A156713 Positive numbers y such that y^2 is of the form x^2+(x+16807)^2 with integer x.

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

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

Table of n, a(n) for n=1..32.

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