A176443 -id:A176443 - cp's OEIS frontend

A041894 Numerators of continued fraction convergents to sqrt(469).

21, 22, 43, 65, 693, 4223, 42923, 47146, 90069, 137215, 5853099, 5990314, 11843413, 17833727, 190180683, 1158917825, 11779358933, 12938276758, 24717635691, 37655912449, 1606265958549, 1643921870998, 3250187829547, 4894109700545, 52191284834997

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 274430, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A041895, A176443, A040447.

Numerator[Convergents[Sqrt[469], 30]] (* Harvey P. Dale, Mar 28 2013 *)

G.f.: -(x^19 -21*x^18 +22*x^17 -43*x^16 +65*x^15 -693*x^14 +4223*x^13 -42923*x^12 +47146*x^11 -90069*x^10 -137215*x^9 -90069*x^8 -47146*x^7 -42923*x^6 -4223*x^5 -693*x^4 -65*x^3 -43*x^2 -22*x -21) / (x^20 -274430*x^10 +1). - Colin Barker, Nov 26 2013

More terms from Colin Barker, Nov 26 2013

A041895 Denominators of continued fraction convergents to sqrt(469).

Original entry on oeis.org

1, 1, 2, 3, 32, 195, 1982, 2177, 4159, 6336, 270271, 276607, 546878, 823485, 8781728, 53513853, 543920258, 597434111, 1141354369, 1738788480, 74170470529, 75909259009, 150079729538, 225988988547, 2409969615008, 14685806678595, 149268036400958

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 274430, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A041894, A176443, A040447.

I:=[1,1,2,3,32,195,1982,2177,4159,6336,270271, 276607,546878,823485,8781728,53513853,543920258, 597434111,1141354369,1738788480]; [n le 20 select I[n] else 274430*Self(n-10)-Self(n-20): n in [1..50]]; // Vincenzo Librandi, Dec 26 2013

Denominator[Convergents[Sqrt[469], 30]] (* Vincenzo Librandi, Dec 26 2013 *)
LinearRecurrence[{0,0,0,0,0,0,0,0,0,274430,0,0,0,0,0,0,0,0,0,-1},{1,1,2,3,32,195,1982,2177,4159,6336,270271,276607,546878,823485,8781728,53513853,543920258,597434111,1141354369,1738788480},30] (* Harvey P. Dale, Mar 23 2023 *)

G.f.: -(x^18 -x^17 +2*x^16 -3*x^15 +32*x^14 -195*x^13 +1982*x^12 -2177*x^11 +4159*x^10 -6336*x^9 -4159*x^8 -2177*x^7 -1982*x^6 -195*x^5 -32*x^4 -3*x^3 -2*x^2 -x -1) / (x^20 -274430*x^10 +1). - Colin Barker, Nov 26 2013

a(n) = 274430*a(n-10) - a(n-20) for n>19. - Vincenzo Librandi, Dec 26 2013

More terms from Colin Barker, Nov 26 2013

A176442 Decimal expansion of (21+sqrt(469))/6.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 19 2010

Continued fraction expansion of (21+sqrt(469))/6 is A010729.

			(21+sqrt(469))/6 = 7.10940130461795253363...

Cf. A176443 (decimal expansion of sqrt(469)), A010729 (repeat 7, 9).

A176520 Decimal expansion of (63+3*sqrt(469))/14.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 23 2010

Continued fraction expansion of (63+3*sqrt(469))/14 is A010729 preceded by 9.

			(63+3*sqrt(469))/14 = 9.14065882022308182895...

Cf. A176443 (decimal expansion of sqrt(469)), A010729 (repeat 7, 9).

RealDigits[(63+3*Sqrt[469])/14,10,120][[1]] (* Harvey P. Dale, Jan 20 2013 *)

(3*sqrt(469)+63)/14 \\ Charles R Greathouse IV, Apr 25 2016

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A041894 Numerators of continued fraction convergents to sqrt(469).

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A041895 Denominators of continued fraction convergents to sqrt(469).

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A176442 Decimal expansion of (21+sqrt(469))/6.

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A176520 Decimal expansion of (63+3*sqrt(469))/14.

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