A093954 -id:A093954 - cp's OEIS frontend

A275496 a(n) = n^2(2n^2 + (-1)^n).

0, 1, 36, 153, 528, 1225, 2628, 4753, 8256, 13041, 20100, 29161, 41616, 56953, 77028, 101025, 131328, 166753, 210276, 260281, 320400, 388521, 468996, 559153, 664128, 780625, 914628, 1062153, 1230096, 1413721, 1620900, 1846081, 2098176, 2370753, 2673828

Offset: 0

Daniel Poveda Parrilla, Jul 30 2016

All terms of this sequence are triangular numbers. Graphically, for each term of the sequence, one corner of the square of squares (4th power) will be part of the corresponding triangle's hypotenuse if the term is an odd number. Otherwise, it will not be part of it.
a(A000129(n)) is a square triangular number.
a(2^((A000043(n) - 1)/2)) - 2^A000043(n) is a perfect number.

			a(5) = 5^4 + Sum_{k=0..(5^2 - (5 mod 2))} 2k = 625 + Sum_{k=0..(25 - 1)} 2k = 625 + 600 = 1225.
a(12) = 12^4 + Sum_{k=0..(12^2 - (12 mod 2))} 2k = 20736 + Sum_{k=0..(144 - 0)} 2k = 20736 + 20880 = 41616.

Colin Barker and Daniel Poveda Parrilla, Table of n, a(n) for n = 0..46340 [n = 1 through 1000 by Colin Barker, Aug 02 2016; and n=1001 to 46340 by Daniel Poveda Parrilla, Aug 04 2016]
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A000040, A000129, A000217, A001110, A077221, A093954, A275543.

Table[n^2 ((-1)^n + 2 n^2), {n, 0, 34}] (* or *)
CoefficientList[Series[x (1 + 34 x + 79 x^2 + 156 x^3 + 79 x^4 + 34 x^5 +
x^6)/((1 - x)^5 (1 + x)^3), {x, 0, 34}], x] (* Michael De Vlieger, Aug 01 2016 *)
LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{0,1,36,153,528,1225,2628,4753},40] (* Harvey P. Dale, Sep 10 2016 *)

a(n)=n=n^2; if(n%2,2*n-1,2*n+1)*n \\ Charles R Greathouse IV, Jul 30 2016

concat(0, Vec(x*(1+34*x+79*x^2+156*x^3+79*x^4+34*x^5+x^6)/((1-x)^5*(1+x)^3) + O(x^100))) \\ Colin Barker, Aug 01 2016

a(n) = n^4 + Sum_{k=0..(n^2 - (n mod 2))} 2k.
a(n) = A275543(n)*(n^2).
From Colin Barker, Aug 01 2016 and Aug 04 2016: (Start)
a(n) = n^2*(2*n^2 + (-1)^n).
a(n) = 2*n^4 + n^2 for n even.
a(n) = 2*n^4 - n^2 for n odd.
G.f.: x*(1 +34*x +79*x^2 +156*x^3 +79*x^4 +34*x^5 +x^6) / ((1-x)^5*(1+x)^3).
(End)
a(n) = n^2*A000217(2n-1) + 2n*A000217(n-(n mod 2)) for n > 0.
E.g.f.: x*(2*(1 + 7*x + 6*x^2 + x^3)*exp(x) - exp(-x)). - G. C. Greubel, Aug 05 2016
a(n) = A000217(A077221(n)).
a(n) = (A001844(A077221(n)) - 1)/4.
Sum_{n>=1} 1/a(n) = 1 - Pi^2/12 + (tan(c) - coth(c))*c, where c = Pi/(2*sqrt(2)) is A093954. - Amiram Eldar, Aug 21 2022

New name from Colin Barker, Aug 04 2016

A326919 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Nov 19 2019

Let Chi() be a primitive character modulo d, the so-called Dirichlet L-series L(s,Chi) is the analytic continuation (see the functional equations involving L(s,Chi) in the MathWorld link entitled Dirichlet L-Series) of the sum Sum_{k>=1} Chi(k)/k^s, Re(s)>0 (if d = 1, the sum converges requires Re(s)>1).
If s != 1, we can represent L(s,Chi) in terms of the Hurwitz zeta function by L(s,Chi) = (Sum_{k=1..d} Chi(k)*zeta(s,k/d))/d^s.
L(s,Chi) can also be represented in terms of the polylog function by L(s,Chi) = (Sum_{k=1..d} Chi'(k)*polylog(s,u^k))/(Sum_{k=1..d} Chi'(k)*u^k), where Chi' is the complex conjugate of Chi, u is any primitive d-th root of unity.
If m is a positive integer, we have L(m,Chi) = (Sum_{k=1..d} Chi(k)*polygamma(m-1,k/d))/((-d)^m*(m-1)!).
In this sequence we have Chi = A175629 and s = 1.

			1 + 1/2 - 1/3 + 1/4 - 1/5 - 1/6 + 1/8 + 1/9 - 1/10 + 1/11 - 1/12 - 1/13 + ... = Pi/sqrt(7) = 1.1874104117...

Sergio A. Carrillo, Where did the examples of Abel's continuity theorem go?, arXiv:2010.10290 [math.HO], 2020. See p. 11.
Eric Weisstein's World of Mathematics, Class Number.
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
Eric Weisstein's World of Mathematics, Polygamma Function.

Cf. A175629.
Decimal expansion of Sum_{k>=1} Kronecker(d,k)/k, where d is a fundamental discriminant: A093954 (d=-8), this sequence (d=-7), A003881 (d=-4), A073010 (d=-3), A086466 (d=5), A196525 (d=8), A196530 (d=12).
Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k^s: this sequence (s=1), A103133 (s=2), A327135 (s=3).

RealDigits[Pi/Sqrt[7], 10, 102] // First

PARI
```
default(realprecision, 100); Pi/sqrt(7)
```

Equals Pi/sqrt(7). This is related to the class number formula: if d<0 is the fundamental discriminant of an imaginary quadratic number field, Chi(k) = Kronecker(d,k), then L(1,Chi) = Sum_{k>=1} Kronecker(d,k)/k = 2*Pi*h(d)/(sqrt(|d|)*w(d)), where h(d) is the class number of K = Q[sqrt(d)], w(d) is the number of elements in K whose norms are 1 (w(d) = 6 if d = -3, 4 if d = -4 and 2 if d < -4). Here d = -7, h(d) = 1, w(d) = 2.
Equals (polylog(1,u) + polylog(1,u^2) - polylog(1,u^3) + polylog(1,u^4) - polylog(1,u^5) - polylog(1,u^6))/sqrt(-7), where u = exp(2*Pi*i/7) is a 7th primitive root of unity, i = sqrt(-1).
Equals (polygamma(0,1/7) + polygamma(0,2/7) - polygamma(0,3/7) + polygamma(0,4/7) - polygamma(0,5/7) - polygamma(0,6/7))/49.
Equals 1/Product_{p prime} (1 - Kronecker(-7,p)/p), where Kronecker(-7,p) = 0 if p = 7, 1 if p == 1, 2 or 4 (mod 7) or -1 if p == 3, 5 or 6 (mod 7). - Amiram Eldar, Dec 17 2023

A346908 Decimal expansion of 2 - Pi / (2*sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 06 2021

			0.889279265460408438246029752484826575346344577656...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (326).

Index entries for transcendental numbers

Cf. A346907, A346850, A346849, A093954.

evalf(2-Pi/sqrt(8), 140);  # Alois P. Heinz, Aug 06 2021

RealDigits[2 - Pi / (2 * Sqrt[2]), 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

2 - Pi/(2*sqrt(2)) \\ Michel Marcus, Aug 06 2021

Equals 1 + Sum_{k>=1} ( (-1)^k/(4*k-1) - (-1)^k/(4*k+1) ).

A384509 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = ((5/n+1) + (5/n-1)i)/(nsqrt(2)) + 1/4 + (1/2)*i to reach |z| > 2, starting with z = 0.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 51, 51, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 71, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Luke Bennet, May 31 2025

nonn

a(n)/n seems to converge to Pi/(2*sqrt(2)).

a(n) counts the escape time of points outside the Mandelbrot set that converge to the Mandelbrot set's 1/4 period bulb.

Luke Bennet, Table of n, a(n) for n = 1..10001
Thies Brockmöller, Oscar Scherz, and Nedim Srkalović, Pi in the Mandelbrot set everywhere, arXiv preprint arXiv:2505.07138 [math.DS], 2025.
Aaron Klebanoff, Pi in the Mandelbrot Set, Fractals 9 (2001), nr. 4, p. 393-402.

Cf. A093954, A097486, A383750, A384513.

c(n) = ((5/n+1) + (5/n-1)*I)/(n*sqrt(2)) + 1/4 + (1/2)*I;
a(n) = my(z=0, k=0, c=c(n)); while(norml2(z)<=4, z = z^2 + c; k++); k; \\ Michel Marcus, Jun 01 2025

import mpmath
from mpmath import iv
def a(n):
    dps = 1
    while True:
        mpmath.iv.dps = dps
        c = iv.mpc(iv.mpf(5) / n + 1, iv.mpf(5) / n - 1)
        c = c / (n * iv.sqrt(2)) + 0.25 + 0.5j
        z = iv.mpc(0, 0)
        counter = 0
        while (z.real**2 + z.imag**2).b <= 4:
            z = z ** 2 + c
            counter += 1
        if (z.real**2 + z.imag**2).a > 4:
            return counter
        dps *= 2

A125096 Expansion of -1 + (phi(q) * phi(q^2) + phi(-q^2) * phi(q^4)) / 2 in powers of q.

Original entry on oeis.org

1, 0, 2, 2, 0, 0, 0, 2, 3, 0, 2, 4, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 4, 1, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 6, 0, 0, 0, 0, 2, 0, 2, 4, 0, 0, 0, 4, 1, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 6, 2, 0, 2, 4, 0, 0, 0, 0, 5, 0, 2, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 4, 2, 0, 6, 2, 0, 0, 0, 0, 0

Offset: 1

Michael Somos, Nov 20 2006

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002324, A002325, A033761, A093954, A112603.

f[p_, e_] := If[MemberQ[{1, 3}, Mod[p, 8]], e + 1, (1 + (-1)^e)/2]; f[2, e_] := If[e > 1, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 13 2022 *)

{a(n) = if( n<1, 0, qfrep([1, 0; 0, 8], n)[n] + qfrep([3, 1; 1, 3], n)[n])}

a(n) is multiplicative with a(2) = 0, a(2^e) = 2 if e>1, a(p^e) = e+1 if p == 1, 3 (mod 8), a(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8).
a(4*n + 2) = a(8*n + 5) = a(8*n + 7) = 0. a(4*n) = 2 * A002325(n). a(8*n + 1) = A112603(n). a(8*n + 3) = A033761(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(2)) = 1.110720... (A093954). - Amiram Eldar, Oct 13 2022

A161684 Continued fraction for Pi/(2*sqrt(2)).

Original entry on oeis.org

1, 9, 31, 1, 1, 17, 2, 3, 3, 2, 3, 1, 1, 2, 2, 1, 4, 9, 1, 3, 1, 1, 3, 2, 3, 3, 2, 10, 7, 1, 5, 1, 9, 1, 13, 1, 1, 1, 1, 1, 4, 3, 4, 8, 1, 3, 7, 1, 15, 1, 3, 1, 3, 5, 2, 1, 1, 5, 1, 1, 5, 1, 3, 3, 2, 33, 1, 4, 3, 111, 3, 1, 3, 4, 1, 5, 1, 5, 31, 1, 8, 1, 2, 2, 1, 1, 12, 1, 5, 3, 2, 1, 1, 1, 1, 147, 3, 2, 3, 8

Offset: 0

Harry J. Smith, Jun 17 2009

			1.11072073453959156175397024... = 1 + 1/(9 + 1/(31 + 1/(1 + 1/(1 + ...))))

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A093954 Decimal expansion.

ContinuedFraction[Pi/(2Sqrt[2]),100] (* Harvey P. Dale, Oct 22 2011 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi*sqrt(2)/4); for (n=0, 20000, write("b161684.txt", n, " ", x[n+1])); }

A206161 Decimal expansion of the Fresnel integral int_{x=0..infinity} cos(x^4) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

			0.83740669676908648308360272218083226...

Wikipedia, Fresnel Integral

Cf. A204067.

evalf(Pi*cos(Pi/8)/GAMMA(3/4)/2^(3/2)) ;

RealDigits[ Sqrt[2 + Sqrt[2]]*Gamma[1/4]/8, 10, 72] // First (* Jean-François Alcover, Feb 20 2013 *)

Equals A093954 * A144981 / A068465 .

A275543 A081585 and A069129 interleaved.

Original entry on oeis.org

1, 1, 9, 17, 33, 49, 73, 97, 129, 161, 201, 241, 289, 337, 393, 449, 513, 577, 649, 721, 801, 881, 969, 1057, 1153, 1249, 1353, 1457, 1569, 1681, 1801, 1921, 2049, 2177, 2313, 2449, 2593, 2737, 2889, 3041, 3201, 3361, 3529, 3697, 3873, 4049, 4233, 4417, 4609

Offset: 0

Daniel Poveda Parrilla, Aug 01 2016

a(A000129(n)) is a square.
(n^2)*a(n) = A275496(n) which is a triangular number.
(A000129(n)^2)*a(A000129(n)) = A275496(A000129(n)) = A001110(n) which is a square triangular number.
a(2n+1)/a(2n) is convergent to 1.

			a(1) = A275496(1) = 1.
a(5) = A275496(5)/25 = 1225/25 = 49.
a(7) = A275496(7)/49 = 4753/49 = 97.
a(12) = A275496(12)/144 = 41616/144 = 289.

Daniel Poveda Parrilla, Table of n, a(n) for n = 0..500000
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A081585(n) = a(2n), A069129(n) = a(2n + 1).

Cf. A000129, A000217, A000290, A001110, A077221, A093954, A275496.

CoefficientList[Series[(1 - x + 7 x^2 + x^3)/((1 - x)^3 (1 + x)), {x, 0, 48}], x] (* or as defined *)
Riffle[LinearRecurrence[{3, -3, 1}, {1, 9, 33}, #], FoldList[#1 + #2 &, 1, 16 Range@ #]] &@ 25 (* Michael De Vlieger, Aug 01 2016, after Vincenzo Librandi at A081585 and Robert G. Wilson v at A069129 *)

a(n)=(-1)^n + 2*n^2 \\ Charles R Greathouse IV, Aug 03 2016

Vec((1-x+7*x^2+x^3)/((1-x)^3*(1+x)) + O(x^100)) \\ Colin Barker, Aug 21 2016

a(0) = 1; a(n) = A275496(n)/(n^2) for n > 0.
From Colin Barker, Aug 01 2016: (Start)
a(n) = (2*n^2 + (-1)^n).
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4) for n > 3.
G.f.: (1 -x +7*x^2 +x^3) / ((1 - x)^3*(1 + x)).
(End)
From Daniel Poveda Parrilla, Aug 18 2016: (Start)
a(2n) = A077221(2n) + 1.
a(2n + 1) = A077221(2n + 1). (End)
Sum_{n>=0} 1/a(n) = (1 + (tan(c) + coth(c))*c)/2, where c = Pi/(2*sqrt(2)) is A093954. - Amiram Eldar, Aug 21 2022

A335828 Numerators of coefficients in a power series expansion of the distance between two bodies falling freely towards each other along a straight line under the influence of their mutual gravitational attraction.

Original entry on oeis.org

1, 1, 11, 73, 887, 136883, 7680089, 26838347, 14893630313, 1908777537383, 2422889987331397, 233104477447558811, 2430782624763507659, 14420190617640617313953, 4515429325405165295004389, 812454316441781379614873497, 166481868581561511154267399013

Offset: 1

Amiram Eldar, Jun 25 2020

Consider two point objects with masses m_1 and m_2 that are starting to fall towards each other from rest at time t = 0 and initial distance r_0. Foong (2008) gave the solution for the distance as a function of time, r(t) = r_0 * f(t/t_0), where t_0 = sqrt(r_0^3/(G*(m1+m2))), G is the gravitational constant (A070058), and f(x) = 1 - Sum_{n>=1} c(n) * x^(2*n) is a dimensionless function. c(n) are the rational coefficients whose numerators are given in this sequence. The denominators are given in A335829. The collision occurs when f(x) = 0, at x = Pi/(2*sqrt(2)) (A093954), which corresponds to the time t = (Pi/(2*sqrt(2))) * t_0.

A similar expansion was given by Ernst Meissel in his study of the three-body problem in 1882. In Meissel's expansion the coefficients are c(n)/2^n.

			The series begins with f(x) = 1 - (1/2)*x^2 - (1/12)*x^4 - (11/360)*x^6 - ...

Sudhir Ranjan Jain, Mechanics, Waves and Thermodynamics: An Example-based Approach, Cambridge University Press, 2016. See page 97.
Ernst Meissel, Über Reihen, denen man bei der numerischen Lösung des Problems der Dreikörperproblems begegnet, wenn die Anfangsgeschwindigkeiten Null sind, in: Jahresbericht über die Realschule in Kiel: Während des Schuljahres 1881/82, A. F. Jensen, Kiel, 1882, pp. 1-11.

Amiram Eldar, Table of n, a(n) for n = 1..243
S. K. Foong, From Moon-fall to motions under inverse square laws, European journal of physics, Vol. 29, No. 5 (2008), pp. 987-1003, alternative link.
Jaak Peetre, Ernst Meissel and the Pythagorean problem - the Drei-Körper-Problem in the Nachlass Meissel, draft, 1997.

Cf. A070058, A093954, A202623, A335829 (denominators).

c[1] = 1/2; c[n_] := c[n] = (2*Sum[(n - k)*(2*n - 2*k - 1)*c[n - k]*c[k], {k, 1, n - 1}] - Sum[(n - m)*(2*n - 2*m - 1)*c[n - m]*c[m - k]*c[k], {m, 2, n - 1}, {k, 1, m - 1}])/(n*(2*n - 1)); Numerator @ Array[c, 17]
(* or *)
Quiet[-Numerator @ CoefficientList[AsymptoticDSolveValue[{y[x]*y'[x]^2 == 2*(1-y[x]), y[0] == 1}, y[x], {x, 0, 25}], x][[3;;-1;;2]]] (* requires Mathematica 11.3+ *)

a(n) = numerator(c(n)), c(1) = 1/2, c(n) = (2 * Sum_{k=1..n-1} (n-k)*(2*n-2*k-1)*c(n-k)*c(k) - Sum_{m=2..n-1} (n-m)*(2*n-2*m-1)*c(n-m) * Sum_{k=1..m-1} c(m-k)*c(k))/(n*(2*n - 1)).

c(n) ~ c_0 * n^(-5/3) * (Pi/(2*sqrt(2)))^(-2*n), where c_0 = (3*Pi)^(2/3) / (18*Gamma(4/3)) = 0.277587...

A206769 Decimal expansion of the Fresnel integral Integral_{x=0..oo} sin(x^4) dx.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 10 2013

Imaginary part associated with A206161.

			0.3468652110238094960420351000471...

Wikipedia, Fresnel Integral.

Cf. A068465, A068467, A093954, A182168, A206161.

evalf(Pi*sin(Pi/8)/GAMMA(3/4)/2^(3/2)) ;

RealDigits[Pi * Sin[Pi/8] / (2^(3/2) * Gamma[3/4]), 10, 120][[1]] (* Amiram Eldar, Aug 23 2024 *)

Equals A093954 * A182168 / A068465.

(this constant)^2 + A206161 ^2 = A068467 ^2.

cp's OEIS Frontend

A275496 a(n) = n^2*(2*n^2 + (-1)^n).

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A326919 Decimal expansion of Sum_{k>=1} Kronecker(-7,k)/k.

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A346908 Decimal expansion of 2 - Pi / (2*sqrt(2)).

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A384509 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = ((5/n+1) + (5/n-1)*i)/(n*sqrt(2)) + 1/4 + (1/2)*i to reach |z| > 2, starting with z = 0.

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A125096 Expansion of -1 + (phi(q) * phi(q^2) + phi(-q^2) * phi(q^4)) / 2 in powers of q.

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A161684 Continued fraction for Pi/(2*sqrt(2)).

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A206161 Decimal expansion of the Fresnel integral int_{x=0..infinity} cos(x^4) dx.

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A275496 a(n) = n^2(2n^2 + (-1)^n).

A384509 a(n) = number of iterations of z -> z^2 + c(n) with c(n) = ((5/n+1) + (5/n-1)i)/(nsqrt(2)) + 1/4 + (1/2)*i to reach |z| > 2, starting with z = 0.