{n -> infinity} a(n)/a(n-1) = (83+18*sqrt(2))/79 for n mod 3 = {0, 2}. Lim_{n -> infinity} a(n)/a(n-1) = (10659+6110*sqrt(2))/79^2 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 >= 5, the x values are given by the sequence defined by a(n) = 6*a(n-3) - a(n-6) + 2*p with a(1)=0, a(2) = 2*m + 2, a(3) = 3*m^2 - 10*m + 8, a(4) = 3*p, a(5) = 3*m^2 + 10*m + 8, a(6) = 20*m^2 - 58*m + 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 + 2*m + 2, b(3) = 5*m^2 - 14*m + 10, b(4)= 5*p, b(5) = 5*m^2 + 14*m + 10, b(6) = 29*m^2 - 82*m + 58. - Mohamed Bouhamida (bhmd95(AT)yahoo.fr), Sep 09 2009 LINKS Harvey P. Dale, Table of n, a(n) for n = 1..1000 Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1). FORMULA a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=65, a(2)=79, a(3)=101, a(4)=289, a(5)=395, a(6)=541. G.f.: (1-x)*(65+144*x+245*x^2+144*x^3+65*x^4) / (1-6*x^3+x^6). a(3*k-1) = 79*A001653(k) for k >= 1. EXAMPLE (-16, a(1)) = (-16, 65) is a solution: (-16)^2 + (-16+79)^2 = 256+3969 = 4225 = 65^2. (A118676(1), a(2)) = (0, 79) is a solution: 0^2 + (0+79)^2 = 6241 = 79^2. (A118676(3), a(4)) = (161, 289) is a solution: 161^2 + (161+79)^2 = 25921 + 57600 = 83521 = 289^2. MATHEMATICA RecurrenceTable[{a[1]==65, a[2]==79, a[3]==101, a[4]==289, a[5]==395, a[6]== 541, a[n]==6a[n-3]-a[n-6]}, a[n], {n, 30}] (* or *) LinearRecurrence[ {0, 0, 6, 0, 0, -1}, {65, 79, 101, 289, 395, 541}, 30] (* Harvey P. Dale, Oct 03 2011 *) PROG (PARI) {forstep(n=-16, 10000000, [1, 3], if(issquare(2*n^2+158*n+6241, &k), print1(k, ", ")))} (MAGMA) I:=[65, 79, 101, 289, 395, 541]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 22 2018 CROSSREFS Cf. A118676, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159759 (decimal expansion of (83+18*sqrt(2))/79), A159760 (decimal expansion of (10659+6110*sqrt(2))/79^2). Sequence in context: A060877 A113688 A214484 * A056693 A164282 A025312 Adjacent sequences:  A159755 A159756 A159757 * A159759 A159760 A159761 KEYWORD nonn,easy AUTHOR Klaus Brockhaus, Apr 30 2009 STATUS approved

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Last modified November 20 16:38 EST 2018. Contains 317412 sequences. (Running on oeis4.)