A100050 -id:A100050 - cp's OEIS frontend

A100051 A Chebyshev transform of 1,1,1,...

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

Offset: 0

Paul Barry, Oct 31 2004

1, followed by period 6: repeat [1, -1, -2, -1, 1, 2]. - Joerg Arndt, Aug 28 2024
A Chebyshev transform of 1/(1-x): if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)).
Transform of 1/(1+x) under the mapping g(x)->((1+x)/(1-x))g(x/(1-x)^2). - Paul Barry, Dec 01 2004
Multiplicative with a(p^e) = -1 if p = 2; -2 if p = 3; 1 otherwise. - David W. Wilson, Jun 10 2005

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (1,-1).

Cf. A011655, A057079, A087204, A099837, A099443, A100047, A100048, A100050.

Row sums of array A127677.

CoefficientList[Series[(1 - x^2)/(1 - x + x^2), {x,0,50}], x] (* G. C. Greubel, May 03 2017 *)
LinearRecurrence[{1,-1},{1,1,-1},80] (* Harvey P. Dale, Mar 25 2019 *)

{a(n) = - (n == 0) + [2, 1, -1, -2, -1, 1][n%6 + 1]}; /* Michael Somos, Mar 21 2011 */

From Paul Barry, Dec 01 2004: (Start)
G.f.: (1-x^2)/(1-x+x^2).
a(n) = a(n-1) - a(n-2), n>2.
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)/(n-k).
a(n) = Sum_{k=0..n} binomial(n+k, 2k)*(2n/(n+k))*(-1)^k, n>1. (End)
Moebius transform is length 6 sequence [1, -2, -3, 0, 0, 6].
Euler transform of length 6 sequence [1, -2, -1, 0, 0, 1].
a(n) = a(-n). a(n) = c_6(n) if n>1 where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
a(n) = A087204(n), n>0. - R. J. Mathar, Sep 02 2008
a(n) = A057079(n+1), n>0. Dirichlet g.f. zeta(s) *(1-2^(1-s)-3^(1-s)+6^(1-s)). - R. J. Mathar, Apr 11 2011

A091684 a(n) = 0 if n is divisible by 3, otherwise a(n) = n.

Original entry on oeis.org

0, 1, 2, 0, 4, 5, 0, 7, 8, 0, 10, 11, 0, 13, 14, 0, 16, 17, 0, 19, 20, 0, 22, 23, 0, 25, 26, 0, 28, 29, 0, 31, 32, 0, 34, 35, 0, 37, 38, 0, 40, 41, 0, 43, 44, 0, 46, 47, 0, 49, 50, 0, 52, 53, 0, 55, 56, 0, 58, 59, 0, 61, 62, 0, 64, 65, 0, 67, 68, 0, 70, 71, 0, 73, 74, 0, 76, 77, 0, 79, 80

Offset: 0

Paul Barry, Jan 28 2004

Multiplicative with a(3^e) = 0, a(p^e) = p^e otherwise. - Mitch Harris, Jun 09 2005

Completely multiplicative with a(3) = 0, a(p) = p otherwise. - Charles R Greathouse IV, Feb 21 2011

			x + 2*x^2 + 4*x^4 + 5*x^5 + 7*x^7 + 8*x^8 + 10*x^10 + 11*x^11 + 13*x^13 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).

Cf. A008585, A100050.

&cat[[0,3*n+1,3*n+2]: n in [0..26]];  // Bruno Berselli, Aug 29 2011

f[n_] := If[ Mod[n, 3] == 0, 0, n] (* Or *) n (Fibonacci[n] - 2 Floor[ Fibonacci[n]/2]); Array[f, 78, 0] (* Robert G. Wilson v *)
{#,0,#}[[Mod[#-1,3,1]]]&/@Range[0,99] (* Federico Provvedi, Jun 15 2021 *)

a(n)=if(n%3,n) \\ Charles R Greathouse IV, Feb 21 2011

{a(n) = n * sign( n%3)} /* Michael Somos, Mar 19 2011 */

a(n) = Product_{k=0..2} Sum_{j=1..n} w(3)^(k*j), w(3)=e^(2*Pi*i/3), i=sqrt(-1).
a(n) = 2*n/3 - n*sin(2*Pi*n/3 + Pi/3)/sqrt(3) - n*cos(2*Pi*n/3 + Pi/3)/3.
G.f.: x*(x^4 + 2*x^3 + 2*x + 1)/((x^2 + x + 1)^2*(x - 1)^2). - Ralf Stephan, Jan 29 2004
a(n) = n^3 mod 3n = A000027(n)*A011655(n). - Paul Barry, Apr 13 2005
Dirichlet g.f.: zeta(s-1)*(1-1/3^(s-1)). - R. J. Mathar, Feb 10 2011
a(3*n) = 0, a(3*n + 1) = 3*n + 1, a(3*n + 2) = 3*n + 2. a(-n) = -a(n). - Michael Somos, Mar 19 2011
a(n) = n * sign(n mod 3). - Wesley Ivan Hurt, Sep 24 2017

A165202 Expansion of (1+x)/(1 - x + x^2)^2.

Original entry on oeis.org

1, 3, 3, -1, -6, -6, 1, 9, 9, -1, -12, -12, 1, 15, 15, -1, -18, -18, 1, 21, 21, -1, -24, -24, 1, 27, 27, -1, -30, -30, 1, 33, 33, -1, -36, -36, 1, 39, 39, -1, -42, -42, 1, 45, 45, -1, -48, -48, 1, 51, 51, -1, -54, -54, 1, 57, 57, -1, -60, -60, 1

Offset: 0

Paul Barry, Sep 07 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-3,2,-1).

Cf. A100050 (first differences).

Hankel transform of A165201.

a:=[1,3,3,-1];; for n in [5..70] do a[n]:=2*a[n-1]-3*a[n-2]+ 2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jul 18 2019

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1+x)/(1-x+x^2)^2 )); // G. C. Greubel, Jul 18 2019

LinearRecurrence[{2,-3,2,-1}, {1,3,3,-1}, 70] (* G. C. Greubel, Jul 18 2019 *)
(-1)^Quotient[#-1,3]{1,1+#,#}[[Mod[#,3,1]]]&/@Range[0, 10] (* Federico Provvedi, Jul 18 2021 *)

my(x='x+O('x^70)); Vec((1+x)/(1-x+x^2)^2) \\ G. C. Greubel, Jul 18 2019

((1+x)/(1-x+x^2)^2).series(x, 70).coefficients(x, sparse=False) # G. C. Greubel, Jul 18 2019

a(n) = cos(Pi*n/3) + sin(Pi*n/3)*(2n/3 + 1)*sqrt(3).

a(n) = A099254(n) + A099254(n-1). - R. J. Mathar, May 02 2013

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A100051 A Chebyshev transform of 1,1,1,...

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A091684 a(n) = 0 if n is divisible by 3, otherwise a(n) = n.

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A165202 Expansion of (1+x)/(1 - x + x^2)^2.

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