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

Author

Klaus Brockhaus, Feb 17 2009

Keywords

• nonn

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}.

Examples

			(-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.
		

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).

Crossrefs

Cf. A118576, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7).

Programs

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 *)

PARI

{forstep(n=-7220, 700000, [1, 3], if(issquare(2*n^2+33614*n+282475249, &k),print1(k, ",")))}

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).