A160212 -id:A160212 - cp's OEIS frontend

A129975 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2.

0, 132, 2295, 2859, 3535, 15792, 19060, 22984, 94363, 113407, 136275, 552292, 663288, 796572, 3221295, 3868227, 4645063, 18777384, 22547980, 27075712, 109444915, 131421559, 157811115, 637894012, 765983280, 919792884, 3717921063

Offset: 1

Mohamed Bouhamida, Jun 13 2007

Also values x of Pythagorean triples (x, x+953, y).
Corresponding values y of solutions (x, y) are in A160212.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (969+124*sqrt(2))/953 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1947891+1218490*sqrt(2))/953^2 for n mod 3 = 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A160212, A001652, A129974, A156035 (decimal expansion of 3+2*sqrt(2)), A160213 (decimal expansion of (969+124*sqrt(2))/953), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,132,2295,2859,3535,15792,19060},30] (* Harvey P. Dale, Apr 12 2013 *)

{forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1906*n+908209), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+1906 for n > 6; a(1)=0, a(2)=132, a(3)=2295, a(4)=2859, a(5)=3535, a(6)=15792.
G.f.: x*(132+2163*x+564*x^2-116*x^3-721*x^4-116*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 953*A001652(k) for k >= 0.
a(1)=0, a(2)=132, a(3)=2295, a(4)=2859, a(5)=3535, a(6)=15792, a(7)=19060, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Apr 12 2013

Edited and two terms added by Klaus Brockhaus, May 18 2009

A160213 Decimal expansion of (969+124*sqrt(2))/953.

Original entry on oeis.org

1, 2, 0, 0, 8, 0, 0, 0, 8, 5, 7, 6, 5, 2, 2, 9, 5, 7, 6, 1, 2, 9, 4, 9, 5, 6, 9, 9, 6, 8, 7, 3, 0, 5, 9, 4, 0, 9, 6, 8, 1, 4, 1, 9, 8, 5, 8, 8, 3, 2, 5, 4, 5, 2, 3, 6, 8, 7, 2, 0, 7, 0, 1, 7, 3, 2, 5, 1, 3, 2, 0, 3, 2, 8, 7, 4, 3, 9, 7, 9, 7, 7, 7, 7, 2, 7, 6, 8, 1, 5, 7, 6, 6, 6, 9, 7, 5, 3, 9, 8, 7, 6, 0, 9, 0

Offset: 1

Klaus Brockhaus, May 18 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129975.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160212.

			(969+124*sqrt(2))/953 = 1.20080008576522957612...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A129975, A160212, A002193 (decimal expansion of sqrt(2)), A160214 (decimal expansion of (1947891+1218490*sqrt(2))/953^2).

(969 +124*Sqrt(2))/953; // G. C. Greubel, Apr 08 2018

RealDigits[(969 +124*Sqrt[2])/953, 10, 100][[1]] (* G. C. Greubel, Apr 08 2018 *)

(969 +124*sqrt(2))/953 \\ G. C. Greubel, Apr 08 2018

Equals (31+2*sqrt(2))/(31-2*sqrt(2)).

A160214 Decimal expansion of (1947891+1218490*sqrt(2))/953^2.

Original entry on oeis.org

4, 0, 4, 2, 1, 2, 6, 9, 5, 9, 3, 4, 0, 8, 4, 8, 4, 0, 1, 6, 5, 0, 2, 4, 7, 5, 6, 8, 0, 8, 4, 3, 0, 1, 0, 9, 3, 4, 2, 2, 7, 2, 4, 8, 1, 7, 1, 1, 5, 9, 4, 7, 3, 8, 4, 0, 1, 0, 7, 8, 6, 6, 0, 7, 4, 2, 6, 6, 2, 4, 9, 4, 8, 3, 1, 1, 7, 7, 9, 3, 4, 3, 4, 8, 6, 8, 0, 6, 1, 2, 7, 9, 9, 7, 9, 4, 7, 5, 8, 6, 9, 1, 2, 1, 3

Offset: 1

Klaus Brockhaus, May 18 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A129975.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160212.

			(1947891+1218490*sqrt(2))/953^2 = 4.04212695934084840165...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A129975, A160212, A002193 (decimal expansion of sqrt(2)), A160213 (decimal expansion of (969+124*sqrt(2))/953).

(1947891 +1218490*Sqrt(2))/953^2; // G. C. Greubel, Apr 08 2018

RealDigits[(1947891 +1218490*Sqrt[2])/953^2, 10, 100][[1]] (* G. C. Greubel, Apr 08 2018 *)

(1947891 +1218490*sqrt(2))/953^2 \\ G. C. Greubel, Apr 08 2018

Equals (1690+721*sqrt(2))/(1690-721*sqrt(2)).

Equals (3+2*sqrt(2))*(31-2*sqrt(2))^2/(31+2*sqrt(2))^2.

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A129975 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+953)^2 = y^2.

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A160213 Decimal expansion of (969+124*sqrt(2))/953.

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A160214 Decimal expansion of (1947891+1218490*sqrt(2))/953^2.

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