A156835 - OEIS

A156835

Positive numbers y such that y^2 is of the form x^2+(x+833)^2 with integer x.

593, 595, 623, 637, 697, 707, 733, 833, 965, 1015, 1037, 1225, 1295, 1547, 1585, 1973, 2023, 2443, 2597, 3145, 3227, 3433, 4165, 5057, 5383, 5525, 6713, 7147, 8687, 8917, 11245, 11543, 14035, 14945, 18173, 18655, 19865, 24157, 29377, 31283, 32113

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

(-368, a(1)), (-357, a(2)), (-273, a(3)), (-245, a(4)), (-153, a(5)), (-140, a(6)), (-108, a(7)) and (A129010(n), a(n+7)) are solutions (x, y) to the Diophantine equation x^2+(x+833)^2 = y^2.

lim_{n -> oo} a(n)/a(n-15) = 3+2*sqrt(2).

lim_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^4/((3+2*sqrt(2))*((19+6*sqrt(2))/17)^2) for n mod 15 = 1.

lim_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))*((19+6*sqrt(2))/17)/((9+4*sqrt(2))/7)^3 for n mod 15 = {0, 2, 6, 11}.

lim_{n -> oo} a(n)/a(n-1) = ((9+4*sqrt(2))/7)^2*((19+6*sqrt(2))/17)/(3+2*sqrt(2)) for n mod 15 = {3, 5, 8, 9, 12, 14}.

lim_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))/(((9+4*sqrt(2))/7)*((19+6*sqrt(2))/17)^2) for n mod 15 = {4, 7, 10, 13}.

LINKS

Table of n, a(n) for n=1..41.

FORMULA

a(n) = 6*a(n-15)-a(n-30) for n > 30.

G.f.: (1-x)*(593 +1188*x+1811*x^2+2448*x^3+3145*x^4+3852*x^5+4585*x^6+5418*x^7+6383*x^8+7398*x^9+8435*x^10+9660*x^11+10955*x^12+12502*x^13+14087*x^14+12502*x^15+10955*x^16+9660*x^17+8435*x^18+7398*x^19+6383*x^20+5418*x^21+4585*x^22+3852*x^23+3145*x^24+2448*x^25+1811*x^26+1188*x^27+593*x^28)/(1-6*x^15+x^30).

EXAMPLE

(-368, a(1)) = (-368, 593) is a solution: (-368)^2+(-368+833)^2 = 135424+216225 = 351649 = 593^2.

(A129010(1), a(8)) = (0, 833) is a solution: 0^2+(0+833)^2 = 693889 = 833^2.

(A129010(3), a(10)) = (168, 1015) is a solution: (168)^2+(168+833)^2 = 28224+1002001 = 1030225 = 1015^2.

PROG

(PARI) {forstep(n=-400, 26000, [3, 1], if(issquare(2*n^2+1666*n+693889, &k), print1(k, ", ")))}

CROSSREFS

Cf. A129010, A156035 (decimal expansion of 3+2*sqrt(2)), A156649 (decimal expansion of (9+4*sqrt(2))/7), A156163 (decimal expansion of (19+6*sqrt(2))/17).

Sequence in context: A171724 A195894 A263555 * A129191 A078960 A185515

Adjacent sequences: A156832 A156833 A156834 * A156836 A156837 A156838

KEYWORD

nonn

AUTHOR

Klaus Brockhaus, Feb 17 2009

STATUS

approved