A047538 -id:A047538 - cp's OEIS frontend

A084101 Expansion of (1+x)^2/((1-x)*(1+x^2)).

1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1, 3, 3, 1, 1

Offset: 0

Paul Barry, May 15 2003

Partial sums of A084099. Inverse binomial transform of A000749 (without leading zeros).
From Klaus Brockhaus, May 31 2010: (Start)
Periodic sequence: Repeat 1, 3, 3, 1.
Interleaving of A010684 and A176040.
Continued fraction expansion of (7 + 5*sqrt(29))/26.
Decimal expansion of 121/909.
a(n) = A143432(n+3) + 1 = 2*A021913(n+1) + 1 = 2*A133872(n+3) + 1.
a(n) = A165207(n+1) - 1.
First differences of A047538.
Binomial transform of A084102. (End)
From Wolfdieter Lang, Feb 09 2012: (Start)
a(n) = A045572(n+1) (Modd 5) := A203571(A045572(n+1)), n >= 0.
For general Modd n (not to be confused with mod n) see a comment on A203571. The nonnegative members of the five residue classes Modd 5, called [m] for m=0,1,...,4, are shown in the array A090298 if there the last row is taken as class [0] after inclusion of 0.
(End)

			From _Wolfdieter Lang_, Feb 09 2012: (Start)
Modd 5 of nonnegative odd numbers restricted mod 5:
A045572: 1, 3, 7, 9, 11, 13, 17, 19, 21, 23, ...
Modd 5:  1, 3, 3, 1,  1,  3,  3,  1,  1,  3, ...
(End)

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

Cf. A084102.

Cf. A010684 (repeat 1, 3), A176040 (repeat 3, 1), A178593 (decimal expansion of (7+5*sqrt(29))/26), A143432 (expansion of (1+x^4)/((1-x)*(1+x^2))), A021913 (repeat 0, 0, 1, 1), A133872 (repeat 1, 1, 0, 0), A165207 (repeat 2, 2, 4, 4), A047538 (congruent to 0, 1, 4 or 7 mod 8), A084099 (expansion of (1+x)^2/(1+x^2)), A000749 (expansion of x^3/((1-x)^4-x^4)). - Klaus Brockhaus, May 31 2010

R:=PowerSeriesRing(Integers(), 100); Coefficients(R!( (1+x)^2/((1-x)*(1+x^2)) )); // G. C. Greubel, Feb 28 2019

CoefficientList[Series[(1+x)^2/((1-x)(1+x^2)),{x,0,110}],x] (* or *) PadRight[{},110,{1,3,3,1}] (* Harvey P. Dale, Nov 21 2012 *)

x='x+O('x^100); Vec((1+x)^2/((1-x)*(1+x^2))) \\ Altug Alkan, Dec 24 2015

((1+x)^2/((1-x)*(1+x^2))).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Feb 28 2019

a(n) = binomial(3, n mod 4). - Paul Barry, May 25 2003
From Klaus Brockhaus, May 31 2010: (Start)
a(n) = a(n-4) for n > 3; a(0) = a(3) = 1, a(1) = a(2) = 3.
a(n) = (4 - (1+i)*i^n - (1-i)*(-i)^n)/2 where i = sqrt(-1). (End)
E.g.f.: 2*exp(x) + sin(x) - cos(x). - Arkadiusz Wesolowski, Nov 04 2017
a(n) = 2 - (-1)^(n*(n+1)/2). - Guenther Schrack, Feb 26 2019

A099855 a(n) = n2^n - 2^(n/2)sin(Pi*n/4).

Original entry on oeis.org

0, 1, 6, 22, 64, 164, 392, 904, 2048, 4592, 10208, 22496, 49152, 106560, 229504, 491648, 1048576, 2227968, 4718080, 9960960, 20971520, 44041216, 92276736, 192940032, 402653184, 838856704, 1744822272, 3623870464, 7516192768

Offset: 0

Paul Barry, Oct 28 2004

Related to binomial transform of A002265. Sequence is identical to its fourth differences (cf. A139756, A137221). See also A097064, A135035, A038504, A135356. - Paul Curtz, Jun 18 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-8).

Cf. A001787, A002265, A009545, A038504, A097064, A135035, A135356.
Cf. A137221, A139756.
Binomial transform of A047538.

I:=[0,1,6,22]; [n le 4 select I[n] else 6*Self(n-1) -14*Self(n-2) +16*Self(n-3) -8*Self(n-4): n in [1..51]]; // G. C. Greubel, Apr 20 2023

LinearRecurrence[{6,-14,16,-8},{0,1,6,22},30] (* Harvey P. Dale, Mar 22 2018 *)

@CachedFunction
def a(n): # a = A099855
    if (n<5): return (0,1,6,22,64)[n]
    else: return 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 8*a(n-4)
[a(n) for n in range(51)] # G. C. Greubel, Apr 20 2023

G.f.: x/((1-2*x+2*x^2)*(1-4*x+4*x^2)).
a(n) = Sum_{k=0..n} 2^(k/2)*sin(Pi*k/4)*2^(n-k)*(n-k+1).
a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 8*a(n-4).
a(n) = 2*A001787(n) - A009545(n).

A074231 Numbers n such that Kronecker(8,n) = mu(gcd(8,n)).

Original entry on oeis.org

1, 4, 7, 8, 9, 12, 15, 16, 17, 20, 23, 24, 25, 28, 31, 32, 33, 36, 39, 40, 41, 44, 47, 48, 49, 52, 55, 56, 57, 60, 63, 64, 65, 68, 71, 72, 73, 76, 79, 80, 81, 84, 87, 88, 89, 92, 95, 96, 97, 100, 103, 104, 105, 108, 111, 112, 113, 116, 119, 120, 121, 124, 127, 128, 129

Offset: 1

Jon Perry, Sep 17 2002

nonn

A Chebyshev transform of (1+2x)/(1-2x) (A046055) given by G(x)->(1/(1+x^2))G(x/(1+x^2)). - Paul Barry, Oct 27 2004

Essentially the same as A047538.

for (x=1,200, for (y=1,200,if (kronecker(x,y)==moebius(gcd(x,y)),write("km.txt",x,";",y," : ",kronecker(x,y)))))

[lucas_number1(n+2, 0, 1)+2*n for n in range(1, 66)] # Zerinvary Lajos, Mar 09 2009

From Paul Barry, Oct 27 2004: (Start)
G.f.: (1+x)^2/((1+x^2)*(1-2x+x^2));
e.g.f.: exp(x)(2+2x) - cos(x);
a(n) = 2n + 2 - cos(Pi*n/2);
a(n) = Sum_{k=0..n} (0^k + 4^k)*cos(Pi*(n-k)/2);
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k(2*2^(n-2k)-0^(n-2k));
a(n) = 2a(n-1) - 2a(n-2) + 2a(n-3) - a(n-4). (End)

A199627 G.f.: (1+x)^(2g)(1+x^3)^(3g)/((1-x^2)(1-x^4))-x^(2g)(1+x)^4/((1-x^2)*(1-x^4)) for g=1.

Original entry on oeis.org

1, 2, 1, 1, 2, 2, 4, 7, 8, 9, 12, 15, 16, 17, 20, 23, 24, 25, 28, 31, 32, 33, 36, 39, 40, 41, 44, 47, 48, 49, 52, 55, 56, 57, 60, 63, 64, 65, 68, 71, 72, 73, 76, 79, 80, 81, 84, 87, 88, 89, 92, 95, 96, 97, 100, 103, 104, 105, 108, 111

Offset: 0

N. J. A. Sloane, Nov 08 2011

Expansion of a Poincaré series [or Poincare series] for space of moduli M_2 of stable bundles.

Georg Fischer, Table of n, a(n) for n = 0..999
Bott, Raoul, Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. 7 (1982), no. 2, 331-358; reprinted in Vol. 48 (October, 2011). See Eq. (4.30).
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A047538.

g:=1; m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4))));  // Bruno Berselli, Nov 08 2011

f:=g->(1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4));
s:=g->seriestolist(series(f(g),x,60));
s(1);

Vec((1 + x)^2*(1 - 2*x + 2*x^2 - x^3 - x^4 + 3*x^5 - 2*x^6 + x^7) / ((1 - x)^2*(1 + x^2)) + O(x^70)) \\ Colin Barker, Nov 05 2019

a(n) = A047538(n-3) for n >= 6. - Georg Fischer, Oct 28 2018
From Colin Barker, Nov 05 2019: (Start)
G.f.: (1 + x)^2*(1 - 2*x + 2*x^2 - x^3 - x^4 + 3*x^5 - 2*x^6 + x^7) / ((1 - x)^2*(1 + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>9.
a(n) = (-16 + (-i)^(1+n) + i^(1+n) + 4*n) / 2 for n>5, where i=sqrt(-1).
(End)

A242668 Expansion of 1/(1 - 8x + 16x^2 + x^4 - 4*x^5).

Original entry on oeis.org

1, 8, 48, 256, 1279, 6132, 28576, 130432, 585985, 2599952, 11419808, 49743104, 215163647, 925163500, 3957669648, 16854677312, 71498512897, 302248757272, 1273756836176, 5353050574336, 22440215412223, 93856659402724, 391745066819136, 1631995960879872

Offset: 0

Bruno Berselli, May 20 2014

Subsequence of A047538.

a(n) is divisible by 4^(n mod 4).

C. Mariconda and A. Tonolo, Calcolo discreto, Apogeo (2012), 229-230 (example 9.43).

Bruno Berselli, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (8,-16,0,-1,4).

Cf. A047538.

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-8*x+16*x^2+x^4-4*x^5)));

CoefficientList[Series[1/((1 - 4 x) (1 - 4 x + x^4)), {x, 0, 30}], x]
LinearRecurrence[{8,-16,0,-1,4},{1,8,48,256,1279},40] (* Harvey P. Dale, Aug 10 2021 *)

makelist(coeff(taylor(1/(1-8*x+16*x^2+x^4-4*x^5), x, 0, n), x, n), n, 0, 30);

Vec(1/(1-8*x+16*x^2+x^4-4*x^5)+O(x^30))

m = 30; L. = PowerSeriesRing(ZZ, m); f = 1/(1-8*x+16*x^2+x^4-4*x^5); print(f.coefficients())

G.f.: 1/((1 - 4*x)*(1 - 4*x + x^4)).
a(n) = 8*a(n-1) - 16*a(n-2) - a(n-4) + 4*a(n-5) for n>4.
a(n) = 4*a(n-1) - a(n-4) + 4^n for n>3 (see References, p. 229).
Trisections:
a(3k): 1, 256, 28576, 2599952, 215163647, 16854677312, 1273756836176, ... has g.f. (1+128*x-48*x^2+4*x^3)/((1-64*x)*(1-64*x+48*x^2-12*x^3+x^4));
a(3k+1): 8, 1279, 130432, 11419808, 925163500, 71498512897, ... has g.f. (8+255*x-128*x^2+16*x^3)/((1-64*x)*(1-64*x+48*x^2-12*x^3+x^4));
a(3k+2): 48, 6132, 585985, 49743104, 3957669648, 302248757272, ... has g.f. (48-12*x+x^2)/((1-64*x)*(1-64*x+48*x^2-12*x^3+x^4)).
a(n) ~ 4^(4+n). - Stefano Spezia, Mar 29 2023

cp's OEIS Frontend

A084101 Expansion of (1+x)^2/((1-x)*(1+x^2)).

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A099855 a(n) = n*2^n - 2^(n/2)*sin(Pi*n/4).

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A074231 Numbers n such that Kronecker(8,n) = mu(gcd(8,n)).

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A199627 G.f.: (1+x)^(2*g)*(1+x^3)^(3*g)/((1-x^2)*(1-x^4))-x^(2*g)*(1+x)^4/((1-x^2)*(1-x^4)) for g=1.

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A242668 Expansion of 1/(1 - 8*x + 16*x^2 + x^4 - 4*x^5).

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A099855 a(n) = n2^n - 2^(n/2)sin(Pi*n/4).

A199627 G.f.: (1+x)^(2g)(1+x^3)^(3g)/((1-x^2)(1-x^4))-x^(2g)(1+x)^4/((1-x^2)*(1-x^4)) for g=1.

A242668 Expansion of 1/(1 - 8x + 16x^2 + x^4 - 4*x^5).