A038608 -id:A038608 - cp's OEIS frontend

A166711 Permutation of the integers: two positives, one negative.

0, 1, 2, -1, 3, 4, -2, 5, 6, -3, 7, 8, -4, 9, 10, -5, 11, 12, -6, 13, 14, -7, 15, 16, -8, 17, 18, -9, 19, 20, -10, 21, 22, -11, 23, 24, -12, 25, 26, -13, 27, 28, -14, 29, 30, -15, 31, 32, -16, 33, 34, -17, 35, 36, -18, 37, 38, -19, 39, 40, -20, 41, 42, -21, 43, 44, -22, 45, 46

Offset: 0

Jaume Oliver Lafont, Oct 18 2009

Setting m=2 in
log(m) = Sum_{n>0} (n mod m - (n-1) mod m)/n [1]
yields the sum
log(2) = (1 -1/2) +(1/3 -1/4) +(1/5 -1/6)+...
Substituting every -1/d by 1/d - 2/d we obtain
log(2) = (1+1/2-1)+(1/3+1/4-1/2)+(1/5+1/6-1/3)+...
a(n) is the sequence of denominators of this modified sum with unit numerators, so
Sum_{k>0} 1/a(k) = log(2)
Substituting -1/d by -2/d + 1/d would yield another permutation (one positive, one negative, one positive) with the same sum of inverses.
Similar sequences (m positives, one negative) may be obtained for the logarithm of any integer m>0. A001057 is the case m=1, with sum of inverses log(1).
Equation [1] is a result of expanding log( Sum_{0<=k<=m-1} x^k ) at x=1 (see comment to A061347.)

G. C. Greubel, Table of n, a(n) for n = 0..10000
Wikipedia, Riemann series theorem
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A001057, A002162, A038608. Signed and shifted version of A009947.

LinearRecurrence[{0, 0, 2, 0, 0, -1}, {0, 1, 2, -1, 3, 4}, 100] (* G. C. Greubel, May 24 2016 *)
Join[{0},With[{nn=50},Riffle[Range[nn],Range[-1,-nn/2,-1],3]]] (* Harvey P. Dale, May 15 2023 *)

PARI
```
a(n)=(2*(n+1)\3)*(1-3/2*!(n%3))
```

a(n)=if(n>=0,[ -n\3, 2*(n\3)+1, 2*(n\3)+2][n%3+1]) \\ Jaume Oliver Lafont, Nov 14 2009

G.f.: (x*(1+2*x-x^2+x^3)/((1-x)^2*(1+x+x^2)^2)).
a(0)=0, a(1)=1, a(2)=2, a(3)=-1, a(4)=3, a(5)=4, a(n)=2*a(n-3)-a(n-6), n>=6.
a(n) = (n+1)/3 +2*A049347(n)/3 -(-1)^n*A076118(n+1). - R. J. Mathar, Oct 30 2009

Corrected by Jaume Oliver Lafont, Oct 22 2009

frac keyword removed by Jaume Oliver Lafont, Nov 02 2009

A097141 Expansion of x(1+2x)/(1+x)^2.

Original entry on oeis.org

0, 1, 0, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59, 60

Offset: 0

Paul Barry, Jul 29 2004

Partial sums of A097140.
Binomial transform is x(1+x)/(1-x), or {0,1,2,2,2,2,....}.
Second binomial transform is x/((1-x)^2(1 - 2x)), or Eulerian numbers A000295(n+1).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Cf. A000295, A038608, A040000, A097140, A099570.

[0] cat [(n-2)*(-1)^n : n in [1..100]]; // Wesley Ivan Hurt, Dec 11 2016

A097141:=n->(n-2)*(-1)^n: 0, seq(A097141(n), n=1..100); # Wesley Ivan Hurt, Dec 11 2016

CoefficientList[Series[x (1 + 2 x)/(1 + x)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Mar 11 2014 *)

a(n)=if(n, (n-2)*(-1)^n, 0) \\ Charles R Greathouse IV, Dec 13 2016

G.f.: x*(1+2*x)/(1+x)^2.
a(n) = (n-2)*(-1)^n + 2*0^n.
a(n) = -2*a(n-1) - a(n-2) for n > 2.
a(n) = A099570(n) for n > 1. - R. J. Mathar, Dec 15 2008
a(n) = (Sum_{k=1..n} k*(-1)^(n-k)*binomial(n-1,k-1)*binomial(2*n-k-1,n-1))/n, n>0, a(0)=0. - Vladimir Kruchinin, Mar 09 2014
a(n) = A038608(n-2) for n > 2. - Georg Fischer, Oct 06 2018
E.g.f.: 2 - exp(-x)*(2 + x). - Stefano Spezia, Mar 07 2023

A099570 Expansion of ((1+x)^2 - x^3)/(1+x)^2.

Original entry on oeis.org

1, 0, 0, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59, 60, -61, 62, -63, 64, -65, 66, -67, 68, -69, 70, -71, 72, -73

Offset: 0

Paul Barry, Oct 22 2004

Row sums of number triangle A099569.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Cf. A038608, A099569.

[1,0] cat [(-1)^n*(n-2): n in [2..100]]; // G. C. Greubel, Jul 25 2022

Table[(-1)^n*(n-2) -Boole[n==1]+3*Boole[n==0], {n,0,100}] (* G. C. Greubel, Jul 25 2022 *)

[(-1)^n*(n-2) -bool(n==1) +3*bool(n==0) for n in (0..100)] # G. C. Greubel, Jul 25 2022

a(n) = 2*0^n + (-1)^n*(n-2) - Sum_{k=0..n} k*binomial(n, k)*(-1)^(n-k).
a(n) = -2*a(n-1) - a(n-2), n>3, with a(0) = 1, a(1) = 0, a(2) = 0, a(3) = -1.
From G. C. Greubel, Jul 25 2022: (Start)
a(n) = (-1)^n*(n-2) - [n=1] + 3*[n=0].
a(n) = A038608(n-2), for n >= 2, with a(0) = 1, a(1) = 0.
E.g.f.: 3 - x - (x+2)*exp(-x). (End)

A076040 a(n) = (-1)^n * (3^n - 1)/2.

Original entry on oeis.org

0, -1, 4, -13, 40, -121, 364, -1093, 3280, -9841, 29524, -88573, 265720, -797161, 2391484, -7174453, 21523360, -64570081, 193710244, -581130733, 1743392200, -5230176601, 15690529804, -47071589413, 141214768240, -423644304721, 1270932914164

Offset: 0

M. F. Hasler, Oct 21 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-4,-3).

Cf. A003462, A033999.

Table[(-1)^n*(3^n -1)/2, {n, 0, 30}] (* G. C. Greubel, Jun 18 2021 *)

concat(0, Vec(-x/((x+1)*(3*x+1)) + O(x^100))) \\ Colin Barker, Oct 22 2014

[(-1)^n*(3^n -1)/2 for n in (0..30)] # G. C. Greubel, Jun 18 2021

Equals A038608 o A003462 =  A033999 * A003462, i.e., a(n) = (-1)^n*A003462(n) = (-1)^A003462(n)*A003462(n) = A038608(A003462(n)). - M. F. Hasler, Oct 21 2014
From Colin Barker, Oct 22 2014: (Start)
a(n) = -4*a(n-1) - 3*a(n-2).
G.f.: -x / ((1+x)*(1+3*x)). (End)
E.g.f.: (-1)*exp(-2*x)*sinh(x). - G. C. Greubel, Jun 18 2021

Former duplicate of A003462 changed to the signed variant by M. F. Hasler, Oct 21 2014

A062534 Table by antidiagonals of coefficient of x^k in expansion of 1/((1+x)^2*(1-x)^n).

Original entry on oeis.org

1, -2, 1, 3, -1, 1, -4, 2, 0, 1, 5, -2, 2, 1, 1, -6, 3, 0, 3, 2, 1, 7, -3, 3, 3, 5, 3, 1, -8, 4, 0, 6, 8, 8, 4, 1, 9, -4, 4, 6, 14, 16, 12, 5, 1, -10, 5, 0, 10, 20, 30, 28, 17, 6, 1, 11, -5, 5, 10, 30, 50, 58, 45, 23, 7, 1, -12, 6, 0, 15, 40, 80, 108, 103, 68, 30, 8, 1, 13, -6, 6, 15, 55, 120, 188, 211, 171, 98, 38, 9, 1, -14, 7, 0, 21, 70, 175

Offset: 0

Henry Bottomley, Jun 25 2001

Rows are effectively (with minor adjustments): A038608, A001057, A027656, A008805, A006918, A002624, A028346. Cf. A058394 which (adjusting for signs and an overlap of three rows) is effectively the continuation of this table for negative n.

Each row is partial sum of preceding row, i.e. T(n, k)=T(n-1, k)+T(n, k-1) with T(0, k)=(k+1)*(-1)^k and T(n, 0)=1.

A176624 a(n) = prime(n) + n*(-1)^n.

Original entry on oeis.org

1, 5, 2, 11, 6, 19, 10, 27, 14, 39, 20, 49, 28, 57, 32, 69, 42, 79, 48, 91, 52, 101, 60, 113, 72, 127, 76, 135, 80, 143, 96, 163, 104, 173, 114, 187, 120, 201, 128, 213, 138, 223, 148, 237, 152, 245, 164, 271, 178, 279, 182, 291, 188, 305, 202, 319, 212, 329, 218

Offset: 1

Juri-Stepan Gerasimov, Apr 22 2010

nonn

			a(1)=1 because prime(1) + 1*(-1)^1 = 1.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000040, A038608, A176628.

[NthPrime(n) +(-1)^n*n: n in [1..60]]; // G. C. Greubel, Jul 01 2021

A176624 := proc(n) ithprime(n)+n*(-1)^n ; end proc: seq(A176624(n),n=1..120) ; # R. J. Mathar, Apr 27 2010

Table[Prime[n]+(-1)^n*n, {n,60}] (* G. C. Greubel, Jul 01 2021 *)

[nth_prime(n) +(-1)^n*n for n in (1..60)] # G. C. Greubel, Jul 01 2021

a(n) = A000040(n) + A038608(n+1).

Entries checked by R. J. Mathar, Apr 27 2010

A195084 a(2n-1) = 2-n, a(2n) = 2+n.

Original entry on oeis.org

2, 1, 3, 0, 4, -1, 5, -2, 6, -3, 7, -4, 8, -5, 9, -6, 10, -7, 11, -8, 12, -9, 13, -10, 14, -11, 15, -12, 16, -13, 17, -14, 18, -15, 19, -16, 20, -17, 21, -18, 22, -19, 23, -20, 24, -21, 25, -22, 26, -23, 27, -24, 28, -25, 29, -26, 30, -27, 31, -28, 32, -29, 33

Offset: 0

Dave Durgin, Sep 08 2011

Start with a(0)=2, subtract 1, add 2, subtract 3, add 4, subtract 5 and so on.

A permutation of all integers. - Ruud H.G. van Tol, Sep 21 2024

Index entries for sequences that are permutations of the integers
Index entries for linear recurrences with constant coefficients, signature (-1,1,1).

Cf. A001057, A168230.

From Bruno Berselli, Sep 12 2011: (Start)
G.f.: (2*x^2+3*x+2)/((1-x)*(1+x)^2).
a(n) = a(-n-1) = -((2*n+1)*a(n-1)-7*n)/(2*n-1) = -a(n-1)+a(n-2)+a(n-3).
a(n) = ((2*n+1)*(-1)^n+7)/4.
a(n) = 2 - A001057(n).
a(n)-a(n-1) = A038608(n); a(n)+a(n-1) = A010702(n-1).
Sum(n=1..n, a(i)) = ((n+1)*(-1)^n+7*n-1)/4, i.e. A016777 and A008586 (>0) alternately. (End)
a(n+2) = a(n) + (-1)^n. - Vincenzo Librandi, Sep 12 2011
E.g.f.: ((4 - x)*cosh(x) + (3 + x)*sinh(x))/2. - Stefano Spezia, Sep 22 2024

Definition corrected by Omar E. Pol, Sep 11 2011

a(0)=2 prepended by Ruud H.G. van Tol, Sep 21 2024

A274922 a(n) = (-1)^n * n if n>0, a(0) = 1.

Original entry on oeis.org

1, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59

Offset: 0

Michael Somos, Dec 28 2016

This is a divisibility sequence.

			G.f. = 1 - x + 2*x^2 - 3*x^3 + 4*x^4 - 5*x^5 + 6*x^6 - 7*x^7 + 8*x^8 + ...

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

Cf. A004277, A005408, A028310, A060576, A106510.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + x+x^2)/(1+2*x+x^2))); // G. C. Greubel, Jul 29 2018

a[ n_] := If[ n < 1, Boole[n == 0], (-1)^n n];
a[ n_] := SeriesCoefficient[ (1 + x + x^2) / (1 + 2*x + x^2), {x, 0, n}];
LinearRecurrence[{-2,-1},{1,-1,2},60] (* Harvey P. Dale, Mar 30 2019 *)

PARI
```
{a(n) = if( n<1, n==0, (-1)^n * n)};
```

{a(n) = if( n<0, 0, polcoeff( (1 + x + x^2) / (1 + 2*x + x^2) + x * O(x^n), n))};

Euler transform of length 3 sequence [-1, 2, -1].
a(n) = -b(n) where b() is multiplicative with b(2^e) = -(2^e) if e>0, b(p^e) = p^e otherwise.
E.g.f.: 1 - x * exp(-x).
G.f.: (1 + x + x^2) / (1 + 2*x + x^2).
G.f.: (1 - x) * (1 - x^3) / (1 - x^2)^2.
G.f.: 1 / (1 + x / (1 + x / (1 - x / (1 + x)))).
G.f.: 1 - x / (1 + x)^2 = 1 - x /(1 - x)^2 + 4*x^2 / (1 - x^2)^2.
a(n) = (-1)^n * A028310(n). a(2*n) = A004277(n). a(2*n + 1) = - A005408(n).
Convolution inverse of A106510.
A060576(n) = Sum_{k=0..n} binomial(n, k) * a(k).
A028310(n) = Sum_{k=0..n} binomial(n+1, k+1) * a(k).
a(n) = A038608(n), n>0. - R. J. Mathar, May 25 2020

A374894 Obverse convolution (n)**(n*(-1)^n); see Comments.

Original entry on oeis.org

0, -1, 0, -27, -256, 1875, 0, 252105, 4718592, -55801305, 0, -18415376595, -515978035200, 8479520844795, 0, 5195043369140625, 193953851139686400, -4086470036057634225, 0, -4014187339733760264075, -187280916480000000000000, 4816074142389727429287075, 0

Offset: 0

Clark Kimberling, Sep 14 2024

sign

See A374848 for the definition of obverse convolution and a guide to related sequences.

Cf. A000027, A038608, A374848.

s[n_] := n; t[n_] := (-1)^n n;
u[n_] := Product[s[k] + t[n - k], {k, 0, n}]
Table[u[n], {n, 0, 26}]

A227832 Sum of odd numbers starting with 1, alternating signs (beginning with plus).

Original entry on oeis.org

1, 4, -1, 6, -3, 8, -5, 10, -7, 12, -9, 14, -11, 16, -13, 18, -15, 20, -17, 22, -19, 24, -21, 26, -23, 28, -25, 30, -27, 32, -29, 34, -31, 36, -33, 38, -35, 40, -37, 42, -39, 44, -41, 46, -43, 48, -45, 50, -47, 52, -49

Offset: 1

D.Wilde, Aug 02 2013

1st,3rd,5th (odd terms) increase by 2, 2nd,4th,6th,8th (even terms) decrease by 2 each time.

			(1+3)=4 (4-5)=-1 (-1+7)=6 (6-9)=-3 (-3+11)=8 (8-13)=-5 (-5+15)=10.

Index entries for linear recurrences with constant coefficients, signature (-1,1,1).

Cf. A065164 (absolute values).

a(n) = n*(-1)^n + 2; \\ Michel Marcus, Feb 08 2016

a(n) = -a(n-1) + a(n-2) + a(n-3). - Charles R Greathouse IV, Aug 02 2013

G.f.: x*(2*x^2 + 5*x + 1)/((1-x)*(1+x)^2). a(n) = n*(-1)^n + 2 = A038608(n) + 2. - Ralf Stephan, Aug 07 2013

cp's OEIS Frontend

A166711 Permutation of the integers: two positives, one negative.

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A097141 Expansion of x*(1+2*x)/(1+x)^2.

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A099570 Expansion of ((1+x)^2 - x^3)/(1+x)^2.

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A076040 a(n) = (-1)^n * (3^n - 1)/2.

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A062534 Table by antidiagonals of coefficient of x^k in expansion of 1/((1+x)^2*(1-x)^n).

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A176624 a(n) = prime(n) + n*(-1)^n.

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A195084 a(2n-1) = 2-n, a(2n) = 2+n.

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A097141 Expansion of x(1+2x)/(1+x)^2.