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
KEYWORD
nonn
AUTHOR
Klaus Brockhaus, Feb 17 2009
STATUS
approved