A265288 -id:A265288 - cp's OEIS frontend

A153386 Decimal expansion of Sum_{n>=1} 1/Fibonacci(2*n).

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

Offset: 1

Eric W. Weisstein, Dec 25 2008

			1.535370508836252985029852896651599006367...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.14.1, p. 358.

Richard André-Jeannin, Problem B-745, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 31, No. 3 (1993), p. 277; Fun with Unit Fractions, Solution to Problem B-745 by Paul S. Bruckman, ibid., Vol. 33, No. 1 (1995), p. 86.
A. F. Horadam, Elliptic Functions and Lambert Series in the Summation of Reciprocals in Certain Recurrence-Generated Sequences, The Fibonacci Quarterly, Vol. 26, No. 2 (1988), pp. 98-114.

Cf. A000045, A001906 (Fibonacci(2*n)), A065563.

Cf. A000796, A001113, A002163, A002390, A016628, A079586, A153387, A256178, A265288, A360928.

rd[k_] := rd[k] = RealDigits[ N[ Sum[ 1/Fibonacci[2*n], {n, 1, 2^k}], 105]][[1]]; rd[k = 4]; While[ rd[k] != rd[k - 1], k++]; rd[k] (* Jean-François Alcover, Oct 29 2012 *)
RealDigits[Sqrt[5] * (Log[5] + 2*QPolyGamma[0, 1, 1/GoldenRatio^4] - 4*QPolyGamma[0, 1, 1/GoldenRatio^2]) / (8*ArcCsch[2]), 10, 105][[1]] (* Vaclav Kotesovec, Feb 26 2023 *)

sumpos(n=1, 1/fibonacci(2*n)) \\ Michel Marcus, Sep 04 2021

Equals sqrt(5) * (L((3-sqrt(5))/2) - L((7-3*sqrt(5))/2)), where L(x) = Sum_{k>=1} x^k/(1-x^k) (Horadam, 1988, equation (4.6)). - Amiram Eldar, Oct 04 2020
From Gleb Koloskov, Sep 04 2021: (Start)
Equals 1/2 + (sqrt(5)/log(phi))*(log(5)/8 + 3*Integral_{x=0..infinity} sin(x)/((4*sin(x)^2+5)*(exp(Pi*x/log(phi))-1)) dx), where phi = (1+sqrt(5))/2 = A001622.
Equals 1/2 + (A002163/A002390)*(A016628/8 + 3*Integral_{x=0..infinity} sin(x)/((4*sin(x)^2+5)*(A001113^(A000796*x/A002390)-1)) dx). (End)
Equals 1 + Sum_{n>=1} 1/A065563(2*n-1) (André-Jeannin, 1993). - Amiram Eldar, Jan 15 2022
From Peter Bala, Aug 17 2022: (Start)
Equals 5/3 - 3*Sum_{n >= 1} 1/(F(2*n)*F(2*n+2)*F(2*n+4)), where F(n) = Fibonacci(n).
Conjecture: Equals 151/96 - 6*Sum_{n >= 1} 1/(F(2*n)*F(2*n+4)*F(2*n+6)). (End)
Equals A360928 * sqrt(5). - Kevin Ryde, Feb 27 2023

A256178 Expansion of exp( Sum_{n >= 1} L(2n)L(4n)x^n/n ), where L(n) = A000032(n) is a Lucas number.

Original entry on oeis.org

1, 21, 385, 6930, 124410, 2232594, 40062659, 718896255, 12900072515, 231482415780, 4153783429236, 74536619356836, 1337505365115205, 24000559953034665, 430672573790340805, 7728105768275278134, 138675231255170368494

Offset: 0

Peter Bala, Mar 18 2015

Let L(n) = A000032(n) denote the n-th Lucas number.
For a fixed positive integer k, the power series expansion of exp( Sum_{n >= 1} L(k*n)x^n/n ) has integer coefficients given by the formula F(k*n)/F(k), where F(n) = A000045(n) [Johnson, 2.22].
The power series expansion of exp( Sum_{n >= 1} L(k*n)*L(2*k*n) *x^n/n ) has integer coefficients given by ( F(k*(n + 1))*F(k*(n + 2))*F(k*(n + 3)) )/( F(k)*F(2*k)*F(3*k) )
The present sequence is the particular case k = 2. See A001655 for the case k = 1.

B. Johnson, Fibonacci Identities by Matrix Methods and Generalisation to Related Sequences
Eric Weisstein's World of Mathematics, Fibonacci Number
Index entries for linear recurrences with constant coefficients, signature (21,-56,21,-1).

Cf. A000032, A000045, A001655, A010048, A153386, A265288.

seq((1/24)*fibonacci(2*n+2)*fibonacci(2*n+4)*fibonacci(2*n+6), n = 0 .. 16);

Table[1/8 * Sum[Fibonacci[2*k + 2]*Fibonacci[6*n - 6*k + 6], {k, 0, n}], {n, 0, 17}] (* or *) RecurrenceTable[{a[n] == 21*a[n - 1] - 56*a[n - 2] + 21*a[n - 3] - a[n - 4], a[1] == 1, a[2] == 21, a[3] == 385, a[4] == 6930}, a, {n, 17}] (* Michael De Vlieger, Mar 18 2015 *)

a(n) = ( F(2*n + 2)*F(2*n + 4)*F(2*n + 6) )/( F(2)*F(4)*F(6) ).
a(n) = (1/8) * Sum_{k = 0..n} F(2*k + 2)*F(6*n - 6*k + 6).
O.g.f.: 1/( (1 - 3*x + x^2)*(1 - 18*x + x^2) ) = 1/8 * Sum_{n >= 0} F(2*n + 2)*x^n * Sum_{n >= 0} F(6*n + 6)*x^n.
O.g.f. also equals exp( Sum_{n >= 1} trace( M^(2*n) + M^(6*n) )*x^n/n ), where M is the 2X2 matrix [ 1, 1; 1, 0 ].
Recurrences: a(n) = 21*a(n-1) - 56*a(n-2) + 21*a(n-3) - a(n-4).
Also a(0) = 1 and for n >= 1, a(n) = (1/n)*Sum_{k = 1..n} L(2*k)*L(4*k)*a(n-k).
From Peter Bala, Aug 19 2022: (Start)
Sum_{n >= 0} 1/a(n) = 40/3 - 8*Sum_{n >= 1} 1/F(2*n) = 40/3 - 8*A153386.
Sum_{n >= 0} (-1)^n/a(n) = - 88/3 + 40*Sum_{n >= 1} (-1)^(n+1)/F(2*n). Cf. A265288. (End)

A265290 Decimal expansion of Sum_{n>=1} |phi - c(n)|, where phi is the golden ratio (A001622) and c(n) are the convergents to phi.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 06 2015

Define the deviance of x > 0 by dev(x) = Sum_{n>=1} |x - c(n,x)|, where c(n,x) = n-th convergent to x. The greatest value of dev(x) occurs when x = golden ratio, so that this constant is the maximal deviance.

			1.195955786017513596003474800021...

Cf. A000045, A001622, A290565.

Cf. A265288, A265289.

x := (3 - sqrt(5))/2:
evalf(sqrt(5)*add(x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), n = 1..16), 100); # Peter Bala, Aug 21 2022

x = GoldenRatio; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265288 *)
RealDigits[s2, 10, 120][[1]]  (* A265289 *)
RealDigits[s1 + s2, 10, 120][[1]] (* A265290, dev(x) *)
d[x_] := If[IntegerQ[1000!*x], Total[Abs[x - Convergents[x]]],
  Total[Abs[x - Convergents[x, 30]]]]
Plot[{d[x], 1.195}, {x, 0, 1}]

Equals Sum_{n>=1} 1/(F(2*n-1)*F(2*n)), where F(n) is the n-th Fibonacci number (A000045).
From Amiram Eldar, Oct 05 2020: (Start)
Equals Sum_{k>=1} 1/(phi^k * F(k)).
Equals sqrt(5) * Sum_{k>=1} 1/(phi^(2*k) - (-1)^k) = sqrt(5) * Sum_{k>=1} (-1)^(k+1)/(phi^(2*k) + (-1)^k).
Equals (A290565 + 1/phi)/2. (End)
A rapidly converging series for the constant is sqrt(5)*Sum_{k >= 1} x^(k^2)*(1 + x^(2*k))/(1 - x^(2*k)), where x = (3 - sqrt(5))/2. See A112329. - Peter Bala, Aug 21 2022

A265291 Decimal expansion of Sum_{n >= 1} (x - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(2).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 06 2015

			sum = 0.4288603380680959830021113676132723...

Cf. A000129, A001227, A002193, A265292, A265293, A265288 (guide).

x := 2*sqrt(2) - 3:
evalf(2*sqrt(2)*add( x^(n*(n+1)/2)/(x^n - 1), n = 1..16), 100); # Peter Bala, Aug 21 2022

x = Sqrt[2]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265291 *)
RealDigits[s2, 10, 120][[1]]  (* A265292 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265293 *)

From Peter Bala, Aug 20 2022: (Start)
Equals Sum_{n >= 1} (-1)^(n+1)/Pell(2*n), where Pell(n) = A000129(n).
Equals 2*sqrt(2)*Sum_{n >= 1} x^(n*(n+1)/2)/(x^n - 1), where x = 2^sqrt(2) - 3. (End)

A265292 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(2).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 06 2015

			sum = 0.0883138821525759032178529847253...

Cf. A000005, A000129, A002193, A265291, A265293, A265288 (guide).

x := 17 - 12*sqrt(2) :
evalf(2*sqrt(2)*add( x^(n^2)*(1 + x^n)/(1 - x^n), n = 1..8), 100); # Peter Bala, Aug 23 2022

x = Sqrt[2]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265291 *)
RealDigits[s2, 10, 120][[1]]  (* A265292 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265293 *)

From Peter Bala, Aug 23 2022: (Start)
Equals Sum_{n >= 1} 1/( (1 + sqrt(2))^(2*n)*Pell(2*n) ), where Pell(n) = A000129(n).
Equals Sum_{n >= 1} 1/( (1 + sqrt(2))^(4*n) - 1).
A more rapidly converging series for the constant is 2*sqrt(2)*Sum_{n >= 1} x^(n^2)*(1 + x^n)/(1 - x^n), where x = 17 - 12*sqrt(2) = 0.029437.... See A000005. (End)

A265293 Decimal expansion of Sum_{n >= 1} (c(2n) - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(2).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 06 2015

			sum = 0.51717422022067188621996435233866923610552...

Cf. A000129, A002193, A112329, A265291, A265292, A265288 (guide).

x := 3 - 2*sqrt(2):
evalf(2*sqrt(2)*add(x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), n = 1..12), 100); # Peter Bala, Aug 20 2022

x = Sqrt[2]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265291 *)
RealDigits[s2, 10, 120][[1]]  (* A265292 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265293 *)

From Peter Bala, Aug 20 2022: (Start)
Constant equals Sum_{n >= 1} 1/((1 + sqrt(2))^n*Pell(n)) =  2*sqrt(2)*Sum_{n >= 1} 1/( (3 + 2*sqrt(2))^n - (-1)^n ), where Pell(n) = A000129(n).
A more rapidly converging series for the constant is 2*sqrt(2)*Sum_{n >= 1} x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), where x = 3 - 2*sqrt(2). See A112329. (End)

A265294 Decimal expansion of Sum_{n>=1} (x - c(2n-1)), where c = convergents to (x = sqrt(3)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 07 2015

			sum = 0.8025830908035148343778741812630...

Cf. A002194, A002530, A002531, A265295, A265296, A265288 (guide).

x := sqrt(3) - 2:
evalf(2*sqrt(3)*add( x^(n*(n+1)/2)/(x^n - 1), n = 1..18), 100); # Peter Bala, Aug 24 2022

x = Sqrt[3]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265294 *)
RealDigits[s2, 10, 120][[1]]  (* A265295 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)

From Peter Bala, Aug 24 2022: (Start)
Equals 2*sqrt(3)*Sum_{n >= 1} 1/( 1 + (2+sqrt(3))^(2*n-1) ).
A more rapidly converging series for the constant is 2*sqrt(3)*Sum_{n >= 1} x^(n*(n+1)/2)/(x^n - 1), where x = sqrt(3) - 2. See A001227. (End)

A265295 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(3).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 07 2015

			sum = 0.28728008008348839351145153987668331682390...

Cf. A002194, A002530, A002531, A265294, A265296, A265288 (guide).

x := 7 - 4*sqrt(3):
evalf(2*sqrt(3)*add( x^(n^2)*(1 + x^n)/(1 - x^n), n = 1..10), 100); # Peter Bala, Aug 24 2022

x = Sqrt[3]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265294 *)
RealDigits[s2, 10, 120][[1]]  (* A265295 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)

Equals 2*sqrt(3)*Sum_{n >= 1} x^(n^2)*(1 + x^n)/(1 - x^n), where x = 7 - 4*sqrt(3). - Peter Bala, Aug 24 2022

A265296 Decimal expansion of Sum_{n >= 1} (c(2n) - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(3).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Dec 07 2015

			sum = 1.0898631708870032278893257211397258128825141977596999...

Cf. A002194, A002530, A002531, A265294, A265295, A265288 (guide).

x := 2 - sqrt(3):
evalf(2*sqrt(3)*add(x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), n = 1..13), 100); # Peter Bala, Aug 24 2022

x = Sqrt[3]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265294 *)
RealDigits[s2, 10, 120][[1]]  (* A265295 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)

Equals 2*sqrt(3)*Sum_{n >= 1} x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), where x = 2 - sqrt(3). - Peter Bala, Aug 24 2022

A265297 Decimal expansion of sum{x - c(2n-1), n=1,2,...}, where c = convergents to (x = sqrt(5)).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Dec 07 2015

			sum = 0.236844248570148187595380178229901194760980420...

Cf. A002163, A265298, A265299, A265288 (guide).

x = Sqrt[5]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265297 *)
RealDigits[s2, 10, 120][[1]]  (* A265298 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265299 *)

cp's OEIS Frontend

A153386 Decimal expansion of Sum_{n>=1} 1/Fibonacci(2*n).

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A256178 Expansion of exp( Sum_{n >= 1} L(2*n)*L(4*n)*x^n/n ), where L(n) = A000032(n) is a Lucas number.

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A265290 Decimal expansion of Sum_{n>=1} |phi - c(n)|, where phi is the golden ratio (A001622) and c(n) are the convergents to phi.

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A265291 Decimal expansion of Sum_{n >= 1} (x - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(2).

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A265292 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(2).

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A265293 Decimal expansion of Sum_{n >= 1} (c(2*n) - c(2*n-1)), where c(n) = the n-th convergent to x = sqrt(2).

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A265294 Decimal expansion of Sum_{n>=1} (x - c(2n-1)), where c = convergents to (x = sqrt(3)).

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A265295 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(3).

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A256178 Expansion of exp( Sum_{n >= 1} L(2n)L(4n)x^n/n ), where L(n) = A000032(n) is a Lucas number.

A265293 Decimal expansion of Sum_{n >= 1} (c(2n) - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(2).

A265296 Decimal expansion of Sum_{n >= 1} (c(2n) - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(3).