Roman Witula - cp's OEIS frontend

A254666 Decimal expansion of the right Alzer's constant.

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

Offset: 0

Roman Witula, Feb 04 2015

The right Alzer's constant x is defined to be the best constant in the right Alzer's inequality: abs(cos a + sin a) <= x*abs(cos(cos a) + cos(sin a)), where a is any real number.

			0.9301043163036166822994530624072616003335332058073463854808289441...

Horst Alzer, A trigonometric double-inequality, Elemente der Mathematik 65 (2010), 45-48.

Cf. A254615.

RealDigits[1/(Sqrt[2]*Cos[1/Sqrt[2]]), 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

1/(sqrt(2)*cos(1/sqrt(2))) \\ Michel Marcus, Feb 05 2015

Equals (sqrt(2)*cos(1/sqrt(2)))^(-1).

a(99) corrected by Georg Fischer, Aug 12 2021

A254615 Decimal expansion of the left Alzer's constant x.

Original entry on oeis.org

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

Offset: 1

Roman Witula, Feb 03 2015

The left Alzer's constant x is defined to be the best constant in the left Alzer's inequality: x*abs(sin(cos a) + sin(sin a)) <= abs(cos a + sin a), where a is any real number.

			x = 1.088464554044397392026605399544901779407224058765958312439431735...

Horst Alzer, A trigonometric double-inequality, Elemente der Mathematik 65 (2010), 45-48.

RealDigits[(Sqrt[2] Sin[1/Sqrt[2]])^(-1), 10, 100][[1]] (* Bruno Berselli, Feb 03 2015 *)

1/(sqrt(2)*sin(1/sqrt(2))) \\ Michel Marcus, Feb 03 2015

x = (sqrt(2)*sin(1/sqrt(2)))^(-1).

x = Sum_{k=-oo..oo} (-1)^k/(1 - 2*(Pi*k)^2). - Bruno Berselli, Feb 03 2015

A254604 Decimal expansion of the half of a single half-wave constant.

Original entry on oeis.org

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

Offset: 0

Roman Witula, Feb 02 2015

By the definition this constant takes the value 0 < x < Pi such that the horizontal line y = sin x divides the region bounded by the sine curve and the interval [0,Pi] on the x-axis into two shapes of equal area.

			x = 0.368305390596661399946991689932336338080760158209792727276899496553...

FindRoot[2 Cos[x] - (Pi - 2 x) Sin[x] - 1 == 0, {x, 0, Pi}, WorkingPrecision -> 150] (* Bruno Berselli, Feb 03 2015 *)

solve(x=0, Pi, 2*cos(x) - (Pi - 2*x)*sin(x) - 1) \\ Michel Marcus, Feb 03 2015

2*cos(x) - (Pi - 2*x)*sin(x) = 1, 0 < x < Pi.

A251926 The Faulhaber-Knuth a(0,n) sequence.

Original entry on oeis.org

2, 1, 1, 1, 1, 0, 0, 1, 37, -60, -5, 37, 174, -955, -10545, 38610, 176297, -322740, -205420, 4512655, 56820585, -104019264, -25907081, 94854194, 1141847218, -2090335775, -414239903275, 6066664425833, 85621405759989, -156743813184120, -4337631088920, 47644406040193, 1265208493396175131, -2316168508680582540, -192288633159406495

Offset: 4

Roman Witula, Dec 11 2014

sign

a(n) is equal to the remainder when dividing the polynomial T_n(x) by x^2 + x - 1. T_n(x) (in Z[x]) is the positive integer multiplicity of the modified Faulhaber polynomial T*_n(x), coefficients of which have GCD equal to 1. We have T*_n(x) = S(n;x)/x^2(x+1)^2 if n is odd, and T*_n(x) = S(n;x)/x(x+1)(2x+1) if n is even, n >= 4, where S(n;x) denotes the n-th Faulhaber polynomial, i.e., S(n;x) = 1/(n+1) sum{taken over i=0,1,...,n} Bin(n+1,i)Bern(i)x^(n+1-i), and Bern(i) denotes the i-th Bernoulli number with Bern(1)=1/2.

We note that every T_n(x) is a polynomial in the variable (x^2 + x - 1), for example T_7(x) = 3(x^2 + x - 1)^2 + 2(x^2 + x - 1) + 1. Furthermore, every T_n(x) is a polynomial in (x^2 + x + a) for each complex a. But only for a = -1 is the element a(n) also equal to the remainder when dividing S(n;x) by x^2 + x + a if n is odd and S(n;x)/(2x+1) by x^2 + x + a if n is even.

			We have: T_4(x) = 3x^2 + 3x - 1, T_4(x) - T_5(x) = x^2 + x, T_6(x) - T_7(x) = x^2 + x - 1, T_9(x) = (x^2 + x - 1)(2x^4 + 4x^3 - x^2 - 3x + 3) and T_15(x) - T_12(x) is divisible by (x^2 + x - 1), which implies a(0)=2, a(1)=1, a(2)=a(3), a(5)=0 and a(8)=a(11).

Edyta Hetmaniok, Piotr Lorenc, Mariusz Pleszczyński, and Roman Wituła, Iterated integrals of polynomials, Applied Mathematics and Computation, Volume 249, 15 December 2014, Pages 389-398.
D. E. Knuth, Johann Faulhaber and sums of powers, Math. Comput. 203 (1993), 277-294.
Piotr Lorenc, Jakub Jan Ludew, Mariusz Pleszczyński, Alicja Samulewicz, and Roman Wituła, Iterated integrals of Faulhaber polynomials and some properties of their roots, 2018.

Cf. A093556, A093557.

coeffFaulh[n_] := Module[{t, tab = {}, s, p, x},
  If[n < 4, Return["Give n greater than 3."]];
  t = Table[1, {n + 2}];
  Do[t[[i + 1]] = BernoulliB[i], {i, 1, n + 1}];
  t[[2]] = 1/2;
  s[m_, x_] := (Sum[Binomial[m + 1, i]t[[ i + 1]] x^(m + 1 - i),{i,0,m}])/(m + 1);
  Do[If[Mod[i, 2] == 0,
    p = PolynomialRemainder[FactorList[Factor[s[i, x]] (i + 1)/(x (x + 1) (2 x + 1))][[2,1]], -1 + x + x^2, x],
    p = PolynomialRemainder[FactorList[Factor[s[i, x]] (i + 1)/(x^2 (x + 1)^2)][[2,1]], -1 + x + x^2, x]];
   tab = Append[tab, p], {i, 4, n}];
  tab]

A219246 Decimal expansion of the maximum M(5) of the ratio (Sum_{k=1..5} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(5)) taken over x(1), ..., x(5) > 0.

Original entry on oeis.org

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

Offset: 1

Roman Witula, Nov 16 2012

The maximum M(n) of the ratio (Sum_{k=1..n} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(n)) taken over x(1), ..., x(n) > 0 is discussed in A219245 - see also the paper of Witula et al. for the proofs.

The decimal expansions of M(4) and M(6) are A219245 and A219336, respectively.

			1.486353228963....

R. Witula, D. Jama, D. Slota, E. Hetmaniok, Finite version of Carleman's and Knopp's inequalities, Zeszyty naukowe Politechniki Slaskiej (Gliwice, Poland) 92 (2010), 93-96.

Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]
Yu-Dong Wu, Zhi-Hua Zhang and Zhi-Gang Wang, The Best Constant for Carleman's Inequality of Finite Type, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 24, No. 2, 2008

Cf. A219245, A219336, A249403.

RealDigits[c5/.FindRoot[{1+x2/2+x3/3+x4/4+x5/5==c5, x2/2+x3/3+x4/4+x5/5==c5*x2^2, x3/3+x4/4+x5/5==c5*x3^3/x2^2, x4/4+x5/5==c5*x4^4/x3^3, x5/5==c5*x5^5/x4^4},{{c5,3/2},{x2,1/2},{x3,1/2},{x4,1/2},{x5,1/2}},WorkingPrecision->120],10,105][[1]] (* Vaclav Kotesovec, Oct 27 2014 *)

A219245 Decimal expansion of the maximum M(4) of the ratio (Sum_{k=1..4} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(4)) taken over x(1), ..., x(4) > 0.

Original entry on oeis.org

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

Offset: 1

Roman Witula, Nov 16 2012

We note that the maximum M(n) of the ratio (Sum_{k=1..n} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(n)) taken over x(1), ..., x(n) > 0 is equal to (1+sqrt(2))/2 for n=2 and 4/3 for n=3. Moreover it can be proved that M(n) < (1 + 1/n)^(n-1) - it is a finite version of Carleman's inequality (see the paper of Witula et al. for the proof). The sequence M(n), n=2,3,..., is increasing.

The decimal expansions of M(5) and M(6) are A219246 and A219336, respectively.

			M(4) = 1.42084438540961...

R. Witula, D. Jama, D. Slota, E. Hetmaniok, Finite version of Carleman's and Knopp's inequalities, Zeszyty naukowe Politechniki Slaskiej (Gliwice, Poland) 92 (2010), 93-96.

Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]
Yu-Dong Wu, Zhi-Hua Zhang and Zhi-Gang Wang, The Best Constant for Carleman's Inequality of Finite Type, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 24, No. 2, 2008.

Cf. A219246, A219336, A249403.

RealDigits[N[Root[387420489 + 22039921152*#1 + 373658292864*#1^2 + 12841816536576*#1^3 + 274965186525696*#1^4 - 201976270848000*#1^5 + 42624005978423296*#1^6 + 342213608420278272*#1^7 + 660475521813381120*#1^8 - 2629784260986273792*#1^9 + 41447678188009291776*#1^10 + 427447433656163893248*#1^11 - 198705178996352483328*#1^12 - 2098418839125516877824*#1^13 + 16905530303693690241024*#1^14 + 14417509185682352898048*#1^15 - 20033038006659651207168*#1^16 - 149735761790067869220864*#1^17 + 18738444188050884919296*#1^18 + 361130725214496730644480*#1^19 + 220843507713085418766336*#1^20 - 1387347813563214701002752*#1^21 + 1472163837099830446915584*#1^22 - 654295038711035754184704*#1^23 + 109049173118505959030784*#1^24 & , 4], 105]][[1]] (* Vaclav Kotesovec, Oct 26 2014 *)

A219336 Decimal expansion of the maximum M(6) of the ratio (Sum_{k=1..6} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(6)) taken over x(1), ..., x(6) > 0.

Original entry on oeis.org

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

Offset: 1

Roman Witula, Nov 18 2012

The maximum M(n) of the ratio (Sum_{k=1..n} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(n)) taken over x(1), ..., x(n) > 0 is discussed in A219245 - see also the paper of Witula et al. for the proofs.

The decimal expansions of M(4) and M(5) are A219245 and A219246, respectively.

			1.537937556520034931368158716...

R. Witula, D. Jama, D. Slota, E. Hetmaniok, Finite version of Carleman's and Knopp's inequalities, Zeszyty naukowe Politechniki Slaskiej (Gliwice, Poland) 92 (2010), 93-96.

Steven R. Finch, Carleman's inequality, 2013. [Cached copy, with permission of the author]
Yu-Dong Wu, Zhi-Hua Zhang and Zhi-Gang Wang, The Best Constant for Carleman's Inequality of Finite Type, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis, Vol. 24, No. 2, 2008.

Cf. A219245, A219246, A249403.

RealDigits[c6/.FindRoot[{1 + x2/2 + x3/3 + x4/4 + x5/5 + x6/6 == c6, x2/2 + x3/3 + x4/4 + x5/5 + x6/6 == c6*x2^2, x3/3 + x4/4 + x5/5 + x6/6 == c6*x3^3/x2^2, x4/4 + x5/5 + x6/6 == c6*x4^4/x3^3, x5/5 + x6/6 == c6*x5^5/x4^4, x6/6 == c6*x6^6/x5^5},{{c6,3/2},{x2,1/2},{x3,1/2},{x4,1/2},{x5,1/2},{x6,1/2}},WorkingPrecision->120],10,105][[1]] (* Vaclav Kotesovec, Oct 27 2014 *)

A218664 Coefficients of cubic polynomials p(x+n), where p(x) = x^3 + x^2 - 2*x - 1.

Original entry on oeis.org

1, 1, -2, -1, 1, 4, 3, -1, 1, 7, 14, 7, 1, 10, 31, 29, 1, 13, 54, 71, 1, 16, 83, 139, 1, 19, 118, 239, 1, 22, 159, 377, 1, 25, 206, 559, 1, 28, 259, 791, 1, 31, 318, 1079, 1, 34, 383, 1429, 1, 37, 454, 1847, 1, 40, 531, 2339, 1, 43, 614, 2911, 1, 46, 703, 3569, 1, 49, 798, 4319

Offset: 0

Roman Witula, Nov 04 2012

sign

We have p(x) = (x - c(1))*(x - c(2))*(x - c(4)), where c(j) := 2*cos(2*Pi*j/7). We note that c(4) = c(3) = -c(1/2), c(1) = s(3) and c(2) = -s(1), where s(j) := 2*sin(Pi*j/14). Moreover we obtain -p(-x) = x^3 - x^2 - 2*x + 1 = (x + c(1))*(x + c(2))*(x + c(4)), q(x) := -x^3*p(1/x) = x^3 + 2*x^2 + x - 1 = (x - c(1)^(-1))*(x - c(2)^(-1))*(x - c(4)^(-1)), and -q(-x) = x^3 - 2*x^2 + x + 1 = (x + c(1)^(-1))*(x + c(2)^(-1))*(x + c(4)^(-1)).
We also have p(x+2) =  x^3 + 7*x^2 + 14*x + 7 = (x + s(2)^2)*(x + s(4)^2)*(x + s(6)^2). The polynomial -p(-x-2) = x^3 - 7*x^2 + 14*x - 7 = (x - s(2)^2)*(x - s(4)^2)*(x - s(6)^2) is known as Johannes Kepler's cubic polynomial (see Witula's book).
Let us set r(x) := p(x+1). It can be verified that -x^3*r(1/x) = x^3 - 3*x^2 - 4*x - 1 = (x - c(1)/c(4))*(x - c(4)/c(2))*(x - c(2)/c(1)); for example, we have c(1)^3 + c(1)^2 - 2*c(1) - 1 = 0 which implies that c(1)^2 + 2*c(1) = 1/(c(1) - 1), and then c(1)^2 + 2*c(1) = c(4)/c(2) since c(4)/c(2) = (c(1)^4 - 4*c(1)^2 + 2)/(c(1)^2 - 2).
The polynomials p(x+n) and the ones obtained as above (i.e., after simple algebraic transformations) are the characteristic polynomials of many sequences in the OEIS; see crossrefs.

R. Witula, Complex Numbers, Polynomials and Partial Fraction Decomposition, Part 3, Wydawnictwo Politechniki Slaskiej, Gliwice 2010 (Silesian Technical University publishers).

Cf. A094648, A096975, A033304, A214683, A006053, A215076, A215100, A215007, A215008, A215143, A215493, A215494, A215510.

We have a(4*k) = 1, a(4*k + 1) = 3*k + 1, a(4*k + 2) = 3*k^2 + 2*k - 2, a(4*k + 3) = k^3 + k^2 - 2*k - 1. Further, the following relations hold true: b(k+1) = b(k) + 3, c(k+1) = 2*b(k) -2*c(k) + 3, d(k+1) = b(k) - 2*c(k) - d(k) + 1, where p(x + k) = x^3 + b(k)*x^2 + c(k)*x + d(k).

Empirical g.f.: -(x^15 - x^14 - 2*x^13 + x^12 - 5*x^11 + 10*x^10 + 3*x^9 - 3*x^8 - 3*x^7 - 11*x^6 + 3*x^4 + x^3 + 2*x^2 - x - 1) / ((x-1)^4*(x+1)^4*(x^2+1)^4). - Colin Barker, May 17 2013

A211988 The Berndt-type sequence number 9 for the argument 2*Pi/13.

Original entry on oeis.org

0, -6, -37, 676, 2882, 12502, -196209, -856850, -3740697, 58876883, 257003504, 1121852777, -17656510365, -77073076671, -336434457597, 5295048110651, 23113603862267, 100894018986142, -1587942800101489, -6931585922526870, -30257313674299627, 476211413709501353

Offset: 0

Roman Witula, Oct 25 2012

sign

a(n) + A218655(n)*sqrt(13) = A(2*n+1)*13^((1+floor(n/3))/2)*sqrt(2*(13 + 3*sqrt(13))/13), where A(n) is defined below.
The sequence A(n) from the name of a(n) is defined by the relation A(n) = s(1)^(-n) + s(3)^(-n) + s(9)^(-n), where s(j) := 2*sin(2*Pi*j/13). The sequence with respective positive powers is discussed in A216508 (see sequence Y(n) in Comments to A216508).
It follows that A(n) = sqrt((13-3*sqrt(13))/2)*A(n-1) + (sqrt(13)-3)*A(n-2)/2 - sqrt((13-3*sqrt(13))/26)*A(n-3), with A(-1) = sqrt((13-3*sqrt(13))/2), A(0)=3, and A(1) = sqrt((13-3*sqrt(13))/2).
We note that s(1) + s(3) + s(9) = s(1)^(-1) + s(3)^(-1) + s(9)^(-1) = sqrt((13-3*sqrt(13))/2), sqrt(2*sqrt(13))*(s(1)^(-3) + s(3)^(-3) + s(9)^(-3)) = sqrt(97*sqrt(13)-339), and s(1)^(-9) + s(3)^(-9) + s(9)^(-9) = (131/13)*sqrt(2834 - 786*sqrt(13)).
The numbers of other Berndt-type sequences for the argument 2*Pi/13 in crossrefs are given.

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, Thirteenth International Conference on Fibonacci Numbers and their Applications, Congressus Numerantium, 201 (2010), 89-107.
R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.

Cf. A216605, A216486, A216508, A216597, A216540, A161905, A217548, A217549, A216450.

A218655 The Berndt-type sequence number 10 for the argument 2*Pi/13.

Original entry on oeis.org

2, 4, 13, -176, -786, -3452, 54483, 237722, 1037569, -16329149, -71279530, -311145495, 4897036897, 21376227709, 93310132523, -1468582101731, -6410560285891, -27982966049682, 440416091468393, 1922476035761802, 8391868916275609

Offset: 0

Roman Witula, Nov 04 2012

sign

A211988(n) + a(n)*sqrt(13) = A(2*n+1)*13^((1 + floor(n/3))/2)*sqrt(2*(13 + 3*sqrt(13))/13), where A(n) is defined below.
The sequence A(n) from the name of a(n) is defined  by the relation A(n) = s(1)^(-n) + s(3)^(-n) + s(9)^(-n), where s(j) := 2*sin(2*Pi*j/13). The sequence with respective positive powers is discussed in A216508 (see sequence Y(n) in comments to A216508).
It could be deduced that A(n) = sqrt((13-3*sqrt(13))/2)*A(n-1) + (sqrt(13)-3)*A(n-2)/2 - sqrt((13-3*sqrt(13))/26)*A(n-3), with A(-1) = sqrt((13-3*sqrt(13))/2), A(0)=3, and A(1) = sqrt((13-3*sqrt(13))/2).
The numbers of other Berndt-type sequences for the argument 2*Pi/13 in crossrefs are given.

			Let us put b(n) = A211988(n) + a(n)*sqrt(13). Then we get b(0) = 2*sqrt(13), b(1) = -6 + 4*sqrt(13), b(2) = -37 + 13*sqrt(13), b(3) = 676 - 176*sqrt(13), b(4) = 2882 - 786*sqrt(13), b(5) = 12502 - 3452*sqrt(13).

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, Thirteenth International Conference on Fibonacci Numbers and their Applications, Congressus Numerantium, 201 (2010), 89-107.
R. Witula, On some applications of formulas for sums of the unimodular complex numbers, Wyd. Pracowni Komputerowej Jacka Skalmierskiego, Gliwice 2011 (in Polish).

R. Witula and D. Slota, Quasi-Fibonacci numbers of order 13, (abstract) see p. 15.

Cf. A211988, A216605, A216486, A216508, A216597, A216540, A161905, A217548, A217549, A216450.

cp's OEIS Frontend

User: Roman Witula

A254666 Decimal expansion of the right Alzer's constant.

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A254615 Decimal expansion of the left Alzer's constant x.

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A254604 Decimal expansion of the half of a single half-wave constant.

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A251926 The Faulhaber-Knuth a(0,n) sequence.

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A219246 Decimal expansion of the maximum M(5) of the ratio (Sum_{k=1..5} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(5)) taken over x(1), ..., x(5) > 0.

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A219245 Decimal expansion of the maximum M(4) of the ratio (Sum_{k=1..4} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(4)) taken over x(1), ..., x(4) > 0.

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A219336 Decimal expansion of the maximum M(6) of the ratio (Sum_{k=1..6} (x(1)*x(2)*...*x(k))^(1/k))/(x(1) + ... + x(6)) taken over x(1), ..., x(6) > 0.

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A218664 Coefficients of cubic polynomials p(x+n), where p(x) = x^3 + x^2 - 2*x - 1.

A219246 Decimal expansion of the maximum M(5) of the ratio (Sum_{k=1..5} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(5)) taken over x(1), ..., x(5) > 0.

A219245 Decimal expansion of the maximum M(4) of the ratio (Sum_{k=1..4} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(4)) taken over x(1), ..., x(4) > 0.

A219336 Decimal expansion of the maximum M(6) of the ratio (Sum_{k=1..6} (x(1)x(2)...*x(k))^(1/k))/(x(1) + ... + x(6)) taken over x(1), ..., x(6) > 0.