A206739 -id:A206739 - cp's OEIS frontend

A285407 G.f.: 1/(1 - x^2/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^11/(1 - ... - x^prime(k)/(1 - ... ))))))), a continued fraction.

1, 0, 1, 0, 1, 1, 1, 2, 2, 3, 5, 5, 9, 11, 15, 23, 28, 43, 57, 78, 113, 149, 214, 293, 403, 569, 774, 1086, 1502, 2072, 2896, 3986, 5548, 7691, 10636, 14797, 20459, 28400, 39386, 54542, 75724, 104886, 145468, 201733, 279545, 387786, 537472, 745233, 1033383, 1432415, 1986394

Offset: 0

Ilya Gutkovskiy, Apr 18 2017

nonn

			G.f.: A(x) = 1 + x^2 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 5*x^10 + ...

Robert Israel, Table of n, a(n) for n = 0..5000

Cf. A000040, A206739, A206741, A206742, A206743, A227310, A269353.

R:= 1:
for i from numtheory:-pi(50) to 1 by -1 do
  R:= series(1-x^ithprime(i)/R, x, 51);
od:
R:= series(1/R, x, 51):
seq(coeff(R,x,j),j=0..50); # Robert Israel, Apr 20 2017

nmax = 50; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^Prime[k], 1, {k, 1, nmax}]), {x, 0, nmax}], x]

a(n) ~ c * d^n, where d = 1.3864622092472465020397266918102624708859968795203700659786636158522760956... and c = 0.15945087310540003725148530084775272562567007586487061850065597143186... - Vaclav Kotesovec, Aug 25 2017

A206740 G.f.: 1/(1 - x/(1 - x^3/(1 - x^6/(1 - x^10/(1 - x^15/(1 - x^21/(1 -...- x^(n*(n+1)/2)/(1 -...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 9, 13, 20, 30, 44, 66, 99, 147, 220, 329, 490, 732, 1095, 1634, 2440, 3646, 5444, 8130, 12146, 18139, 27089, 40463, 60434, 90258, 134811, 201349, 300721, 449153, 670844, 1001939, 1496467, 2235080, 3338227, 4985868, 7446739, 11122179

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 6*x^7 + 9*x^8 +...

Cf. A206739.

nmax = 50; CoefficientList[Series[1/Fold[(1 - #2/#1) &, 1, Reverse[x^(Range[nmax + 1]*(Range[nmax + 1] + 1)/2)]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2017 *)

{a(n)=local(CF=1+x*O(x^n),M=sqrtint(n+1)); for(k=0, M, CF=1/(1-x^((M-k+1)*(M-k+2)/2)*CF)); polcoeff(CF, n, x)}
for(n=0,55,print1(a(n),", "))

G.f.: 1/Q(0) , where Q(k) = 1 - x^((2*k+1)*(2*k+2)/2)/(1 - x^((2*k+2)*(2*k+3)/2)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 10 2013

a(n) ~ c * d^n, where d = 1.49356638691558702616975760297981328... and c = 0.35853801643147450974166770910994348... - Vaclav Kotesovec, Aug 24 2017

A285408 Expansion of 1/(1 + x/(1 + x^4/(1 + x^9/(1 + x^16/(1 + x^25/(1 + ... + x^(k^2)/(1 + ...))))))), a continued fraction.

Original entry on oeis.org

1, -1, 1, -1, 1, 0, -1, 2, -3, 3, -2, 0, 3, -6, 7, -6, 2, 5, -12, 17, -17, 9, 6, -24, 40, -45, 32, -1, -44, 89, -112, 97, -34, -72, 189, -272, 273, -153, -84, 380, -637, 723, -526, 22, 703, -1427, 1824, -1593, 575, 1126, -3041, 4423, -4461, 2562, 1251, -6096

Offset: 0

Ilya Gutkovskiy, Apr 18 2017

sign

			G.f.: A(x) = 1 - x + x^2 - x^3 + x^4 - x^6 + 2*x^7 - 3*x^8 + 3*x^9 - 2*x^10 +  ...

Cf. A000290, A007325, A206739.

nmax = 55; CoefficientList[Series[1/(1 + ContinuedFractionK[x^k^2, 1, {k, 1, nmax}]), {x, 0, nmax}], x]

A290771 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the continued fraction 1/(1 - x/(1 - x^(2^k)/(1 - x^(3^k)/(1 - x^(4^k)/(1 - x^(5^k)/(1 - ...)))))).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 1, 2, 14, 1, 1, 1, 1, 3, 42, 1, 1, 1, 1, 1, 5, 132, 1, 1, 1, 1, 1, 2, 9, 429, 1, 1, 1, 1, 1, 1, 3, 15, 1430, 1, 1, 1, 1, 1, 1, 1, 4, 26, 4862, 1, 1, 1, 1, 1, 1, 1, 1, 5, 45, 16796, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 78, 58786, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 10, 135, 208012

Offset: 0

Ilya Gutkovskiy, Aug 10 2017

			Square array begins:
   1,  1,  1,  1,  1,  1, ...
   1,  1,  1,  1,  1,  1, ...
   2,  1,  1,  1,  1,  1, ...
   5,  2,  1,  1,  1,  1, ...
  14,  3,  1,  1,  1,  1, ...
  42,  5,  2,  1,  1,  1, ...

Columns k = 0..5 give A000108, A005169, A206739, A291146, A291149, A291168.

Table[Function[k, SeriesCoefficient[1/(1 + ContinuedFractionK[-x^(i^k), 1, {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: 1/(1 - x/(1 - x^(2^k)/(1 - x^(3^k)/(1 - x^(4^k)/(1 - x^(5^k)/(1 - ...)))))), a continued fraction.

A290975 Expansion of 1 - x/(1 - x^4/(1 - x^9/(1 - x^16/(1 - x^25/(1 - x^36/(1 - ... - x^(n^2)/(1 - ...))))))), a continued fraction.

Original entry on oeis.org

1, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, -1, 0, 0, -1, -2, 0, 0, -1, -3, -1, 0, -1, -4, -3, 0, -1, -6, -6, -1, -1, -8, -10, -4, -1, -10, -17, -10, -2, -12, -27, -20, -6, -15, -40, -38, -16, -19, -56, -68, -36, -27, -79, -114, -75, -45, -109, -180, -147, -84

Offset: 0

Seiichi Manyama, Aug 16 2017

sign

			G.f. = 1 - x - x^5 - x^9 - x^13 - x^14 - x^17 - 2*x^18 - ...

Cf. A206739.

Convolution inverse of A206739.

A291146 Expansion of 1/(1 - x/(1 - x^8/(1 - x^27/(1 - x^64/(1 - x^125/(1 - x^216/(1 - ... - x^(n^3)/(1 - ...)))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 23, 29, 36, 44, 53, 64, 78, 96, 119, 148, 184, 228, 281, 345, 423, 519, 639, 788, 973, 1202, 1484, 1830, 2254, 2774, 3415, 4206, 5183, 6390, 7880, 9717, 11979, 14762, 18188, 22408, 27609, 34022, 41931

Offset: 0

Seiichi Manyama, Aug 18 2017

nonn

			G.f. = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 3*x^10 + 4*x^11 + 5*x^12 + ...

Column k=3 of A290771.

Cf. A005169, A206739.

A295073 Expansion of 1/(1 - x/(1 - x^5/(1 - x^14/(1 - x^30/(1 - x^55/(1 - ... - x^(k(k+1)(2*k+1)/6)/(1 - ...))))))), a continued fraction.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 34, 45, 60, 80, 107, 142, 188, 249, 330, 439, 584, 776, 1030, 1366, 1813, 2408, 3199, 4249, 5642, 7490, 9944, 13204, 17534, 23285, 30920, 41056, 54514, 72384, 96116, 127631, 169478, 225042, 298819, 396783, 526869, 699608, 928981, 1233552

Offset: 0

Ilya Gutkovskiy, Nov 13 2017

nonn

Cf. A000330, A206739, A285408, A295072.

nmax = 53; CoefficientList[Series[1/(1 + ContinuedFractionK[-x^(k (k + 1) (2 k + 1)/6), 1, {k, 1, nmax}]), {x, 0, nmax}], x]

G.f.: 1/(1 - x/(1 - x^5/(1 - x^14/(1 - x^30/(1 - x^55/(1 - ... - x^A000330(k)/(1 - ...))))))), a continued fraction.

a(n) ~ c * d^n, where d = 1.327852426419013789340602526081665378868516025761586390361772232517175463... and c = 0.366619510178622647108505347089605503045273798338613615745637268621... - Vaclav Kotesovec, Sep 18 2021

cp's OEIS Frontend

A285407 G.f.: 1/(1 - x^2/(1 - x^3/(1 - x^5/(1 - x^7/(1 - x^11/(1 - ... - x^prime(k)/(1 - ... ))))))), a continued fraction.

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A206740 G.f.: 1/(1 - x/(1 - x^3/(1 - x^6/(1 - x^10/(1 - x^15/(1 - x^21/(1 -...- x^(n*(n+1)/2)/(1 -...))))))), a continued fraction.

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A285408 Expansion of 1/(1 + x/(1 + x^4/(1 + x^9/(1 + x^16/(1 + x^25/(1 + ... + x^(k^2)/(1 + ...))))))), a continued fraction.

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A290771 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of the continued fraction 1/(1 - x/(1 - x^(2^k)/(1 - x^(3^k)/(1 - x^(4^k)/(1 - x^(5^k)/(1 - ...)))))).

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A290975 Expansion of 1 - x/(1 - x^4/(1 - x^9/(1 - x^16/(1 - x^25/(1 - x^36/(1 - ... - x^(n^2)/(1 - ...))))))), a continued fraction.

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A291146 Expansion of 1/(1 - x/(1 - x^8/(1 - x^27/(1 - x^64/(1 - x^125/(1 - x^216/(1 - ... - x^(n^3)/(1 - ...)))))))), a continued fraction.

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A295073 Expansion of 1/(1 - x/(1 - x^5/(1 - x^14/(1 - x^30/(1 - x^55/(1 - ... - x^(k*(k+1)*(2*k+1)/6)/(1 - ...))))))), a continued fraction.

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A295073 Expansion of 1/(1 - x/(1 - x^5/(1 - x^14/(1 - x^30/(1 - x^55/(1 - ... - x^(k(k+1)(2*k+1)/6)/(1 - ...))))))), a continued fraction.