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 A156713 Positive numbers y such that y^2 is of the form x^2+(x+16807)^2 with integer x. 1
 12005, 12467, 12985, 14063, 15025, 16807, 19073, 20923, 24157, 26747, 31213, 40817, 48055, 53753, 63455, 71077, 84035, 99413, 111475, 131957, 148015, 175273, 232897, 275863, 309533, 366667, 411437, 487403, 577405, 647927, 767585, 861343 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS (-7203, a(1)), (-5740, a(2)), (-4704, a(3)), (-3087, a(4)), (-1903, a(5)), and (A118576(n), a(n+5)) are solutions (x, y) to the Diophantine equation x^2+(x+16807)^2 = y^2. lim_{n -> infinity} a(n)/a(n-11) = 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 11 = 1. lim_{n -> infinity} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^5 / (3+2*sqrt(2))^2 for n mod 11 = {0, 2, 4, 6, 7, 9}. lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))^3 / ((9+4*sqrt(2))/7)^7 for n mod 11 = {3, 5, 8, 10}. LINKS Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1). FORMULA a(n) = 6*a(n-11)-a(n-22) for n > 22; a(1) = 12005, a(2) = 12467, a(3) = 12985, a(4) = 14063, a(5) = 15025, a(6) = 16807, a(7) = 19073, a(8) = 20923, a(9) = 24157, a(10) = 26747, a(11) = 31213, a(12) = 40817, a(13) = 48055, a(14) = 53753, a(15) = 63455, a(16) = 71077, a(17) = 84035, a(18) = 99413, a(19) = 111475, a(20) = 131957, a(21) = 148015, a(22) = 175273. G.f.: (1-x)*(12005 +24472*x+37457*x^2+51520*x^3+66545*x^4+83352*x^5+102425*x^6+123348*x^7+147505*x^8+174252*x^9+205465*x^10+174252*x^11+147505*x^12+123348*x^13+102425*x^14+83352*x^15+66545*x^16+51520*x^17 +37457*x^18+24472*x^19+12005*x^20)/(1-6*x^11+x^22). EXAMPLE (-7203, a(1)) = (-7203, 12005) is a solution: (-7203)^2+(-7203+16807)^2 = 51883209+92236816 = 144120025 = 12005^2. (A118576(1), a(6)) = (0, 16807) is a solution: 0^2+(0+16807)^2 = 258791569 = 16807^2. (A118576(3), a(8)) = (3773, 20923) is a solution: 3773^2+(3773+16807)^2 = 14235529+423536400 = 437771929 = 20923^2. MATHEMATICA CoefficientList[Series[(1-x)(12005+24472x+37457x^2+51520x^3+66545x^4+83352x^5+ 102425x^6+123348x^7+147505x^8+ 174252x^9+205465x^10+ 174252x^11+ 147505x^12+ 123348x^13+ 102425x^14+83352x^15+66545x^16+51520x^17+ 37457x^18+ 24472x^19+ 12005x^20)/(1-6x^11+x^22), {x, 0, 40}], x] (* or *) LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1}, {12005, 12467, 12985, 14063, 15025, 16807, 19073, 20923, 24157, 26747, 31213, 40817, 48055, 53753, 63455, 71077, 84035, 99413, 111475, 131957, 148015, 175273}, 40] (* Harvey P. Dale, Oct 02 2021 *) PROG (PARI) {forstep(n=-7220, 700000, [1, 3], if(issquare(2*n^2+33614*n+282475249, &k), print1(k, ", ")))} CROSSREFS Cf. A118576, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7). Sequence in context: A235074 A251163 A236607 * A183059 A287046 A235308 Adjacent sequences:  A156710 A156711 A156712 * A156714 A156715 A156716 KEYWORD nonn AUTHOR Klaus Brockhaus, Feb 17 2009 STATUS approved

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Last modified August 18 19:27 EDT 2022. Contains 356215 sequences. (Running on oeis4.)