A001609 -id:A001609 - cp's OEIS frontend

A146485 Decimal expansion of Product_{n>=2} (1 - 1/(n^2*(n-1))).

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

Offset: 0

R. J. Mathar, Feb 13 2009

Product of Artin's constant of rank 2 and the equivalent almost-prime products.

			0.6739173633763... = (1 - 1/4)*(1 - 1/18)*(1 - 1/48)*(1 - 1/100)*...

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], table 3, first line with r=2.

Cf. A065414.

r := 2 : ni := fsolve( (n+1)^r*n-1,n,complex) : 1.0/mul(GAMMA(1-d),d=ni) ; # R. J. Mathar, Feb 20 2009

g[k_] := Gamma[Root[-3 + 8# - 5#^2 + #^3 & , k]]; RealDigits[1/(g[1]*g[2]*g[3]) // Re, 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)

The logarithm is -Sum_{s>=2} Sum_{j=1..floor(s/(1+r))} binomial(s-r*j-1, j-1)*(1-Zeta(s))/j at r = 2.
s*Sum_{j=1..floor(s/3)} binomial(s-2j-1, j-1)/j = A001609(s)-1.
Equals 1/Product_{k=1..3} Gamma(1-x_k), where x_k are the 3 roots of the polynomial x*(x+1)^2-1. [R. J. Mathar, Feb 20 2009]

More terms from Jean-François Alcover, Feb 12 2013

A218438 G.f.: 1 / ( (1 + x^2 - x^3)^2 * (1 - x - 2*x^2 - x^3) ).

Original entry on oeis.org

1, 1, 1, 6, 12, 19, 48, 110, 218, 470, 1040, 2208, 4710, 10184, 21879, 46879, 100767, 216570, 464952, 998613, 2145312, 4607724, 9896436, 21257196, 45658624, 98068864, 210642412, 452440320, 971794317, 2087314717, 4483345053, 9629771966, 20683772420, 44426659559

Offset: 0

Paul D. Hanna, Oct 28 2012

nonn

			G.f.: A(x) = 1 + x + x^2 + 6*x^3 + 12*x^4 + 19*x^5 + 48*x^6 + 110*x^7 +...
where
log(A(x)) = x + x^2/2 + 16*x^3/3 + 25*x^4/4 + 36*x^5/5 + 100*x^6/6 + 225*x^7/7 +...+ A001609(n)^2*x^n/n +...

Index entries for linear recurrences with constant coefficients, signature (1, 0, 5, 1, 1, -3, -2, 0, 1).

Cf. A218439, A001609.

CoefficientList[Series[1/((1+x^2-x^3)^2(1-x-2x^2-x^3)),{x,0,40}],x] (* or *) LinearRecurrence[{1,0,5,1,1,-3,-2,0,1},{1,1,1,6,12,19,48,110,218},40] (* Harvey P. Dale, Jan 23 2013 *)

{a(n)=polcoeff(1/((1 + x^2 - x^3)^2*(1 - x*(1+x)^2+x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

Logarithmic derivative yields A218439, where A218439(n) = A001609(n)^2.

a(0)=1, a(1)=1, a(2)=1, a(3)=6, a(4)=12, a(5)=19, a(6)=48, a(7)=110, a(8)=218, a(n)=a(n-1)+5*a(n-3)+a(n-4)+a(n-5)-3*a(n-6)-2*a(n-7)+a(n-9). - Harvey P. Dale, Jan 23 2013

A382641 a(n) = round(c^n), where c is the supergolden ratio A092526.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 21, 31, 46, 67, 98, 144, 211, 309, 453, 664, 973, 1426, 2090, 3063, 4489, 6579, 9642, 14131, 20710, 30352, 44483, 65193, 95545, 140028, 205221, 300766, 440794, 646015, 946781, 1387575, 2033590, 2980371, 4367946, 6401536, 9381907, 13749853

Offset: 0

Jwalin Bhatt, Apr 01 2025

Wikipedia, Supergolden ratio
Index entries for linear recurrences with constant coefficients, signature (1,0,1).

Cf. A000930, A001609, A092526, A169985, A205579.

r = Root[x^3-x^2-1, 1]; Table[Round[r^i], {i,0,120 }]
CoefficientList[Series[(1+x^2+x^4-x^8)/(1-x-x^3), {x,0,120}], x]

G.f.: (1 + x^2 + x^4 - x^8)/(1 - x - x^3).
a(n) = a(n-1) + a(n-3) for n>=9.
a(n) = round(((2/3)*cos((1/3)*arccos(29/2))+1/3)^n) = round(A092526^n).
a(n) = A001609(n) for n>=6.

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A146485 Decimal expansion of Product_{n>=2} (1 - 1/(n^2*(n-1))).

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A218438 G.f.: 1 / ( (1 + x^2 - x^3)^2 * (1 - x - 2*x^2 - x^3) ).

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A382641 a(n) = round(c^n), where c is the supergolden ratio A092526.

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