A005590 -id:A005590 - cp's OEIS frontend

A229660 10-section a(10n+k) gives k-th differences of a for k=0..9 with a(n)=0 for n<9 and a(9)=1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -9, 0, 0, 0, 0, 0, 0, 0, 1, -8, 36, 0, 0, 0, 0, 0, 0, 1, -7, 28, -84, 0, 0, 0, 0, 0, 1, -6, 21, -56, 126, 0, 0, 0, 0, 1, -5, 15, -35, 70, -126, 0, 0, 0, 1, -4, 10, -20, 35, -56, 84, 0, 0, 1, -3, 6, -10

Offset: 0

Alois P. Heinz, Sep 27 2013

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A005590, A229653, A229654, A229655, A229656, A229657, A229658, A229659.

a:= proc(n) option remember; local m, q;
      m:= irem(n, 10, 'q'); `if`(n<10, `if`(n=9, 1, 0),
      add(a(q+m-j)*(-1)^j*binomial(m, j), j=0..m))
    end:
seq(a(n), n=0..100);

a(10*n+k) = Sum_{j=0..k} (-1)^j * C(k,j) * a(n+k-j) for k=0..9.

A309048 Expansion of Product_{k>=0} (1 + x^(3^k) + x^(2*3^k) - x^(3^(k+1))).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 1, -1, 0, 0, 1, 1, 1, 0, 1, 1, -1, 0, 0, 1, 1, 1, 0, 1, 1, -2, -1, -1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, -1, 0, 0, 1, 1, 1, 0, 1, 1, -2, -1, -1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, -1, 0, 0, 1, 1, 1, 0, 1, 1, -3, -2, -2, 1, -1, -1, 0, -1, -1, 2, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 0

Ilya Gutkovskiy, Jul 09 2019

sign

Cf. A005590, A054390, A309047.

nmax = 109; CoefficientList[Series[Product[(1 + x^(3^k) + x^(2 3^k) - x^(3^(k + 1))), {k, 0, Floor[Log[3, nmax]] + 1}], {x, 0, nmax}], x]
nmax = 109; A[] = 1; Do[A[x] = (1 + x + x^2 - x^3) A[x^3] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := Switch[Mod[n, 3], 0, a[n/3] - a[(n - 3)/3], 1, a[(n - 1)/3], 2, a[(n - 2)/3]]; Table[a[n], {n, 0, 109}]

G.f. A(x) satisfies: A(x) = (1 + x + x^2 - x^3) * A(x^3).

a(0) = 1; a(3*n) = a(n) - a(n-1), a(3*n+1) = a(n), a(3*n+2) = a(n).

A309020 Expansion of x * Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1)) - x^(2^(k+2))).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 3, 2, 2, 2, 1, 1, 0, 1, 2, 2, 4, 3, 3, 2, 1, 2, 2, 2, 1, 1, 0, 1, 0, 0, 0, 1, 3, 2, 3, 2, 4, 4, 5, 3, 2, 3, 2, 2, 0, 1, 1, 2, 3, 2, 2, 2, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, -1, 0, 0, 0, 1, 1, 4, 3, 4, 2, 2, 3, 3, 2, 3, 4, 6, 4, 5, 5, 4, 3, 0, 2

Offset: 0

Ilya Gutkovskiy, Jul 06 2019

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Cf. A002487, A005590, A309019, A309021, A309022.

nmax = 90; CoefficientList[Series[x Product[(1 + x^(2^k) + x^(2^(k + 1)) - x^(2^(k + 2))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], a[(n - 1)/2] + a[(n + 1)/2] - a[(n - 3)/2]]; Table[a[n], {n, 0, 90}]

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

A309021 Expansion of x * Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1)) - x^(2^(k+2))).

Original entry on oeis.org

0, 1, 1, 0, 1, -2, 0, 0, 1, -3, -2, 1, 0, 2, 0, 1, 1, -4, -3, 0, -2, 6, 1, 1, 0, 1, 2, -2, 0, -1, 1, 0, 1, -6, -4, 0, -3, 7, 0, 1, -2, 8, 6, -3, 1, -6, 1, -2, 0, 0, 1, 1, 2, -5, -2, 0, 0, 1, -1, 2, 1, 0, 0, 0, 1, -7, -6, 1, -4, 10, 0, 1, -3, 10, 7, -4, 0, -6, 1, -3, -2, 9, 8, 0, 6, -17, -3, -2, 1, -4, -6

Offset: 0

Ilya Gutkovskiy, Jul 06 2019

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Cf. A002487, A005590, A309019, A309020, A309022.

nmax = 90; CoefficientList[Series[x Product[(1 + x^(2^k) - x^(2^(k + 1)) - x^(2^(k + 2))), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], a[(n + 1)/2] - a[(n - 1)/2] - a[(n - 3)/2]]; Table[a[n], {n, 0, 90}]

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

A277778 If n is even, a(n) = a(n/2-1), and if n is odd, a(n) = a((n-1)/2) - a((n+1)/2), with a(1) = a(2) = 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, -1, 0, 0, 1, 0, 1, 2, 1, -1, -1, 0, 0, -1, 0, 1, 1, -1, 0, -1, 1, 1, 2, 2, 1, 0, -1, -1, -1, 0, 0, 1, 0, -1, -1, -1, 0, 0, 1, 2, 1, -1, -1, 1, 0, -2, -1, 0, 1, -1, 1, 0, 2, 1, 2, 1, 1, 1, 0, 0, -1, 0, -1, -1, -1, 0, 0, -1, 0, 1, 1, 1, 0, 0

Offset: 1

Tristan Cam, Oct 30 2016

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This sequence grows very slowly, in both positive and negative directions. The first 3 in the list is a(221), the first 12 is a(122333), and the first -13 is a(980851).
First occurrences: a(1) = 1, a(3) = 0, a(7) = -1, a(13) = 2, a(51) = -2, a(221) = 3, a(477) = 4, a(845) = -3, a(1907) = -4, a(3549) = 5, a(7389) = 6, a(7645) = 7, a(13533) = -5, a(27101) = -6, a(30579) = -7, a(56797) = 8, a(61157) = 9, a(117981) = 10, a(122333) = 12, a(216541) = -9, a(216805) = -8, a(236509) = 11, a(245213) = 13, a(433629) = -11, a(471923) = -10, a(489331) = -12, a(978533) = 14, a(978661) = 15, a(980851) = -13, a(1818077) = 16, a(1887709) = 17, a(1957341) = 20, a(3464669) = -15, a(3469029) = -14, a(3755485) = -16, a(3775453) = 18, a(3914701) = 19, a(3914717) = 21, a(3915229) = 22, a(3923421) = 23, a(6938077) = -19, a(7511517) = -17, a(7773661) = -18, a(7829363) = -20, a(15658725) = 25, a(15658867) = -21, a(15660915) = -22, a(15693683) = -23, a(28949981) = 24, a(29089245) = 28, a(29220317) = 27, a(30199005) = 26, a(30203357) = 29, a(31313117) = 30, a(31317469) = 33. - Charles R Greathouse IV, Oct 30 2016
From Robert Israel, Nov 10 2016: (Start)
a(2^k-2) = 1.
a(2^k-1) = A010892(k).
a(2^k) = A010892(k-1).
a(13*(16^k-1)/15) = A000045(k+2) for k >= 1.
Using this, there is c>0 such that a(n) > c n^d for infinitely many n, where d = log_16((1+sqrt(5))/2) = 0.1735604784...
(End)

			The first two terms are a(1) = a(2) = 1. To get the next two terms, subtract the second from the first to get a(3) = a(1) - a(2) = 0 and copy the first term as a(4) = a(1) = 1.
To find a(5) and a(6), start over using a(2) and a(3); then for a(7) and a(8), use a(3) and a(4); and so on.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert G. Wilson v, First occurrence of k.

Cf. A000045, A005590, A010892, A190610.

N:= 200: # to get a(1) .. a(N)
A[1]:= 1: A[2]:= 1:
for j from 1 to (N-1)/2 do
  A[2*j+1]:= A[j] - A[j+1];
  A[2*j+2]:= A[j];
od:
seq(A[i],i=1..N); # Robert Israel, Nov 10 2016

a[n_] := a[n] = If[ OddQ[n], a[(n - 1)/2] - a[(n + 1)/2], a[n/2 - 1]]; a[1] = a[2] = 1; Array[a, 100] (* Robert G. Wilson v, Nov 11 2016 *)

a(n)=if(n<7, return(n!=3)); if(n%2, a(n\2) - a(n\2+1), a(n/2-1)) \\ Charles R Greathouse IV, Oct 30 2016

G.f. satisfies A(x) = (x^2 + x - 1/x) * A(x^2) + 2*x + x^2. - Andrey Zabolotskiy, Oct 30 2016

|a(n)| << n^0.71. - Charles R Greathouse IV, Nov 01 2016

A145865 a(0) = 0, a(1) = 1, a(2n) = a(n), a(2n+1) = a(n) - a(n+1).

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 0, -1, 1, 0, 1, 1, 0, 1, -1, -2, 1, 1, 0, -1, 1, 0, 1, 1, 0, -1, 1, 2, -1, 1, -2, -3, 1, 0, 1, 1, 0, 1, -1, -2, 1, 1, 0, -1, 1, 0, 1, 1, 0, 1, -1, -2, 1, -1, 2, 3, -1, -2, 1, 3, -2, 1, -3, -4, 1, 1, 0, -1, 1, 0, 1, 1, 0, -1, 1, 2, -1, 1, -2, -3, 1, 0, 1, 1, 0, 1, -1, -2, 1, 1, 0

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Oct 22 2008

Variation on Stern's Diatomic Series

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A002487, A005590.

a[0] = 0; a[1] = 1; a[n_] := a[n] = If[EvenQ@ n, a[n/2], a[#] - a[# + 1] &[(n - 1)/2]]; Table[a@ n, {n, 0, 85}] (* Michael De Vlieger, Dec 21 2016 *)

From Chai Wah Wu, Dec 20 2016: (Start)
a(2^k*n+1) = a(n+1) if k is even
a(2^k*n+1) = a(n)-a(n+1) = a(2n+1) if k is odd
a(2^k*n+2^k-1) = a(n) - k*a(n+1)
a(2^k*n+2^k-3) = a(n+1) for k >= 2
a(2^k*n+2^k-5) = (k-1)*a(n+1)-a(n) for k >= 3
a(2^k*n+2^k-7) = a(n) - (k-2)*a(n+1) for k >= 3
This implies that
a(2^k+1) = 1 if k is even
a(2^k+1) = 0 if k is odd
a(2^k-1) = 2 - k for k >= 1
a(2^k-3) = 1 for k >= 2
a(2^k-5) = k - 3 for k >= 3
a(2^k-7) = 4 - k for k >= 3
(End)

A281048 Expansion of x(1 - x)Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1))).

Original entry on oeis.org

1, 0, -1, 1, -2, 1, 1, 0, -3, 1, 2, -1, 1, 0, -1, 1, -4, 1, 3, -2, 3, -1, -2, 1, 1, 0, -1, 1, -2, 1, 1, 0, -5, 1, 4, -3, 5, -2, -3, 1, 4, -1, -3, 2, -3, 1, 2, -1, 1, 0, -1, 1, -2, 1, 1, 0, -3, 1, 2, -1, 1, 0, -1, 1, -6, 1, 5, -4, 7, -3, -4, 1, 7, -2, -5, 3, -4, 1, 3, -2, 5, -1, -4, 3, -5, 2, 3, -1, -4, 1, 3, -2, 3, -1, -2, 1, 1, 0, -1, 1

Offset: 1

Ilya Gutkovskiy, Feb 27 2017

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First differences of A005590.

Michael Gilleland, Some Self-Similar Integer Sequences
Ilya Gutkovskiy, Extended graphical example
R. Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.

Cf. A005590, A070990, A182093.

Rest[CoefficientList[Series[x (1 - x) Product[1 + x^2^k - x^2^(k + 1), {k, 0, 15}], {x, 0, 100}], x]]
Differences[a[0] = 0; a[1] = 1; a[n_] := a[n] = If[OddQ[n], a[(n-1)/2 + 1] - a[(n-1)/2], a[n/2]]; Table[a[n], {n, 0, 100}]]

G.f.: x*(1 - x)*Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1))).

A309044 Expansion of Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1)))^2.

Original entry on oeis.org

1, 2, 1, 2, 0, -2, 3, 2, -1, -4, 0, -4, 5, 10, -3, -2, 0, -6, -1, -6, 3, 8, -8, -8, 9, 18, 1, 10, -8, -26, 11, 2, -1, 4, -8, -12, 5, 10, -11, -10, 8, 18, -1, 10, -13, -32, 8, 0, 9, 34, 1, 18, -8, -34, 27, 18, -17, -36, -8, -36, 29, 74, -35, -18, 8, -6, 7, 10, -13, -24

Offset: 0

Ilya Gutkovskiy, Jul 09 2019

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Cf. A002487, A005590, A309043.

nmax = 69; CoefficientList[Series[Product[(1 + x^(2^k) - x^(2^(k + 1)))^2, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
nmax = 69; A[] = 1; Do[A[x] = (1 + x - x^2)^2 A[x^2] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = (1 + x - x^2)^2 * A(x^2).

a(n) = Sum_{k=0..n} A005590(k+1)*A005590(n-k+1).

A309047 Expansion of Product_{k>=0} (1 + x^(2^k) - x^(3*2^k)).

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 0, -1, 1, 1, 0, -1, 0, 0, -1, -1, 1, 2, 1, 0, 0, -1, -1, -1, 0, 1, 0, 0, -1, -1, -1, 0, 1, 2, 2, 1, 1, -1, 0, -1, 0, 0, -1, -1, -1, 0, -1, 0, 0, 1, 1, 1, 0, -1, 0, 0, -1, -1, -1, 0, -1, 0, 0, 1, 1, 1, 2, 1, 2, 0, 1, -1, 1, 0, -1, -2, 0, 1, -1, -1, 0, 1, 0, 0, -1, -1, -1, 0, -1, 0, 0, 1, -1, -1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, -1, -1, -1, 0, 1

Offset: 0

Ilya Gutkovskiy, Jul 09 2019

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Cf. A002487, A005590, A120562.

nmax = 109; CoefficientList[Series[Product[(1 + x^(2^k) - x^(3 2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
nmax = 109; A[] = 1; Do[A[x] = (1 + x - x^3) A[x^2] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = 1; a[n_] := If[EvenQ[n], a[n/2], a[(n - 1)/2] - a[(n - 3)/2]]; Table[a[n], {n, 0, 109}]

G.f. A(x) satisfies: A(x) = (1 + x - x^3) * A(x^2).

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

A325055 a(0) = 0, a(1) = 1; a(2n) = a(n-1) + a(n), a(2n+1) = a(n+1) - a(n).

Original entry on oeis.org

0, 1, 1, 0, 2, -1, 1, 2, 2, -3, 1, 2, 0, 1, 3, 0, 4, -5, -1, 4, -2, 1, 3, -2, 2, 1, 1, 2, 4, -3, 3, 4, 4, -9, -1, 4, -6, 5, 3, -6, 2, 3, -1, 2, 4, -5, 1, 4, 0, -1, 3, 0, 2, 1, 3, 2, 6, -7, 1, 6, 0, 1, 7, 0, 8, -13, -5, 8, -10, 5, 3, -10, -2, 11, -1, -2

Offset: 0

Ilya Gutkovskiy, Sep 04 2019

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Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000

Cf. A001196 (positions of 0's), A002487, A005590, A075825.

a[0] = 0; a[1] = 1; a[n_] := a[n] = If[EvenQ[n], a[(n - 2)/2] + a[n/2], a[(n + 1)/2] - a[(n - 1)/2]]; Table[a[n], {n, 0, 75}]

a(n) = Sum_{k=1..n} a(2*k-1) = Sum_{k=1..n} (-1)^(n-k) * a(2*k).

a(2^k) = 2^floor(k/2).

cp's OEIS Frontend

A229660 10-section a(10n+k) gives k-th differences of a for k=0..9 with a(n)=0 for n<9 and a(9)=1.

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A309048 Expansion of Product_{k>=0} (1 + x^(3^k) + x^(2*3^k) - x^(3^(k+1))).

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A309020 Expansion of x * Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1)) - x^(2^(k+2))).

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A309021 Expansion of x * Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1)) - x^(2^(k+2))).

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A277778 If n is even, a(n) = a(n/2-1), and if n is odd, a(n) = a((n-1)/2) - a((n+1)/2), with a(1) = a(2) = 1.

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A145865 a(0) = 0, a(1) = 1, a(2n) = a(n), a(2n+1) = a(n) - a(n+1).

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A281048 Expansion of x*(1 - x)*Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1))).

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A309044 Expansion of Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1)))^2.

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A309047 Expansion of Product_{k>=0} (1 + x^(2^k) - x^(3*2^k)).

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A281048 Expansion of x(1 - x)Product_{k>=0} (1 + x^(2^k) - x^(2^(k+1))).

A325055 a(0) = 0, a(1) = 1; a(2n) = a(n-1) + a(n), a(2n+1) = a(n+1) - a(n).