A060997 -id:A060997 - cp's OEIS frontend

A177270 Decimal expansion of (684125+sqrt(635918528029))/1033802.

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

Offset: 1

Klaus Brockhaus, May 07 2010

Continued fraction expansion of (684125+sqrt(635918528029))/1033802 is A177274.

Agrees with A060997 for n < 13.

			(684125+sqrt(635918528029))/1033802 = 1.43312742672437082725...

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A060997, A177271 (decimal expansion of sqrt(635918528029)), A177274 (repeat 1, 2, 3, 4, 5, 6, 7, 8, 9).

First[RealDigits[(684125+Sqrt[635918528029])/1033802,10,120]] (* Paolo Xausa, Jan 09 2024 *)

A034168 Disjoint discriminants (one form per genus) of type 2 (doubled).

Original entry on oeis.org

2, 6, 10, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462

Offset: 1

Jonathan Borwein (jborwein(AT)cecm.sfu.ca), choi(AT)cecm.sfu.ca (Stephen Choi)

J. M. Borwein and P. B. Borwein, Pi and the AGM, page 293.
L. E. Dickson, Introduction to the theory of numbers, Dover, NY, 1929.

J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.
Experimental Mathematics, Home Page

Cf. A000926, A005843, A034169, A055745, A139826. Subsequence of A025052.

noSol = {};
Do[lim = Ceiling[(n-2)/3]; found = False; Do[If[n > a*b && Mod[n - a*b, a+b] == 0 && Quotient[n - a*b, a+b] > b, found = True; Break[]], {a, 1, lim-1}, {b, a+1, lim}]; If[!found, AppendTo[noSol, n]], {n, 1000}];
Select[noSol, EvenQ[#] && SquareFreeQ[#]&] (* Jean-François Alcover, Jul 21 2022, after T. D. Noe in A000926 *)

ok(n)={n%4==2 && issquarefree(n) && !select(t->t<>2, quadclassunit(-4*n).cyc)} \\ Andrew Howroyd, Jun 09 2018

Intersection of A005843 and A139826. - Andrew Howroyd, Jun 09 2018

A103370 Row sums of triangle A095801 (matrix square of the Narayana triangle A001263).

Original entry on oeis.org

1, 3, 12, 57, 303, 1743, 10629, 67791, 448023, 3047745, 21235140, 150969195, 1091936745, 8016114681, 59616180828, 448459155063, 3407842605039, 26131449100821, 202011445055436, 1573171285950639, 12333030718989969

Offset: 1

Paul D. Hanna, Feb 02 2005

nonn

			From _Paul D. Hanna_, Feb 01 2009: (Start)
G.f.: A(x) = 1 + 3*x + 12*x^2/3 + 57*x^3/18 + 303*x^4/180 + 1743*x^5/2700 +...+ a(n)*x^n/[n!*(n+1)!/2^n] +...
A(x) = B(x)^3 where:
B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 + x^5/2700 +...+ x^n/[n!*(n+1)!/2^n] +... (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Jonathan M. Borwein, A short walk can be beautiful, 2015.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
Jonathan M. Borwein, Armin Straub and Christophe Vignat, Densities of short uniform random walks, Part II: Higher dimensions, Preprint, 2015.

Cf. A000108, A001263, A008277, A095801.

RecurrenceTable[{(n + 1) * (n + 2) * a[n] == 2 * (5 * n^2 - 2) * a[n - 1] - 9 * (n - 2) * (n - 1) * a[n - 2], a[1] == 1, a[2] == 3}, a, {n, 21}] (* Vaclav Kotesovec, Oct 17 2012 *)

{a(n)=if(n<1,0,sum(k=1,n,(matrix(n,n,m,j,binomial(m-1,j-1)*binomial(m,j-1)/j)^2)[n,k]))}

{a(n)=local(B=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(B^3,n)*n!*(n+1)!/2^n} \\ Paul D. Hanna, Feb 01 2009

G.f. satisfies: A(x) = B(x)^3 where A(x) = Sum_{n>=0} a(n)*x^n/[n!*(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n]. - Paul D. Hanna, Feb 01 2009
Recurrence: (n+1)*(n+2)*a(n) = 2*(5*n^2-2)*a(n-1) - 9*(n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 3^(2*n+5/2)/(4*Pi*n^3). - Vaclav Kotesovec, Oct 17 2012
G.f.: ((x-1)^2/(4*x*(1-9*x)^(2/3))*(-3*hypergeom([1/3, 1/3],[1],-27*x*(x-1)^2/(9*x-1)^2)+(3*x+1)^3*(9*x-1)^(-2)*hypergeom([4/3, 4/3],[2],-27*x*(x-1)^2/(9*x-1)^2)))-1+1/(2*x). - Mark van Hoeij, May 14 2013
G.f.: -(x-1)^2*hypergeom([1/3, 4/3],[2],-27*x*(x-1)^2/(9*x-1)^2)/(2*x*(1-9*x)^(2/3))-1+1/(2*x). - Mark van Hoeij, Nov 12 2023

A177933 Decimal expansion of (232405+sqrt(71216963807))/348378.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 15 2010

Continued fraction expansion of (232405+sqrt(71216963807))/348378 is A010889.

Agrees with A060997 for n < 14, with A177270 for n < 13, with A177034 for n < 11, with A177160 for n < 9.

			(232405+sqrt(71216963807))/348378 = 1.43312742672229113069...

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A177934 (decimal expansion of sqrt(71216963807)), A010889 (repeat 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), A060997 (decimal representation of continued fraction 1, 2, 3, 4, 5, 6, 7, ...), A177270 (decimal expansion of (684125+sqrt(635918528029))/1033802), A177034 (decimal expansion of (9280+3*sqrt(13493990))/14165), A177160 (decimal expansion of (4502+sqrt(29964677))/6961).

First[RealDigits[(232405+Sqrt[71216963807])/348378,10,120]] (* Paolo Xausa, Jan 09 2024 *)

A247844 Decimal expansion of the value of the continued fraction [1; 1, 2, 3, 4, 5, ...].

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 25 2014

Equals 1+A052119.

			1.697774657964007982006790592551752599486658262998...

MathOverflow, Is any particular algebraic number known to have unbounded continued fraction coefficients?
Eric Weisstein's MathWorld, Continued Fraction Constant
Eric Weisstein's MathWorld, Continued Fraction

Cf. A001040, A001053, A052119, A060997, A070910, A096789.

FromContinuedFraction[Join[{1}, Range[50]]] // RealDigits[#, 10, 105]& // First
(* or *) 1+BesselI[1, 2]/BesselI[0, 2] // RealDigits[#, 10, 105]& // First

1+besseli(1,2)/besseli(0,2) \\ Charles R Greathouse IV, Oct 23 2023

1 + I_1(2) / I_0(2), where I_n(x) gives the modified Bessel function of the first kind.

A253095 Moments of 4-step random walk in 4 dimensions.

Original entry on oeis.org

1, 4, 22, 148, 1144, 9784, 90346, 885868, 9115276, 97578688, 1079676448, 12285725632, 143204046496, 1704422018992, 20660609113186, 254522834851516, 3180935346538684, 40269426101933392, 515743456513546072, 6675036087017279056, 87221496402779437696, 1149701868292524559744

Offset: 0

N. J. A. Sloane, Feb 16 2015

nonn

J. M. Borwein, A short walk can be beautiful, Journal of Humanistic Mathematics, Volume 6 Issue 1 (January 2016), pages 86-109.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
Jonathan M. Borwein, Armin Straub and Christophe Vignat, Densities of short uniform random walks, Part II: Higher dimensions, Preprint, 2015.

W := proc(n,nu,twok)
    option remember;
    local k;
    k := twok/2 ;
    if n = 2 and nu = 1 then
        binomial(2*k+2,k+1)/(k+2) ;
    else
        add( procname(n-1,nu,2*j)*binomial(k,j)*(k+nu)!*nu!/(k-j+nu)!/(j+nu)!,j=0..k) ;
        simplify(%,GAMMA) ;
    end if;
end proc:
A253095 := proc(n)
    W(4,1,n) ;
end proc:
seq(A253095(2*n),n=0..25) ; # R. J. Mathar, Jun 14 2015

W[n_, nu_, twok_] := W[n, nu, twok] = Module[{k}, k = twok/2; If[n == 2 && nu == 1, Binomial[2k+2, k+1]/(k+2), Sum[W[n-1, nu, 2j]*Binomial[k, j]*(k+nu)!*nu!/(k-j+nu)!/(j+nu)!, {j, 0, k}]]];
A253095[n_] := W[4, 1, n];
Table[A253095[2n], {n, 0, 25}] (* Jean-François Alcover, Apr 16 2023, after R. J. Mathar *)

A011248 Twice A000364.

Original entry on oeis.org

2, 2, 10, 122, 2770, 101042, 5405530, 398721962, 38783024290, 4809759350882, 740742376475050, 138697748786275802, 31029068327114173810, 8174145018586247784722, 2504519282807259730936570, 883087786498046209107365642, 355038783159078578873329579330, 161446598471775796124336494906562

Offset: 0

N. J. A. Sloane

nonn

G. Almkvist, Many correct digits of Pi, revisited, Amer. Math. Monthly, 104 (1997), 351-353.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]

Cf. A000364, A009752.

a[n_] := (-1)^n 4 Im[PolyLog[-2 n, I]];
Table[a[n], {n, 0, 17}] (* Peter Luschny, Aug 18 2021 *)

E.g.f.: 2 - 2/Q(0), where Q(k)= 1 - (2*k+1)*(2*k+2)/x + 1/x*(2*k+1)*(2*k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
From Peter Luschny, Aug 18 2021: (Start)
a(n) = (-1)^n*4^(2*n+1)*(Bernoulli(2*n+1, 3/4) - Bernoulli(2*n+1, 1/4))/(2*n+1).
a(n) = (-1)^n*4*Im(PolyLog(-2*n, i)). (End)

A055745 Squarefree numbers not of form ab + bc + ca for 1 <= a <= b <= c (probably the list is complete).

Original entry on oeis.org

1, 2, 6, 10, 22, 30, 42, 58, 70, 78, 102, 130, 190, 210, 330, 462

Offset: 1

N. J. A. Sloane

Maohua Le, A note on positive integer solutions of the equation xy+yz+zx=n, Publ. Math. Debrecen 52 (1998) 159-165; Math. Rev. 98j:11016.

J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
J. Borwein and K.-K. S. Choi, On the representations of xy+yz+zx, Experimental Mathematics, 9 (2000), 153-158.

Cf. A034168, A025052, A034169.

solQ[n_, x_] := Reduce[1 <= y <= z && n == x*y + y*z + z*x, {y, z}, Integers] =!= False; solQ[n_] := Catch[xm = Ceiling[(n-1)/2]; For[x = 1, x <= xm, x++, Which[ solQ[n, x] === True, Throw[True], x == xm, Throw[False]]]] ; solQ[1] = False; Reap[ Do[ If[ SquareFreeQ[n], If[! solQ[n] , Print[n]; Sow[n]]], {n, 1, 500}]][[2, 1]] (* Jean-François Alcover, Jun 15 2012 *)

A218147 Degree of minimal polynomial satisfied by exp(8Piphi_2(1/n,1/n)), where phi_2 is defined in the Comments.

Original entry on oeis.org

2, 2, 4, 4, 12, 8, 18, 8, 30, 16, 36, 24, 32, 32, 64, 36, 90, 32, 96, 60, 132, 64, 100, 72, 162, 96, 196, 64, 240, 128, 240, 128, 192, 144, 324, 180, 288, 128, 400, 192, 462, 240, 288, 264, 552, 256, 588, 200, 512, 288, 676, 324, 480, 384, 720, 392, 870, 256

Offset: 3

Jason Kimberley, Oct 21 2012 and Apr 04 2016

Crandall defines phi_2(r_1,r_2) = (1/Pi^2) Sum_{positive & negative odd m_1, m_2} cos(Pi m_1 r_1) cos(Pi m_2 r_2) / (m_1^2+m_2^2).
Lemma: 4a(n) < n^2. Proof: 4a(2) = 2 < 2^2; 4a(4k+1) = 16k^2 < (4k+1)^2; 4a(4k+3) = (4k+2)(4k+4) = (4k+3)^2-1; 4a(p^2 k) = 4p^2 a(pk) < p^2(pk)^2 = (p^2 k)^2; 4 a(jk) = 4 a(j) 4 a(k) < (jk)^2.
Corollary: a(n) <=  A198442(n).

R. Crandall, The Poisson equation and "natural" Madelung constants, preprint 2012 (see section 2 of BBCZ below).

Jason Kimberley, Table of n, a(n) for n = 3..10000
D. H. Bailey, J. Borwein, R. Crandall and J. Zucker, Lattice sums arising from the Poisson equation, preprint (2012).
D. H. Bailey, J. M. Borwein and J. S. Kimberley, with an appendix by W. B. Ladd, Computer discovery and analysis of large Poisson polynomials, Experimental Mathematics, August 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
OEIS (Plot 2), Plot of (log n, log a(n))
OEIS (Plot 2), Plot of (n, a(n)) and (n, A198442(n))

Cf. A079458, A198442.

A218147 := func; // Jason Kimberley, Oct 23 2012

A218147 := func)/4>; // Jason Kimberley, Nov 14 2015

a(n) = A079458(n) / 4, for n > 2. - Jason Kimberley, Nov 14 2015
Watson Ladd has proved that the sequence satisfies the following recurrence relations, which were conjectured by Jason Kimberley:
a(1) = 1/4, a(2) = 1/2, for notational convenience;
a(4k+1) = (2k)*(2k) for prime 4k+1;
a(4k+3) = (2k+1)*(2k+2) for prime 4k+3;
a(p^2 k) = p^2 * a(p*k) for prime p;
a(jk) = 4*a(j)*a(k) for j coprime to k.

Entry revised by N. J. A. Sloane, May 15 2016, to take into account the fact that the conjectured formula for this sequence has now been established by Watson Ladd.

A119822 Decimal representation of continued fraction 1, 2, 5, 14, 42, ... (Catalan numbers).

Original entry on oeis.org

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

Offset: 1

Alexander Adamchuk, Jul 30 2006

It is the limit of continued fraction of Catalan numbers (1 + 1/(2 + 1/(5 + 1/(14 + 1/(42 + 1/(132 + 1/(429 + 1/(1430 + 1/(4862 + 1/(16796 + ...)))))))))).

			1.45512722857749276356251912344061532491898712686082426222771476844208998388...

Cf. A071897, A060997, A000108.

RealDigits[ Normal[ ContinuedFractionForm[ Table[(2k)!/k!/(k+1)!, {k,1,30} ]]], 10, 130] [[1]]

cp's OEIS Frontend

A177270 Decimal expansion of (684125+sqrt(635918528029))/1033802.

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A034168 Disjoint discriminants (one form per genus) of type 2 (doubled).

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A103370 Row sums of triangle A095801 (matrix square of the Narayana triangle A001263).

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A177933 Decimal expansion of (232405+sqrt(71216963807))/348378.

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A247844 Decimal expansion of the value of the continued fraction [1; 1, 2, 3, 4, 5, ...].

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Mathematica

PARI

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A253095 Moments of 4-step random walk in 4 dimensions.

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Maple

Mathematica

A011248 Twice A000364.

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Formula

A055745 Squarefree numbers not of form ab + bc + ca for 1 <= a <= b <= c (probably the list is complete).

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A218147 Degree of minimal polynomial satisfied by exp(8Piphi_2(1/n,1/n)), where phi_2 is defined in the Comments.