A266539 -id:A266539 - cp's OEIS frontend

A266540 Partial sums of A266539.

0, 0, 1, 2, 3, 4, 7, 10, 11, 12, 15, 18, 23, 28, 35, 42, 43, 44, 47, 50, 55, 60, 67, 74, 83, 92, 103, 114, 127, 140, 155, 170, 171, 172, 175, 178, 183, 188, 195, 202, 211, 220, 231, 242, 255, 268, 283, 298, 315, 332, 351, 370, 391, 412, 435, 458, 483, 508, 535, 562, 591, 620, 651, 682, 683, 684, 687, 690, 695, 700

Offset: 1

Omar E. Pol, Jan 02 2016

nonn

Also A266535 and twice the terms of A256249 interleaved, or in other words A266535 and A266538 interleaved.
It appears that this sequence has a fractal (or fractal-like) behavior.
First differs from both A266510 and A266530 at a(25), with which it shares infinitely many terms.
For an illustration of initial terms consider the diagram of A256249 in the fourth quadrant of the square grid together with a reflected copy in the second quadrant.
Also the third sequence of Betti numbers of the Lie algebra m_0(n) over Z_2. See the Nikolayevsky-Tsartsaflis paper, pages 2 and 6. Note that a(n) is denoted by b_3(m_0(n)).

Robert Israel, Table of n, a(n) for n = 1..10000
Yuri Nikolayevsky and Ioannis Tsartsaflis, Cohomology of N-graded Lie algebras of maximal class over Z_2, arXiv:1512.87676 [math.RA], (2016); see pp. 2 and 6.
Index entries for sequences related to the Josephus Problem

Cf. A006257 (Josephus problem), A256249, A266535, A266510, A266530, A266538, A266539.

A006257[0]:=0: for n from 1 to 100 do A006257[n]:=(A006257[n-1]+1) mod n +1: end do:
ListTools:-PartialSums([seq(A006257[i]$2,i=0..100)]); # Robert Israel, Jan 13 2016

Join[{0, 0}, Table[{k, k}, {n, 1, 6}, {k, 1, 2^n-1, 2}] // Flatten] // Accumulate (* Jean-François Alcover, Sep 19 2018 *)

a(2n-1) = A266535(n).
a(2n) = 2 * A256249(n-1) = A266538(n-1).
a(n) = (a(n-1) + a(n+1))/2, if n is an odd number greater than 1.
G.f.: (x^3+x^5)/(1-2*x+2*x^3-x^4) - x*(1-x)^(-2)*Sum_{k>=1} 2^k*x^(2^(1+k)). - Robert Israel, Jan 13 2016

A266509 Terms of A256263 repeated.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 3, 3, 1, 1, 3, 3, 5, 5, 7, 7, 1, 1, 3, 3, 5, 5, 7, 7, 5, 5, 11, 11, 17, 17, 15, 15, 1, 1, 3, 3, 5, 5, 7, 7, 5, 5, 11, 11, 17, 17, 15, 15, 5, 5, 11, 11, 17, 17, 23, 23, 29, 29, 35, 35, 41, 41, 31, 31, 1, 1, 3, 3, 5, 5, 7, 7, 5, 5, 11, 11, 17, 17, 15, 15, 5, 5, 11, 11, 17, 17, 23, 23, 29, 29

Offset: 1

Omar E. Pol, Jan 02 2016

First differs from A266529 at a(55), with which it shares infinitely many terms.
First differs from A266539 at a(25), with which it shares infinitely many terms.
For an illustration of initial terms consider the diagram of A256263 in the fourth quadrant of the square grid together with a reflected copy in the second quadrant.

			Written as an irregular triangle in which the row lengths are twice the terms of A011782 the sequence begins:
0,0;
1,1;
1,1,3,3;
1,1,3,3,5,5,7,7;
1,1,3,3,5,5,7,7,5,5,11,11,17,17,15,15;
1,1,3,3,5,5,7,7,5,5,11,11,17,17,15,15,5,5,11,11,17,17,23,23,29,29,35,35,41,41,31,31;
...
Row sums give 0 together with A004171.

Ivan Neretin, Table of n, a(n) for n = 1..8192

Partial sums give A266510.

Cf. A004171, A011782, A256263, A256264, A256265, A266529, A266539.

Riffle[#, #] &@ Flatten@Join[{0}, NestList[Join[#, Range[Length[#] - 1]*6 - 1, {2 #[[-1]] + 1}] &, {1}, 5]] (* Ivan Neretin, Feb 14 2017 *)

A266529 Terms of A160552 repeated.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 3, 3, 1, 1, 3, 3, 5, 5, 7, 7, 1, 1, 3, 3, 5, 5, 7, 7, 5, 5, 11, 11, 17, 17, 15, 15, 1, 1, 3, 3, 5, 5, 7, 7, 5, 5, 11, 11, 17, 17, 15, 15, 5, 5, 11, 11, 17, 17, 19, 19, 21, 21, 39, 39, 49, 49, 31, 31, 1, 1, 3, 3, 5, 5, 7, 7, 5, 5, 11, 11, 17, 17, 15, 15, 5, 5, 11, 11, 17, 17, 19, 19, 21, 21

Offset: 1

Omar E. Pol, Jan 02 2016

First differs from A266509 at a(55), with which it shares infinitely many terms.

First differs from A266539 at a(25), with which it shares infinitely many terms.

			Written as an irregular triangle in which the row lengths are twice the terms of A011782 the sequence begins:
0,0;
1,1;
1,1,3,3;
1,1,3,3,5,5,7,7;
1,1,3,3,5,5,7,7,5,5,11,11,17,17,15,15;
1,1,3,3,5,5,7,7,5,5,11,11,17,17,15,15,5,5,11,11,17,17,19,19,21,21,39,39,49,49,31,31;
...
Row sums give 0 together with A004171.

Ivan Neretin, Table of n, a(n) for n = 1..8192

Partial sums give A266530.

Cf. A004171, A011782, A139250, A160552, A255747, A266509, A266539.

Riffle[#, #] &@ Table[SeriesCoefficient[x (1 + 2 x)/(1 + x) + (4 x^2/(1 + 2 x)) (Product[1 + x^(2^k - 1) + 2 x^(2^k), {k, 20}] - 1), {x, 0, n}], {n, 0, 44}] (* Michael De Vlieger, Jan 05 2016, based on Maple by N. J. A. Sloane at A160552 *)

A266535 Sums of two successive terms of A256249, with a(0) = 0.

Original entry on oeis.org

0, 1, 3, 7, 11, 15, 23, 35, 43, 47, 55, 67, 83, 103, 127, 155, 171, 175, 183, 195, 211, 231, 255, 283, 315, 351, 391, 435, 483, 535, 591, 651, 683, 687, 695, 707, 723, 743, 767, 795, 827, 863, 903, 947, 995, 1047, 1103, 1163, 1227, 1295, 1367, 1443, 1523, 1607, 1695, 1787, 1883, 1983, 2087, 2195, 2307, 2423, 2543, 2667, 2731

Offset: 0

Omar E. Pol, Jan 02 2016

nonn

Also bisection of A266540.
It appears that this sequence has a fractal-like behavior (see Plot 2, A139250 vs. this sequence).
First differs from both the toothpick sequence A139250 and A256265 at a(12), with which it shares infinitely many terms.

Michel Marcus, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A006257, A139250, A256249, A256265, A266539, A266540.

Most@ # + Rest@ # &@ Accumulate@ Join[{0, 0}, Flatten@ Table[Range[1, 2^n - 1, 2], {n, 0, 6}]] (* Michael De Vlieger, Jan 05 2016, after Ivan N. Ianakiev at A256249 *)

f(n)=n++; b=#binary(n>>1); (4^b-1)/3+(n-2^b)^2; \\ A256249
a(n) = if (n, f(n)+f(n-1), 0);

A289739 Expansion of solution to dy/dx = y + exp(y).

Original entry on oeis.org

0, 1, 2, 5, 17, 79, 474, 3468, 29799, 293528, 3258373, 40234231, 546921835, 8115147998, 130503876054, 2260929219675, 41979302557200, 831593152814251, 17506400133530765, 390278100156698627, 9185223726173708408, 227578002295869672508, 5921091852493279814589

Offset: 0

Michael Somos, Aug 09 2017

nonn

See A266539 for more details.

			E.g.f. = x + 2*x^2/2! + 5*x^3/3! + 17*x^4/4! + ...

Robert Israel, Table of n, a(n) for n = 0..435

Cf. A266329.

S:= dsolve({diff(y(x),x) = y(x) + exp(y(x)), y(0)=0},y(x),series,order=31):
seq(coeff(rhs(S),x,j)*j!,j=0..30); # Robert Israel, Aug 09 2017

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ InverseSeries[ Series[Integrate[ 1 / (x + Exp[x]), x], {x, 0, n}]], {x, 0, n}]];

{a(n) = if( n<0, 0, my(A = O(x)); for(k=1, n, A = intformal(A + exp(A))); n! * polcoeff(A, n))};

{a(n) = if( n<0, 0, n! * polcoeff( serreverse( intformal( 1 / (exp(x + x * O(x^n)) + x))), n))};

E.g.f. y(x) = log(A(x)) and y'(x) = B(x) where A(x), B(x) are as in A266539.

a(n) ~ c^n * (n-1)!, where c = 1/Integral_{x=0..infinity} 1/(x + exp(x)) dx = 1.2400861064984976662394901721056528110217273471501174317019052800276... - Vaclav Kotesovec, Aug 21 2017

cp's OEIS Frontend

A266540 Partial sums of A266539.

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A266509 Terms of A256263 repeated.

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A266529 Terms of A160552 repeated.

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A266535 Sums of two successive terms of A256249, with a(0) = 0.

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A289739 Expansion of solution to dy/dx = y + exp(y).

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