A002531 -id:A002531 - cp's OEIS frontend

A321119 a(n) = ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2); n-th row common denominator of A321118.

4, 2, 8, 10, 28, 38, 104, 142, 388, 530, 1448, 1978, 5404, 7382, 20168, 27550, 75268, 102818, 280904, 383722, 1048348, 1432070, 3912488, 5344558, 14601604, 19946162, 54493928, 74440090, 203374108, 277814198, 759002504, 1036816702, 2832635908, 3869452610

Offset: 0

Franck Maminirina Ramaharo, Nov 01 2018

			a(0) = ((1 - sqrt(3))^0 + (1 + sqrt(3))^0)/2^floor((0 - 1)/2) = 2*(1 + 1) = 4.

Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967. See p. 47, Table 2.5.2.

Encyclopedia of Mathematics, Quadrature formula
John C. Holladay, A smoothest curve approximation, Math. Comp. Vol. 11 (1957), 233-243.
Peter Köhler, On the weights of Sard's quadrature formulas, CALCOLO Vol. 25 (1988), 169-186.
Leroy F. Meyers and Arthur Sard, Best approximate integration formulas, J. Math. Phys. Vol. 29 (1950), 118-123.
Arthur Sard, Best approximate integration formulas; best approximation formulas, American Journal of Mathematics Vol. 71 (1949), 80-91.
Isaac J. Schoenberg, Spline interpolation and best quadrature formulae, Bull. Amer. Math. Soc. Vol. 70 (1964), 143-148.
Frans Schurer, On natural cubic splines, with an application to numerical integration formulae, EUT report. WSK, Dept. of Mathematics and Computing Science Vol. 70-WSK-04 (1970), 1-32.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A002176 (common denominators of Cotesian numbers).
Cf. A321118, A321120.
Cf. A001834, A002530, A002531, A003500, A005246, A048788, A083336, A125905.

LinearRecurrence[{0, 4, 0, -1}, {4, 2, 8, 10}, 50]

a(n) := ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2)$
makelist(ratsimp(a(n)), n, 0, 50);

a(n) = (((sqrt(2) - sqrt(6))/2)^n + ((sqrt(6) + sqrt(2))/2)^n)*((2 - sqrt(2))*(-1)^n + 2 + sqrt(2))/2.
a(-n) = (-1)^n*a(n).
a(n) = 2*A000034(n+1)*A002531(n).
a(2*n) = 2*A001834(n).
a(2*n+1) = 2*A003500(n).
a(n) = 4*a(n-2) - a(n-4) with a(0) = 4, a(1) = 2, a(2) = 8, a(3) = 10.
a(2*n+3) = a(2*n+1) + a(2*n+2).
a(2*n+2) = a(2*n) + 2*a(2*n+1).
G.f.: 2*(1 - x)*(2 + 3*x - x^2)/(1 - 4*x^2 + x^4).
E.g.f.: (1 + exp(-sqrt(6)*x))*((2 - sqrt(2))*exp(sqrt(2 - sqrt(3))*x) + (2 + sqrt(2))*exp(sqrt(2 + sqrt(3))*x))/2.
Lim_{n->infinity} a(2*n+1)/a(2*n) = (1 + sqrt(3))/2.

A131039 Expansion of (1-x)(2x^2-4x+1)/(1-2x+5x^2-4x^3+x^4).

Original entry on oeis.org

1, -3, -5, 7, 26, 0, -97, -97, 265, 627, -362, -2702, -1351, 8733, 13775, -18817, -70226, 0, 262087, 262087, -716035, -1694157, 978122, 7300802, 3650401, -23596563, -37220045, 50843527, 189750626, 0, -708158977, -708158977, 1934726305, 4577611587, -2642885282, -19726764302, -9863382151

Offset: 0

Creighton Dement, Jun 11 2007

Unsigned bisection gives match to A002316 (Related to Bernoulli numbers). Note that all numbers in A002316 appear to be in A002531 (Numerators of continued fraction convergents to sqrt(3)) as well. a(12*n+5) = (0,0,0,0,...)

Floretion Algebra Multiplication Program, FAMP Code: 2tesseq['i + .5i' + .5j' + .5k' + .5e]

Robert Israel, Table of n, a(n) for n = 0..3492
Index entries for linear recurrences with constant coefficients, signature (2,-5,4,-1).

Cf. A131040, A131041, A131042, A002316, A002531.

f:= gfun:-rectoproc({a(0)=1, a(1)=-3, a(2)=-5, a(3)=7, a(n)=2*a(n-1)-5*a(n-2)+4*a(n-3)-a(n-4)},a(n),remember):
map(f, [$0..100]); # Robert Israel, Dec 25 2016

CoefficientList[Series[(1-x)(2x^2-4x+1)/(1-2x+5x^2-4x^3+x^4),{x, 0, 50}], x] (* or *) LinearRecurrence[{2,-5,4,-1},{1,-3,-5,7},50] (* Harvey P. Dale, Aug 31 2011 *)

a(0)=1, a(1)=-3, a(2)=-5, a(3)=7, a(n)=2*a(n-1)-5*a(n-2)+4*a(n-3)-a(n-4) [Harvey P. Dale, Aug 31 2011]

A131040 a(n) = (1/2+1/2isqrt(11))^n + (1/2-1/2isqrt(11))^n, where i=sqrt(-1).

Original entry on oeis.org

1, -5, -8, 7, 31, 10, -83, -113, 136, 475, 67, -1358, -1559, 2515, 7192, -353, -21929, -20870, 44917, 107527, -27224, -349805, -268133, 781282, 1585681, -758165, -5515208, -3240713, 13304911, 23027050, -16887683, -85968833, -35305784

Offset: 0

Creighton Dement, Jun 11 2007

Generating floretion is 1.5i' + .5j' + .5k' + .5e whereas in A131039 it is 'i + .5i' + .5j' + .5k' + .5e

Essentially the Lucas sequence V(1,3). - Peter Bala, Jun 23 2015

Wikipedia, Lucas sequence

Cf. A131039, A131041, A131042, A002316, A002531.

Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[ 1.5i' + .5j' + .5k' + .5e]

[lucas_number2(n,1,3) for n in range(1, 34)] # Zerinvary Lajos, May 14 2009

a(n) = a(n-1) - 3*a(n-2); G.f. (1 - 6*x)/(1 - x + 3*x^2).

a(n) = [x^n] ( (1 + x + sqrt(1 + 2*x - 11*x^2))/2 )^n. - Peter Bala, Jun 23 2015

A131041 a(n) = 2*a(n-1) - a(n-2) - a(n-4).

Original entry on oeis.org

1, 1, 1, -1, -4, -8, -13, -17, -17, -9, 12, 50, 105, 169, 221, 223, 120, -152, -645, -1361, -2197, -2881, -2920, -1598, 1921, 8321, 17641, 28559, 37556, 38232, 21267, -24257, -107337, -228649, -371228, -489550, -500535, -282871, 306021

Offset: 0

Creighton Dement, Jun 11 2007

Generating floretion is .5i' + .5j' + .5k' + .5e + 'ii' (for A131039 it is 'i + .5i' + .5j' + .5k' + .5e and for A131040 it is 1.5i' + .5j' + .5k' + .5e)

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, -1, 0, -1).

Cf. A131039, A131040, A131042, A002316, A002531.

Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[.5i' + .5j' + .5k' + .5e + 'ii']

LinearRecurrence[{2,-1,0,-1},{1,1,1,-1},40] (* Harvey P. Dale, Oct 14 2012 *)

G.f. (1-x-2*x^3)/(1-2*x+x^2+x^4)

A199710 Expansion of (1+x-14x^2+13x^3)/(1-28x^2+169x^4).

Original entry on oeis.org

1, 1, 14, 41, 223, 979, 3878, 20483, 70897, 408073, 1329734, 7964417, 25250959, 154039339, 482301806, 2967115019, 9237038497, 57046572241, 177128072702, 1095861584537, 3398526529663, 21043253658307, 65224098543926, 404010494645843, 1251923775716881

Offset: 0

Bruno Berselli, Nov 10 2011

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,28,0,-169).

Cf. A002531, A083332.

I:=[1,1,14,41]; [n le 4 select I[n] else 28*Self(n-2)-169*Self(n-4): n in [1..25]];

LinearRecurrence[{0, 28, 0, -169}, {1, 1, 14, 41}, 25]
CoefficientList[Series[(1+x-14x^2+13x^3)/(1-28x^2+169x^4),{x,0,30}],x] (* Harvey P. Dale, Nov 08 2017 *)

makelist(expand(((1+3*sqrt(3))^n+(1-3*sqrt(3))^n)/(2*2^floor(n/2))),n,0,24);

Vec((1+x-14*x^2+13*x^3)/(1-28*x^2+169*x^4)+O(x^25))

G.f.: (1+x-14*x^2+13*x^3)/(1-28*x^2+169*x^4).
a(n) = ((1+3*sqrt(3))^n+(1-3*sqrt(3))^n)/(2*2^floor(n/2)).
a(n) = 28*a(n-2)-169*a(n-4).

A238799 a(0) = 1, a(n+1) = 2a(n)^3 + 3a(n).

Original entry on oeis.org

1, 5, 265, 37220045, 103124220135120334842385, 2193370648451279691104497113491599222165108730278225579497595691360405

Offset: 0

Arkadiusz Wesolowski, Mar 05 2014

a(6) has 209 digits and is too large to include.
Except for the first term, this is a subsequence of A175180.
The squares larger than 1 are in A076445.
If we define u(0) = 1 , u(n+1) = (u(n)/3)*(u(n)^2+9) / (u(n)^2 + 1), then u(n) = a(n) / A378683(n) ; this is Halley's method to calculate sqrt(3). - Robert FERREOL, Dec 21 2024

Wikipedia, Halley's method.

Cf. A175180, A378683.

RecurrenceTable[{a[0] == 1, a[n] == 2*a[n - 1]^3 + 3*a[n - 1]}, a[n], {n, 5}]
NestList[2#^3+3#&,1,5] (* Harvey P. Dale, Mar 22 2023 *)

a=1; print1(a, ", "); for(n=1, 5, b=2*a^3+3*a; print1(b, ", "); a=b);

{ A238799(n) = my(q=Mod(x,x^2-3)); lift( (1+q)*(2+q)^((3^n-1)/2) + (1-q)*(2-q)^((3^n-1)/2) )/2; } \\ Max Alekseyev, Sep 04 2018

a(n) = sqrt(2) * sinh( 3^n * arcsinh(1/sqrt(2)) ) = (1+sqrt(3))/2 * (2+sqrt(3))^((3^n-1)/2) + (1-sqrt(3))/2 * (2-sqrt(3))^((3^n-1)/2). - Max Alekseyev, Sep 04 2018

a(n) = ((1 + sqrt(3))^(3^n) + (1 - sqrt(3))^(3^n))/2^((3^n+1)/2) = A002531(3^n) = A080040(3^n)/2^((3^n+1)/2). - Robert FERREOL, Nov 19 2024

A083336 a(n) = 4*a(n-2) - a(n-4).

Original entry on oeis.org

3, 3, 9, 12, 33, 45, 123, 168, 459, 627, 1713, 2340, 6393, 8733, 23859, 32592, 89043, 121635, 332313, 453948, 1240209, 1694157, 4628523, 6322680, 17273883, 23596563, 64467009, 88063572, 240594153, 328657725, 897909603, 1226567328, 3351044259, 4577611587, 12506267433

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 26 2003

a(n)/A002531(n+1) converges to sqrt(3).

Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

CoefficientList[Series[(3+3x-3x^2)/(1-4x^2+x^4), {x, 0, 30}], x]
Transpose[NestList[Flatten[{Rest[#],4#[[3]]-First[#]}]&, {3,3,9,12}, 50]][[1]]  (* Harvey P. Dale, Mar 26 2011 *)
LinearRecurrence[{0, 4, 0, -1}, {3, 3, 9, 12}, 30] (* T. D. Noe, Mar 26 2011 *)

a(2*n) = A082841(n) = a(2*n-1) + 3*A002531(2*n).
a(2*n+1) = (a(2*n) + 3*A002531(2*n+1)) / 2.
G.f.: (3 + 3*x - 3*x^2) / (1 - 4*x^2 + x^4).

A096146 Prime numerators of the rational convergents to sqrt(3).

Original entry on oeis.org

2, 5, 7, 19, 71, 97, 3691, 191861, 138907099, 708158977, 26947261171

Offset: 1

Cino Hilliard, Jul 24 2004

nonn

Next term is too large to include.

This is the prime subsequence of A002531. See also A086386 for numerators where both numerator and denominator are primes. - Ray Chandler, Aug 01 2004

Amiram Eldar, Table of n, a(n) for n = 1..18

Cf. A002531, A082630, A086386, A096147.

Select[Numerator[Convergents[Sqrt[3],200]],PrimeQ] (* Harvey P. Dale, Nov 08 2022 *)

\\ Continued fraction rational approximation of numeric constants f. m=steps.
cfracnumprime(m,f) = { default(realprecision,3000); cf = vector(m+10); x=f; for(n=0,m, i=floor(x); x=1/(x-i); cf[n+1] = i; ); for(m1=0,m, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n]; ); numer=numerator(r); denom=denominator(r); if(ispseudoprime(numer),print1(numer,", ")); ) }

Offset corrected by Amiram Eldar, Jul 11 2024

A259592 Denominators of the other-side convergents to sqrt(3).

Original entry on oeis.org

1, 2, 4, 7, 15, 26, 56, 97, 209, 362, 780, 1351, 2911, 5042, 10864, 18817, 40545, 70226, 151316, 262087, 564719, 978122, 2107560, 3650401, 7865521, 13623482, 29354524, 50843527, 109552575, 189750626, 408855776, 708158977, 1525870529, 2642885282, 5694626340

Offset: 0

Clark Kimberling, Jul 20 2015

Suppose that a positive irrational number r has continued fraction [a(0), a(1), ...]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:
   p(i)/q(i) = [a(0), a(1), ... a(i)] and
   P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1].
The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0.
Closeness of P(i)/Q(i) to r is indicated by |r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.

			For r = sqrt(3), the first 7 other-side convergents are 4, 25/8, 355/113, 688/219, 104348/33215, 208341/66317, 312689/99532.  A comparison of convergents with other-side convergents:
i   p(i)/q(i)            P(i)/Q(i)   p(i)*Q(i) - P(i)*q(i)
0      1/1  < sqrt(3) <     2/1               -1
1      2/1  > sqrt(3) >     3/2                1
2      5/3  < sqrt(3) <     7/4               -1
3      7/4  > sqrt(3) >    12/7                1
4     19/11 < sqrt(3) <    26/15              -1
5     26/15 > sqrt(3) >    45/26               1

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A002530, A002531, A259593 (numerators).

Cf. A001353 (even bisection), A001075 (odd bisection).

r = Sqrt[3]; a[i_] := Take[ContinuedFraction[r, 35], i];
b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
u = Denominator[t]

Vec(-(x+1)*(x^2-x-1)/(x^4-4*x^2+1) + O(x^50)) \\ Colin Barker, Jul 21 2015

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = Q(i).
a(n) = 4*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015
G.f.: -(x+1)*(x^2-x-1) / (x^4-4*x^2+1). - Colin Barker, Jul 21 2015
a(2n) = A001353(n+1); a(2n+1) = A001075(n+1). - Antonio Alberto Olivares, Jul 23 2021
a(n) = 3^(n/2 + 1/2 - t)*((2 + sqrt(3))^t - (-1)^n*(2 - sqrt(3))^t)/2, where t = floor(n/2) + 1. - Ridouane Oudra, Aug 03 2021

A259593 Numerators of the other-side convergents to sqrt(3).

Original entry on oeis.org

2, 3, 7, 12, 26, 45, 97, 168, 362, 627, 1351, 2340, 5042, 8733, 18817, 32592, 70226, 121635, 262087, 453948, 978122, 1694157, 3650401, 6322680, 13623482, 23596563, 50843527, 88063572, 189750626, 328657725, 708158977, 1226567328, 2642885282, 4577611587

Offset: 0

Clark Kimberling, Jul 20 2015

Suppose that a positive irrational number r has continued fraction [a(0), a(1), ...]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:
   p(i)/q(i) = [a(0), a(1), ..., a(i)] and
   P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1].
The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0.
Closeness of P(i)/Q(i) to r is indicated by |r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.

			For r = sqrt(3), the first 7 other-side convergents are 4, 25/8, 355/113, 688/219, 104348/33215, 208341/66317, 312689/99532. A comparison of convergents with other-side convergents:
i   p(i)/q(i)            P(i)/Q(i)   p(i)*Q(i) - P(i)*q(i)
0      1/1  < sqrt(3) <     2/1               -1
1      2/1  > sqrt(3) >     3/2                1
2      5/3  < sqrt(3) <     7/4               -1
3      7/4  > sqrt(3) >    12/7                1
4     19/11 < sqrt(3) <    26/15              -1
5     26/15 > sqrt(3) >    45/26               1

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-1).

Cf. A002530, A002531, A259592 (denominators).

Cf. A001075 (even bisection), A005320 (odd bisection).

r = Sqrt[3]; a[i_] := Take[ContinuedFraction[r, 35], i];
b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
v = Numerator[t]

Vec(-(x^2-3*x-2)/(x^4-4*x^2+1) + O(x^50)) \\ Colin Barker, Jul 21 2015

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = P(i).
a(n) = 4*a(n-2) - a(n-4) for n>3. - Colin Barker, Jul 21 2015
G.f.: -(x^2-3*x-2) / (x^4-4*x^2+1). - Colin Barker, Jul 21 2015
a(n) = 3^(n/2 - t + 1)*((2 + sqrt(3))^t + (-1)^n*(2 - sqrt(3))^t)/2, where t = floor(n/2) + 1. - Ridouane Oudra, Aug 03 2021

cp's OEIS Frontend

A321119 a(n) = ((1 - sqrt(3))^n + (1 + sqrt(3))^n)/2^floor((n - 1)/2); n-th row common denominator of A321118.

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A131039 Expansion of (1-x)*(2*x^2-4*x+1)/(1-2*x+5*x^2-4*x^3+x^4).

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A131040 a(n) = (1/2+1/2*i*sqrt(11))^n + (1/2-1/2*i*sqrt(11))^n, where i=sqrt(-1).

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A131041 a(n) = 2*a(n-1) - a(n-2) - a(n-4).

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A199710 Expansion of (1+x-14*x^2+13*x^3)/(1-28*x^2+169*x^4).

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A238799 a(0) = 1, a(n+1) = 2*a(n)^3 + 3*a(n).

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A083336 a(n) = 4*a(n-2) - a(n-4).

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A096146 Prime numerators of the rational convergents to sqrt(3).

A131039 Expansion of (1-x)(2x^2-4x+1)/(1-2x+5x^2-4x^3+x^4).

A131040 a(n) = (1/2+1/2isqrt(11))^n + (1/2-1/2isqrt(11))^n, where i=sqrt(-1).

A199710 Expansion of (1+x-14x^2+13x^3)/(1-28x^2+169x^4).

A238799 a(0) = 1, a(n+1) = 2a(n)^3 + 3a(n).