A098178 -id:A098178 - cp's OEIS frontend

A145390 Number of sublattices of index n of a centered rectangular lattice fixed by a reflection.

1, 1, 2, 3, 2, 2, 2, 5, 3, 2, 2, 6, 2, 2, 4, 7, 2, 3, 2, 6, 4, 2, 2, 10, 3, 2, 4, 6, 2, 4, 2, 9, 4, 2, 4, 9, 2, 2, 4, 10, 2, 4, 2, 6, 6, 2, 2, 14, 3, 3, 4, 6, 2, 4, 4, 10, 4, 2, 2, 12, 2, 2, 6, 11, 4, 4, 2, 6, 4, 4, 2, 15, 2, 2, 6, 6, 4, 4, 2, 14, 5, 2, 2, 12, 4, 2, 4, 10, 2, 6, 4, 6, 4, 2, 4, 18, 2, 3, 6, 9, 2

Offset: 1

N. J. A. Sloane, Feb 23 2009, Mar 13 2009

a(n) is the Dirichlet convolution of A000012 and A098178. - Domenico (domenicoo(AT)gmail.com), Oct 21 2009

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] (see Table 3).
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From _N. J. A. Sloane_, Feb 23 2009

Cf. A098178, A060594 (primitive sublattices only), A145391.

nmax := 100 :
L := [1,-1,0,2,seq(0,i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017

m = 101; Drop[ CoefficientList[ Series[ Sum[(1 + Cos[n*Pi/2])*x^n/(1 - x^n), {n, 1, m}], {x, 0, m}], x], 1] (* Jean-François Alcover, Sep 20 2011, after formula *)
f[p_, e_] := e+1; f[2, e_] := 2*e-1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

t1=direuler(p=2,200,1/(1-X)^2)
t2=direuler(p=2,2,1-X+2*X^2,200)
t3=dirmul(t1,t2)

Dirichlet g.f.: (1-2^(-s) + 2*4^(-s))*zeta^2(s).
G.f.: Sum_n (1 + cos(n*Pi/2)) x^n / (1 - x^n). - Domenico (domenicoo(AT)gmail.com), Oct 21 2009
If 4|n then a(n) = d(n) - d(n/2) + 2*d(n/4); else if 2|n then a(n) = d(n) - d(n/2); else a(n) = d(n); where d(n) is the number of divisors of n. [Rutherford] - Andrey Zabolotskiy, Mar 10 2018
a(n) = Sum_{ m: m^2|n } A060594(n/m^2). - Andrey Zabolotskiy, May 07 2018
Sum_{k=1..n} a(k) ~ n*(log(n) - 1 + 2*gamma - log(2)/2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019
Multiplicative with a(2^e) = 2*e-1 and a(p^e) = e+1 for an odd prime p. - Amiram Eldar, Aug 27 2023

New name from Andrey Zabolotskiy, Mar 10 2018

A098180 Odd numbers with twice the odd numbers repeated in order between them.

Original entry on oeis.org

1, 2, 2, 3, 5, 6, 6, 7, 9, 10, 10, 11, 13, 14, 14, 15, 17, 18, 18, 19, 21, 22, 22, 23, 25, 26, 26, 27, 29, 30, 30, 31, 33, 34, 34, 35, 37, 38, 38, 39, 41, 42, 42, 43, 45, 46, 46, 47, 49, 50, 50, 51, 53, 54, 54, 55, 57, 58, 58, 59, 61, 62, 62, 63, 65, 66, 66, 67, 69, 70, 70, 71

Offset: 0

Paul Barry, Aug 30 2004

Partial sums of A098178.
Also A042968 with the even terms repeated. - Michel Marcus, Apr 14 2015
Fixed points are [2,3,6,7,10,11,..] = A042964. - Wesley Ivan Hurt, Oct 13 2015

Iain Fox, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A002265, A042964, A042968, A098178.

[Floor((2*n+1-(-1)^((n+1)*(n+2)/2))/2): n in [0..80]]; // Vincenzo Librandi, Apr 13 2015

A098180:=n->(2*n+1-(-1)^((n+1)*(n+2)/2))/2: seq(A098180(n), n=0..100); # Wesley Ivan Hurt, Apr 12 2015

Table[(2 n + 1 - (-1)^((n + 1) (n + 2)/2))/2, {n, 0, 40}] (* Wesley Ivan Hurt, Apr 12 2015 *)

first(n) = Vec((1+x)*(1-x+x^2)/((1-x)^2*(1+x^2)) + O(x^n)) \\ Iain Fox, Oct 17 2018

a(n) = (2*n+1-(-1)^((n+1)*(n+2)/2))/2 \\ Iain Fox, Oct 17 2018

G.f.: (1+x)(1-x+x^2)/((1-x)^2(1+x^2)).
a(n) = sqrt(2)*sin(Pi*n/2+Pi/4)/2+n+1/2.
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4), n>3.
From Wesley Ivan Hurt, Apr 12 2015, Oct 13 2015: (Start)
a(n) = (2n+1-(-1)^((n+1)*(n+2)/2))/2.
a(n) = n + A002265(n) - A002265(n-2). (End)
E.g.f: (exp(-i*x)*((1+i) + (1-i)*exp(2*i*x) + exp((1+i)*x)*(2+4*x)))/4, where i = sqrt(-1). - Iain Fox, Oct 17 2018

A098179 Expansion of (1-3x+3x^2)/(1-5x+10x^2-10x^3+4x^4).

Original entry on oeis.org

1, 2, 3, 5, 11, 27, 63, 135, 271, 527, 1023, 2015, 4031, 8127, 16383, 32895, 65791, 131327, 262143, 523775, 1047551, 2096127, 4194303, 8390655, 16781311, 33558527, 67108863, 134209535, 268419071, 536854527, 1073741823, 2147516415

Offset: 0

Paul Barry, Aug 30 2004

Partial sums of A038503. Binomial transform of A098178.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-4).

LinearRecurrence[{5,-10,10,-4},{1,2,3,5},40] (* or *) CoefficientList[ Series[(1-3 x+3 x^2)/(1-5 x+10 x^2-10 x^3+4 x^4),{x,0,40}],x] (* Harvey P. Dale, Oct 06 2011 *)

a(n) = 2^n+2^(n/2)cos(pi*n/4)-1; a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-4a(n-4).

A204040 Triangle T(n,k), read by rows, given by (0, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 0, 4, 1, 0, -4, 4, 6, 1, 0, -4, -8, 12, 8, 1, 0, 4, -24, -4, 24, 10, 1, 0, 12, -8, -60, 16, 40, 12, 1, 0, 4, 56, -84, -96, 60, 60, 14, 1, 0, -20, 88, 84, -272, -100, 136, 84, 16, 1, 0, -28, -40

Offset: 0

Philippe Deléham, Jan 27 2012

Antidiagonal sums : periodic sequence 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, ... (see A007877 or A098178).Riordan array (1, x*(1+x)/(1-x+2*x^2)) .

			Triangle begins :
1
0, 1
0, 2, 1
0, 0, 4, 1
0, -4, 4, 6, 1
0, -4, -8, 12, 8, 1
0, 4, -24, -4, 24, 10, 1
0, 12, -8, -60, 16, 40, 12, 1
0, 4, 56, -84, -96, 60, 60, 14, 1
0, -20, 88, 84, -272, -100, 136, 84, 16, 1

Cf. A005408.

T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2, k-1) - 2*T(n-2,k).
G.f.: (1-x+2*x^2)/(1-(1+y)*x + (2-y)*x^2).
T(n,n) = n = A000012(n), T(n+1,n) = 2n = A005843(n), T(n+2,n) = A046092(n-1) for n>0, T(n+1,1) = A078050(n)*(-1)^n.
Sum_{k, 0<=k<=n} T(n,k) = A060747(n) = A005408(n-1).

A323202 Expansion of (1 - x) * (1 - x^3) / (1 - x^4) in powers of x.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jan 06 2019

			G.f. = 1 - x - x^3 + 2*x^4 - x^5 - x^7 + 2*x^8 - x^9 - x^11 + ...

Michael Somos, Rational Function Multiplicative Coefficients.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1).

Cf. A098178.

a[ n_] := (-1)^n + If[Mod[n, 2] == 0, (-1)^(n/2), 0] - Boole[n == 0];
a[ n_] := {-1, 0, -1, 2}[[Mod[n, 4, 1]]] - Boole[n == 0];
a[ n_] := SeriesCoefficient[ (1 - x) (1 - x^3) / (1 - x^4), {x, 0, Abs@n}];
LinearRecurrence[{-1,-1,-1},{1,-1,0,-1},80] (* Harvey P. Dale, May 31 2021 *)

{a(n) = (-1)^n + if(n%2==0, (-1)^(n/2)) - (n==0)};

{a(n) = [2, -1, 0, -1][n%4 + 1] - (n==0)};

{a(n) = n = abs(n); polcoeff( (1 - x) * (1 - x^3) / (1 - x^4) + x * O(x^n), n)};

{a(n) = my(e); n=abs(n); if( n<1, n==0, e=valuation(n, 2); -if( e<2, 1-e, -2))};

a(n) = -b(n) and b() is multiplicative with b(2) = 0, b(2^e) = -2 if e>1, b(p^e) = 1 if p>2.
Euler transform of length 4 sequence [-1, 0, -1, 1].
Moebius transform is length 4 sequence [-1, 1, 0, 2].
G.f.: (1 - x) * (1 - x^3) / (1 - x^4) = -1 + 1 / (1 + x) + 1 / (1 + x^2).
a(n) = a(-n) for all n in Z. a(n+2) = a(n-2) except if n=2 or n=-2.
a(n) = (-1)^n * A098178(n), a(2*n + 1) = -1, a(4*n + 2) = 0 for all n in Z.
E.g.f.: cos(x) + exp(-x) - 1. - Stefano Spezia, Aug 04 2025

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A145390 Number of sublattices of index n of a centered rectangular lattice fixed by a reflection.

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A098180 Odd numbers with twice the odd numbers repeated in order between them.

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A098179 Expansion of (1-3*x+3*x^2)/(1-5*x+10*x^2-10*x^3+4*x^4).

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A204040 Triangle T(n,k), read by rows, given by (0, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A323202 Expansion of (1 - x) * (1 - x^3) / (1 - x^4) in powers of x.

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PARI

PARI

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Formula

A098179 Expansion of (1-3x+3x^2)/(1-5x+10x^2-10x^3+4x^4).