A220466 -id:A220466 - cp's OEIS frontend

A082392 Expansion of (1/x) * Sum_{k>=0} x^2^k / (1 - 2*x^2^(k+1)).

1, 1, 2, 1, 4, 2, 8, 1, 16, 4, 32, 2, 64, 8, 128, 1, 256, 16, 512, 4, 1024, 32, 2048, 2, 4096, 64, 8192, 8, 16384, 128, 32768, 1, 65536, 256, 131072, 16, 262144, 512, 524288, 4, 1048576, 1024, 2097152, 32, 4194304, 2048, 8388608, 2, 16777216

Offset: 0

Ralf Stephan, Jun 07 2003

Seiichi Manyama, Table of n, a(n) for n = 0..5000
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions

Cf. A045654, A220466.

nmax := 48: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := 2^n od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Feb 11 2013
A082392 := proc(n)
    2^A025480(n) ;
end proc:
seq(A082392(n),n=0..100) ; # R. J. Mathar, Jul 16 2020

a[n_] := 2^(((n+1)/2^IntegerExponent[n+1, 2]+1)/2-1);
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 15 2023 *)

for(n=0, 50, l=ceil(log(n+1)/log(2)); t=polcoeff(sum(k=0, l, (x^2^k)/(1-2*x^2^(k+1)))/x + O(x^(n+1)), n); print1(t", ");) ;

a(0) = 1, a(2*n) = 2^n, a(2*n+1) = a(n).
a(n) = 2^A025480(n) = 2^(A003602(n)-1).
a((2*n+1)*2^p-1) = 2^n, p >= 0 and n >= 0. - Johannes W. Meijer, Feb 11 2013

A059134 A hierarchical sequence (W2{3}c - see A059126).

Original entry on oeis.org

11, 32, 11, 53, 11, 32, 11, 74, 11, 32, 11, 53, 11, 32, 11, 95, 11, 32, 11, 53, 11, 32, 11, 74, 11, 32, 11, 53, 11, 32, 11, 116, 11, 32, 11, 53, 11, 32, 11, 74, 11, 32, 11, 53, 11, 32, 11, 95, 11, 32, 11, 53, 11, 32, 11, 74, 11, 32, 11, 53, 11, 32, 11, 137, 11, 32, 11, 53, 11

Offset: 0

Jonas Wallgren, Jan 19 2001

Jonas Wallgren, Hierarchical sequences

Cf. A059126, A059127.

Cf. A059141 (7*p+11), A059144 (9*p+15), A220466. - Johannes W. Meijer, Jan 21 2013

nmax:=68: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := 21*p+11 od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 21 2013

a((2*n+1)*2^p-1) = 21*p+11 for p >= 0. - Johannes W. Meijer, Jan 21 2013

a(n) = Sum_{k=0..6} A059127(7*n+k). - Sean A. Irvine, Sep 12 2022

A048460 Total of odd numbers in the generations from 2 onwards.

Original entry on oeis.org

2, 3, 3, 3, 4, 6, 5, 3, 4, 6, 6, 6, 8, 12, 9, 3, 4, 6, 6, 6, 8, 12, 10, 6, 8, 12, 12, 12, 16, 24, 17, 3, 4, 6, 6, 6, 8, 12, 10, 6, 8, 12, 12, 12, 16, 24, 18, 6, 8, 12, 12, 12, 16, 24, 20, 12, 16, 24, 24, 24, 32, 48, 33, 3, 4, 6, 6, 6, 8, 12, 10, 6, 8, 12, 12, 12, 16, 24, 18, 6, 8, 12, 12, 12, 16

Offset: 2

Patrick De Geest, May 15 1999

nonn

			a(7)=6 because in generation 7 there are six odd numbers: 127,237,403,729,879,1109.

For "Generations" see A048448-A048455. See also A047844.

Cf. A220466.

A048460 := proc(nmax) local par, c, r, prevc, prevl, cpar; par := [[],[1,1]] ; for c from 3 to nmax do prevc := op(-1,par) ; prevl := nops(prevc) ; if nops(prevc) < 2 then cpar := [0] ; else cpar := [op(2,prevc)] ; end if; for r from 2 to prevl-1 do cpar := [op(cpar),( op(r-1,prevc) + op(r+1,prevc)) mod 2] ; end do: cpar := [op(cpar), op(prevl-1,prevc),1] ; par := [op(par),cpar] ; end do: cpar := [] ; for c from 2 to nops(par) do add(r,r=op(c,par)) ; cpar := [op(cpar),%] ; end do: cpar ; end proc: A048460(120) ; # R. J. Mathar, Aug 07 2010
nmax := 86: A001316 := n -> if n <=- 1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 2 to nmax/(p+2) do a((2*n-3)*2^p) := (2^(p-1)+1)*A001316(n-2) od: od: seq(a(n), n=2..nmax); # Johannes W. Meijer, Jan 22 2013

A105321[n_] := Sum[Binomial[1, n-k] Mod[Binomial[k, j], 2], {k, 0, n}, {j, 0, k}];
a[n_] := A105321[n]/2;
Table[a[n], {n, 2, 86}] (* Jean-François Alcover, Oct 25 2023 *)

def A048460(n): return (1<>1 # Chai Wah Wu, Jul 30 2025

It appears that a(n) = A105321(n)/2. - Omar E. Pol, May 29 2010. Proof from Nathaniel Johnston, Nov 07 2010: If you remove every 2nd row from Pascal's triangle then the rule for constructing the parity of the next row from the current row is the same as the rule for constructing generation n+1 of the primes from generation n: add up the previous and next term in the current row.

a((2*n-3)*2^p) = (2^(p-1)+1)*A001316(n-2), p >= 0 and n >= 2. - Johannes W. Meijer, Jan 22 2013

More terms from R. J. Mathar, Aug 07 2010

A065916 Denominator of sigma(8n^2)/sigma(4n^2).

Original entry on oeis.org

7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 2047, 7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 8191, 7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 2047, 7, 31, 7, 127, 7, 31, 7, 511, 7, 31, 7, 127, 7, 31, 7, 32767, 7, 31, 7, 127, 7, 31, 7, 511, 7

Offset: 1

Labos Elemer, Nov 28 2001

The sequence is not periodic. The denominators are always of the form -1+2^s.

			sigma(72)/sigma(36) = 15/7, so a(3) = 7.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions.
Index entries for sequences related to binary expansion of n.

Cf. A000203, A007814, A028982, A065915 (numerators), A083420, A220466.

nmax:=73: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 2*4^(p+1) - 1 od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Feb 12 2013

a[n_] := 2^(2*IntegerExponent[n, 2] + 3) - 1; Array[a, 100] (* Amiram Eldar, Jun 21 2024 *)

a(n) = denominator(sigma(8*n^2)/sigma(4*n^2)) \\ Harry J. Smith, Nov 04 2009

a(n)=2^(2*valuation(n,2)+3)-1 \\ Charles R Greathouse IV, Nov 18 2015

From Johannes W. Meijer, Feb 12 2013: (Start)
a((2*n-1)*2^p) = 2*4^(p+1) - 1 for p >= 0 and n >= 1. Observe that a(2^p) = A083420(p+1).
a(2^(p+3)*n + 2^(p+2) - 1) = a(2^(p+2)*n + 2^(p+1) - 1) for p >= 0. (End)
a(n) = 2^s-1, with s = 2*A007814(n) + 3. Recurrence: a(2n) = 4a(n)+3, a(2n+1) = 7. - Ralf Stephan, Aug 22 2013

A089265 a(1) = 0; thereafter a(2n) = a(n) + 1, a(2n+1) = 2*n.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 6, 3, 8, 5, 10, 4, 12, 7, 14, 4, 16, 9, 18, 6, 20, 11, 22, 5, 24, 13, 26, 8, 28, 15, 30, 5, 32, 17, 34, 10, 36, 19, 38, 7, 40, 21, 42, 12, 44, 23, 46, 6, 48, 25, 50, 14, 52, 27, 54, 9, 56, 29, 58, 16, 60, 31, 62, 6, 64, 33, 66, 18, 68, 35, 70, 11, 72

Offset: 1

Ralf Stephan, Oct 30 2003

In the binary representation of n, swallow all zeros from the right, then add the number of swallowed zeros, and subtract 1. - Ralf Stephan, Aug 22 2013

First differences of A005766.

Cf. A003602, A220466.

nmax:=73: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := p  + 2*(n-1) od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 23 2013

a[n_] := With[{v = IntegerExponent[n, 2]}, v + n/2^v - 1];
Array[a, 100] (* Jean-François Alcover, Feb 28 2019 *)

a(n) = valuation(n,2) + n/2^valuation(n,2) - 1

a(1) = 0; thereafter a(2*n) = a(n) + 1, a(2*n+1) = 2*n.
a(n) = A007814(n) + 2*A025480(n-1) = A007814(n) + A000265(n) - 1.
G.f.: sum(k>=0, (t^2+2t^3-t^4)/(1-t^2)^2, t=(x^2)^k).
a((2*n-1)*2^p) = p + 2*(n-1), p >= 0. - Johannes W. Meijer, Jan 23 2013

A162728 G.f.: x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n.

Original entry on oeis.org

1, 3, 2, 8, 4, 6, 6, 20, 6, 12, 10, 16, 12, 18, 8, 48, 16, 18, 18, 32, 12, 30, 22, 40, 20, 36, 18, 48, 28, 24, 30, 112, 20, 48, 24, 48, 36, 54, 24, 80, 40, 36, 42, 80, 24, 66, 46, 96, 42, 60, 32, 96, 52, 54, 40, 120, 36, 84, 58, 64, 60, 90, 36, 256, 48, 60, 66, 128, 44, 72, 70

Offset: 1

Paul D. Hanna, Jul 12 2009

Dirichlet inverse of A117212. - R. J. Mathar, Jul 15 2010

			x/(1-x) = log(1+x) + 3*log(1+x^2)/2 + 2*log(1+x^3)/3 + 8*log(1+x^4)/4 + 4*log(1+x^5)/5 + 6*log(1+x^6)/6 + 6*log(1+x^7)/7 + 20*log(1+x^8)/8 +...

Paul D. Hanna, Table of n, a(n) for n = 1..10000

Cf. A090739, A091512, A000010 (Euler phi), A220466.

Cf. A007814, A001511, A090740, A085058, A055034, A008683.

nmax:=71: with(numtheory): for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := (p+2)*2^(p-1)*phi(2*n-1) od: od: seq(a(n), n=1..nmax); # Johannes W. Meijer, Jan 26 2013

f[p_, e_] := (p-1)*p^(e-1); f[2, e_] := (e+2)*2^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

/* As the inverse Mobius transform of A091512: */
{a(n)=sumdiv(n,d,moebius(n/d)*valuation((2*d)^d,2))}

/* From a(2n-1)=phi(2n-1); a(2n)=phi(2n)*A090739(n), we get: */
{a(n)=if(n%2==1,eulerphi(n),eulerphi(n)*valuation(3^n-1,2))}

/* From x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n, we get: */
{a(n)=local(A=[1]);for(k=1,n,A=concat(A,0);A[ #A]=#A*(1-polcoeff(sum(m=1,#A,A[m]/m*log(1+x^m +x*O(x^#A)) ),#A)));A[n]}

a(2n-1) = phi(2n-1); a(2n) = phi(2n)*A090739(n), where A090739(n) = exponent of 2 in 3^(2n)-1.
Inverse Mobius transform of A091512, where A091512(n) = exponent of 2 in (2n)^n.
Multiplicative: a(m*n) = a(m)*a(n) when gcd(m,n)=1, with a(p) = p-1 for odd prime p and a(2)=3.
G.f.: x/(1-x)^2 = Sum_{n>=1} a(n)*x^n/(1+x^n). - Paul D. Hanna, Jul 12 2009
Dirichlet g.f.: zeta(s-1)/( zeta(s)*(1-2^(1-s)) ). - R. J. Mathar, Apr 14 2011
a((2*n-1)*2^p) = (p+2)*2^(p-1)* phi(2*n-1), p >= 0. Observe that a(2^p) = A001792(p). - Johannes W. Meijer, Jan 26 2013
Sum_{k=1..n} a(k) ~ 6*n^2 / Pi^2. - Vaclav Kotesovec, Feb 07 2019
Multiplicative with a(2^e) = (e+2)*2^(e-1) and a(p^e) = (p-1)*p^(e-1) for an odd prime p. - Amiram Eldar, Aug 27 2023
From Ridouane Oudra, Jul 05 2025: (Start)
a(n) = Sum_{k=0..A007814(n)} 2^k*phi(n/2^k).
a(n) = Sum_{d|n} mu(n/d)*d*A001511(d).
a(n) = A000010(n)*A090740(n).
a(n) = A085058(n-1)*A055034(n), for n>1. (End)

A193365 a(n) = A220371(n)/(4*A220371(n-1)).

Original entry on oeis.org

15, 126, 143, 1020, 399, 1150, 783, 8184, 1295, 3198, 1935, 9212, 2703, 6270, 3599, 65520, 4623, 10366, 5775, 25596, 7055, 15486, 8463, 73720, 9999, 21630, 11663, 50172, 13455, 28798, 15375, 524256, 17423, 36990

Offset: 1

Johannes W. Meijer, Dec 21 2012

This sequence is, via A220371, related to A220002, which is related to the Catalan numbers.

Information about the peculiar structure of the a(n) can be found in A220466.

Cf. A220466

nmax:= 34: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a(2^p*(2*n-1)) := 2^p*(2^(2*p+4)*(2*n-1)^2-1) od: od: seq(a(n), n=1..nmax);

b[n_] := b[n] = 2^(2n) Product[2i+1, {i, 1, 2n}] GCD[n!, 2^n];
a[n_] := b[n]/(4 b[n-1]);
Array[a, 34] (* Jean-François Alcover, Jun 26 2019 *)

def A193365_list(len):
    a = {}; z = 1; s = 0; p = 1
    while s < len:
        i = s; z += z
        while i < len:
            a[i] = p*((4*i+4)^2-1)
            i += z
        s += s + 1; p += p
    return [a[i] for i in range(len)]
A193365_list(30)  # Peter Luschny, Dec 22 2012

a(n) = A220371(n)/(4*A220371(n-1))

a(2^p*(2*n-1)) = 2^p*(2^(2*p+4)*(2*n-1)^2-1), p >= 0.

A151930 First differences of A001316.

Original entry on oeis.org

1, 0, 2, -2, 2, 0, 4, -6, 2, 0, 4, -4, 4, 0, 8, -14, 2, 0, 4, -4, 4, 0, 8, -12, 4, 0, 8, -8, 8, 0, 16, -30, 2, 0, 4, -4, 4, 0, 8, -12, 4, 0, 8, -8, 8, 0, 16, -28, 4, 0, 8, -8, 8, 0, 16, -24, 8, 0, 16, -16, 16, 0, 32, -62, 2, 0, 4, -4, 4, 0, 8, -12, 4, 0, 8, -8, 8, 0, 16, -28, 4, 0, 8, -8, 8, 0, 16, -24, 8, 0, 16, -16, 16, 0, 32

Offset: 0

N. J. A. Sloane, Aug 10 2009

sign

Net increase in number of ON cells at generation n of 1-D CA using Rule 90.

Index entries for sequences related to cellular automata

Cf. A001316, A220466.

nmax := 94: A001316 := n -> if n<=-1 then 0 else 2^add(i, i=convert(n, base, 2)) fi: for p from 0 to ceil(log[2](nmax))+1 do for n from 0 to nmax/(p+2)+1 do a((2*n+1)*2^p-1) := (2-2^p) * A001316(n) od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 25 2013

a((2*n+1)*2^p-1) = (2-2^p) * A001316(n), p >= 0 and n >=0. - Johannes W. Meijer, Jan 25 2013

G.f.: -1/x + ((1 - x)/x)*Product_{k>=0} (1 + 2*x^(2^k)). - Ilya Gutkovskiy, Feb 28 2017

A182241 a(n) = A161151(2*n)/2.

Original entry on oeis.org

3, 10, 7, 36, 11, 26, 15, 136, 19, 42, 23, 100, 27, 58, 31, 528, 35, 74, 39, 164, 43, 90, 47, 392, 51, 106, 55, 228, 59, 122, 63, 2080, 67, 138, 71, 292, 75, 154, 79, 648, 83, 170, 87, 356, 91, 186, 95, 1552, 99

Offset: 1

Johannes W. Meijer, Dec 24 2012

Cf. A220466

nmax:=49: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 1 to ceil(nmax/(p+2)) do a((2*n-1)*2^p) := 2^(2*p+2)*n - 2*2^(2*p) + 2^p od: od: seq(a(n), n=1..nmax);

a(n) = A161151(2*n)/2

a((2*n-1)*2^p) = 2^(2*p+2)*n - 2*2^(2*p) + 2^p, p >= 0.

cp's OEIS Frontend

A082392 Expansion of (1/x) * Sum_{k>=0} x^2^k / (1 - 2*x^2^(k+1)).

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A059134 A hierarchical sequence (W2{3}c - see A059126).

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A048460 Total of odd numbers in the generations from 2 onwards.

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A065916 Denominator of sigma(8*n^2)/sigma(4*n^2).

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A089265 a(1) = 0; thereafter a(2*n) = a(n) + 1, a(2*n+1) = 2*n.

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A162728 G.f.: x/(1-x) = Sum_{n>=1} a(n)*log(1+x^n)/n.

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A193365 a(n) = A220371(n)/(4*A220371(n-1)).

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A065916 Denominator of sigma(8n^2)/sigma(4n^2).

A089265 a(1) = 0; thereafter a(2n) = a(n) + 1, a(2n+1) = 2*n.