A099838 -id:A099838 - cp's OEIS frontend

A099837 Expansion of (1 - x^2) / (1 + x + x^2) in powers of x.

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

Offset: 0

Paul Barry, Oct 27 2004

A transform of (-1)^n.
Row sums of Riordan array ((1-x)/(1+x), x/(1+x)^2), A110162.
Let b(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)(-1)^(n-2k). Then a(n) = b(n) - b(n-2) = A049347(n) - A049347(n-2) (n > 0). The g.f. 1/(1+x) of (-1)^n is transformed to (1-x^2)/(1+x+x^2) under the mapping G(x)->((1-x^2)/(1+x^2))G(x/(1+x^2)). Partial sums of A099838.
A(n) = a(n+3) (or a(n) if a(0) is replaced by 2) appears, together with B(n) = A049347(n) in the formula 2*exp(2*Pi*n*i/3) = A(n) + B(n)*sqrt(3)*i, n >= 0, with i = sqrt(-1). See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014

			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 + ...

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

Cf. A061347, A100051, A100063, A098554, A110162.

A099837 := proc(n)
    option remember;
    if n <=2 then
        op(n+1,[1,-1,-1]) ;
    else
        -procname(n-1)-procname(n-2) ;
    end if;
end proc:
seq(A099837(n),n=0..80) ; # R. J. Mathar, Apr 26 2022

a[0] = 1; a[n_] := Mod[n+2, 3] - Mod[n, 3]; A099837 = Table[a[n], {n, 0, 71}](* Jean-François Alcover, Feb 15 2012, after Michael Somos *)
LinearRecurrence[{-1, -1}, {1, -1, -1}, 50] (* G. C. Greubel, Aug 08 2017 *)

A099837(n) := block(
        if n = 0 then 1 else [2,-1,-1][1+mod(n,3)]
)$ /* R. J. Mathar, Mar 19 2012 */

{a(n) = [2, -1, -1][n%3 + 1] - (n == 0)}; /* Michael Somos, Jan 19 2012 */

Vec((1-x^2)/(1+x+x^2) + O(x^20)) \\ Felix Fröhlich, Aug 08 2017

G.f.: (1-x^2)/(1+x+x^2).
Euler transform of length 3 sequence [-1, -1, 1]. - Michael Somos, Mar 21 2011
Moebius transform is length 3 sequence [-1, 0, 3]. - Michael Somos, Mar 22 2011
a(n) = -b(n) where b(n) = A061347(n) is multiplicative with b(3^e) = -2 if e > 0, b(p^e) = 1 otherwise. - Michael Somos, Jan 19 2012
a(n) = a(-n). a(n) = c_3(n) if n > 1 where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011
G.f.: (1 - x) * (1 - x^2) / (1 - x^3). a(n) = -a(n-1) - a(n-2) unless n = 0, 1, 2. - Michael Somos, Jan 19 2012
Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s)*(3^(1-s)-1). - R. J. Mathar, Apr 11 2011
a(n+3) = R(n,-1) for n >= 0, with the monic Chebyshev T-polynomials R with coefficient table A127672. - Wolfdieter Lang, Feb 27 2014
For n > 0, a(n) = 2*cos(n*Pi/3)*cos(n*Pi). - Wesley Ivan Hurt, Sep 25 2017
From Peter Bala, Apr 20 2024: (Start)
a(n) is equal to the n-th order Taylor polynomial (centered at 0) of 1/c(x)^(2*n) evaluated at x = 1, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. of the Catalan numbers A000108. Cf. A333093.
Row sums of the Riordan array A110162. (End)

A240438 Greatest minimal difference between numbers of adjacent cells in a regular hexagonal honeycomb of order n with cells numbered from 1 through the total number of cells, the order n corresponding to the number of cells on one side of the honeycomb.

Original entry on oeis.org

0, 1, 5, 11, 18, 28, 40, 53, 69, 87, 106, 128, 152, 177, 205, 235, 266, 300, 336, 373, 413, 455, 498, 544, 592, 641, 693, 747, 802, 860, 920, 981, 1045, 1111, 1178, 1248, 1320, 1393, 1469, 1547, 1626, 1708, 1792, 1877, 1965, 2055, 2146, 2240, 2336, 2433, 2533, 2635

Offset: 1

Jörg Zurkirchen, Apr 05 2014

Difference table of a(n), with a(0)=0 and offset=0:
0,   0,  1,  5, 11, 18, 28, 40, 53, 69, ...
0,   1,  4,  6,  7, 10, 12, 13, 16, 18, ...   =  A047234(n+1)
1,   3,  2,  1,  3,  2,  1,  3,  2,  1, ...   =  A130784
2,  -1, -1,  2, -1, -1,  2, -1, -1,  2, ...   = -A131713(n+1)
-3,  0,  3, -3,  0,  3, -3,  0,  3, -3; ...   =  A099838(n+4)
3,   3, -6,  3,  3, -6,  3,  3, -6,  3, ...
0,  -9,  9,  0, -9,  9,  0, -9,  9,  0, ...
-9, 18, -9, -9, 18, -9, -9, 18, -9, -9, ...
First column: see A057682. - Paul Curtz, Nov 11 2014
Diameter of the chamber graph Γ(Alt(2n+1)). Definition of this graph:
Each vertex v is a sequence (v[1],v[2],...,v[n]) of length n, where each v[i] is a 2-subset of {1,2,...,2n+1} and v[i] and v[j] are disjoint unless i=j.
Vertices u and v are connected iff either:
u and v are identical except for their first elements u[1] and v[1], or
u and v are identical except for some i for which u[i]=v[i+1] and v[i]=u[i+1] - Tim Crinion, 17 Feb 2019

			For n = 3 an example of a honeycomb with the greatest minimal difference of a(3) = 5 is:
.         __
.      __/ 7\__
.   __/15\__/13\__
.  / 4\__/ 2\__/ 1\
.  \__/10\__/ 8\__/
.  /18\__/16\__/14\
.  \__/ 5\__/ 3\__/
.  /12\__/11\__/ 9\
.  \__/19\__/17\__/
.     \__/ 6\__/
.        \__/
.

22ème Championnat des jeux mathématiques et logiques - 1/4 de finale individuels 2008, problème 18, «Les ruches d’Abella»

Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (first 100 terms from Jörg Zurkirchen)
Tim Crinion, Chamber Graphs of some geometries related to the Petersen Graph, 2013.
Fédération Suisse des Jeux Mathématiques, 22nd Championship of Mathematical and Logical Games - Quarter Final 2008, 18 problems in French; problem number 18 was decisive to creating this sequence. See following pdf for an English version of problem 18.
Jörg Zurkirchen, Honeycomb.pdf
Index entries for linear recurrences with constant coefficients, signature (2, -1, 1, -2, 1).

Cf. A042965, A047234, A057682, A099838, A130784, A131713.

[n*(n-1)-Floor((n+1)/3): n in [1..60]]; // Vincenzo Librandi, Nov 12 2014

A240438:=n->n*(n-1)-floor((n+1)/3); seq(A240438(n), n=1..50); # Wesley Ivan Hurt, Apr 08 2014

Table[n (n - 1) - Floor[(n + 1)/3], {n, 50}] (* Wesley Ivan Hurt, Apr 08 2014 *)
CoefficientList[Series[x (x + 1) (2 x + 1) / ((1 - x)^3 (x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Nov 12 2014 *)
LinearRecurrence[{2, -1, 1, -2, 1},{0, 1, 5, 11, 18},52] (* Ray Chandler, Sep 24 2015 *)

a(n) = n*(n-1)-floor((n+1)/3).
G.f.: -x^2*(x+1)*(2*x+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Apr 08 2014
a(n+3) = a(n) + 6*n+5. - Paul Curtz, Nov 11 2014
a(n) = n^2 - (A042965(n+1)=0, 1, 3, 4, ...). - Paul Curtz, Nov 11 2014
a(n+1) = a(n) + A047234(n+1). - Paul Curtz, Nov 11 2014

A158916 Inverse binomial transform of A153130.

Original entry on oeis.org

1, 1, 1, 1, -8, 19, -35, 64, -125, 253, -512, 1027, -2051, 4096, -8189, 16381, -32768, 65539, -131075, 262144, -524285, 1048573, -2097152, 4194307, -8388611, 16777216, -33554429, 67108861, -134217728, 268435459, -536870915, 1073741824, -2147483645

Offset: 0

Paul Curtz, Mar 30 2009

sign

a(n)= A154589(n)+ A099838(n+4).

Index entries for linear recurrences with constant coefficients, signature (-3, -3, -2).

LinearRecurrence[{-3,-3,-2},{1,1,1,1},40] (* Harvey P. Dale, Feb 04 2019 *)

a(n)= -3a(n-1)-3a(n-2)-2a(n-3), n > 3.

G.f.: (4*x+7*x^2+9*x^3+1)/((2*x+1)*(1+x+x^2)). [R. J. Mathar, May 17 2009]

Edited and extended by R. J. Mathar, May 17 2009

A202241 Array F(n,m) read by antidiagonals: F(0,m)=1, F(n,0) = A130713(n), and column m+1 is recursively defined as the partial sums of column m.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 0, 4, 4, 1, 0, 4, 8, 5, 1, 0, 4, 12, 13, 6, 1, 0, 4, 16, 25, 19, 7, 1, 0, 4, 20, 41, 44, 26, 8, 1, 0, 4, 24, 61, 85, 70, 34, 9, 1, 0, 4, 28, 85, 146, 155, 104, 43, 10, 1, 0, 4, 32, 113, 231, 301, 259, 147, 53, 11, 1, 0, 4, 36, 145, 344, 532, 560, 406, 200, 64, 12, 1

Offset: 0

Paul Curtz, Dec 16 2011

The array F(n,m), beginning with row n=0, is:
  1, 1,  1,  1,   1,   1,    1,
  2, 3,  4,  5,   6,   7,    8,
  1, 4,  8, 13,  19,  26,   34,
  0, 4, 12, 25,  44,  70,  104,
  0, 4, 16, 41,  85, 155,  259,
  0, 4, 20, 61, 146, 301,  560,
  0, 4, 24, 85, 231, 532, 1092.
Columns after A130713, A113311, A008574 have signatures (3,-3,1), (4,-6,4,-1), (5,-10,10,-5,1), (6,-15,20,-15,6,-1) (from A135278(n+3)).
Inserting columns of zeros and pushing the columns down, plus alternating sign switches defines the following triangle T(n,2m) = (-1)^(m/2)*F(n-2m,m):
  1,
  2 0,
  1 0 -1,
  0 0 -3 0,
  0 0 -4 0 1,
  0 0 -4 0 4 0,
  0 0 -4 0 8 0 -1
The row sums in the triangle are (-1)^n*A099838(n).
The companion to A201863 is
  1
  1 0
  1 0   0
  1 0  -2 0
  1 0  -4 0  1
  1 0  -6 0  5 0
  1 0  -8 0 13 0   -2
  1 0 -10 0 25 0  -12 0
  1 0 -12 0 41 0  -38 0   4
  1 0 -14 0 61 0  -88 0  28 0
  1 0 -16 0 85 0 -170 0 104 0 -8
5th column: A001844; 7th column: -A035597=-2*A005900(n+1); 9th column: 4*A006325(n+2); 11th column: -8*(1,8,34,104) (from columns 4,5,6,7 of F(n,m)).
As a triangular array, this is the Riordan array ((1+x)^2, x/(1-x)). - Philippe Deléham, Feb 21 2012

			Triangle T(n,k) begins:
  1
  2, 1
  1, 3,  1
  0, 4,  4,  1
  0, 4,  8,  5,   1
  0, 4, 12, 13,   6,   1
  0, 4, 16, 25,  19,   7,   1
  0, 4, 20, 41,  44,  26,   8,  1
  0, 4, 24, 61,  85,  70,  34,  9,  1
  0, 4, 28, 85, 146, 155, 104, 43, 10, 1
- _Philippe Deléham_, Feb 21 2012

Muniru A Asiru, Table of n, a(n) for n = 0..5151

Cf. A130713 (column 0), A113311 (column 1), A008574 (column 2), A001844 (column 3), A005900 (column 4), A006325 (column 5), A033455 (column 6).

Cf. A267633.

Flat(List([0..12],n->List([0..n],k->Binomial(n,n-k)+Binomial(n-1,n-k-1)-Binomial(n-2,n-k-2)-Binomial(n-3,n-k-3)))); # Muniru A Asiru, Mar 22 2018

A130713 := proc(n)
    if n <= 2 and n >=0 then
        op(n+1,[1,2,1]) ;
    else
        0;
    end if;
end proc:
A202241 := proc(n,m)
    option remember;
    if n < 0 then
        0 ;
    elif m = 0 then
        A130713(n);
    else
        procname(n,m-1)+procname(n-1,m) ;
    end if;
end proc:
for d from 0 to 12 do
    for m from 0 to d do
        printf("%d,",A202241(d-m,m)) ;
    end do:
end do: # R. J. Mathar, Dec 22 2011
C := proc (n, k) if 0 <= k and k <= n then factorial(n)/(factorial(k)*factorial(n-k)) else 0 end if end proc:
for n from 0 to 10 do
     seq(C(n, n-k) + C(n-1, n-k-1) - C(n-2, n-k-2) - C(n-3, n-k-3), k = 0..n);
end do; # Peter Bala, Mar 20 2018

rows = 12;
T[0] = PadRight[{1, 2, 1}, rows];
T[n_ /; nJean-François Alcover, Jun 29 2019 *)

def Trow(n): return [binomial(n, n-k) + binomial(n-1, n-k-1) - binomial(n-2, n-k-2) - binomial(n-3, n-k-3) for k in (0..n)]
for n in (0..9): print(Trow(n)) # Peter Luschny, Mar 21 2018

F(1,m) = m+2.
F(2,m) = A034856(m+1).
F(3,m) = A000297(m-1).
Sum_{m=0..d} F(d-m,m) = A116453(d-3), d >= 3 (antidiagonal sums).
As a triangular array T(n,k), 0 <= k <= n, satisfies: T(n,k) = T(n-1,k) + T(n-1,k-1) with T(0,0) = 1, T(1,0) = 2, T(2,0) = 1, T(3,0) = 0. - Philippe Deléham, Feb 21 2012
Unsigned diagonals of A267633 (beginning with its main diagonal) appear to be the reverse rows of this entry's triangle beginning with the fourth row. - Tom Copeland, Jan 26 2016
T(n,k) = C(n, n-k) + C(n-1, n-k-1) - C(n-2, n-k-2) - C(n-3, n-k-3), where C(n, k) = n!/(k!*(n-k)!) if 0 <= k <= n, otherwise 0. - Peter Bala, Mar 20 2018

A158927 a(n) = -3a(n-1) - 3a(n-2) - 2a(n-3), n > 3.

Original entry on oeis.org

2, 2, 2, -7, 11, -16, 29, -61, 128, -259, 515, -1024, 2045, -4093, 8192, -16387, 32771, -65536, 131069, -262141, 524288, -1048579, 2097155, -4194304, 8388605, -16777213, 33554432, -67108867, 134217731, -268435456, 536870909, -1073741821, 2147483648

Offset: 0

Paul Curtz, Mar 31 2009

sign

The inverse binomial transform of A153130, after dropping A153130(0).
The inverse binomial transform of the full A153130 is A158916.
Dropping two initial terms of A153130 yields A158935, dropping three yields essentially a sign-reversed version of A158916, dropping 4 essentially the sequence here.

Muniru A Asiru, Table of n, a(n) for n = 0..600
Index entries for linear recurrences with constant coefficients, signature (-3, -3, -2).

Same recurrence as A131562, A158916, A158926.

a := [2,2,2,-7];; for n in [5..10^3] do a[n] := -3*a[n-1] - 3*a[n-2] - 2*a[n-3]; od; a; # Muniru A Asiru, Jan 27 2018

a := proc(n) option remember: if n=0 then 2 elif n=1 then 2 elif n=2 then 2 elif n=3 then -7 elif n>=4 then -3*procname(n-1) - 3*procname(n-2) - 2*procname(n-3) fi; end:
seq(a(n), n=0..100); # Muniru A Asiru, Jan 27 2018

a(n) = -3a(n-1) - 3a(n-2) - 2a(n-3), with a(0)=a(1)=a(2)=2, a(3)=-7.
a(n) = (-1)^(n+1)*A157823(n) - A099838(n+3).
G.f.: (2+8*x+14*x^2+9*x^3)/((2*x+1)*(1+x+x^2)). - R. J. Mathar, Apr 09 2009
a(0)=2; a(n) = (1/2)*(-2)^n - 3*cos(2*Pi*n/3) + sqrt(3)*sin(2*Pi*n/3) for n >= 1. - Richard Choulet, Apr 23 2009

Edited and extended by R. J. Mathar, Apr 09 2009

A158935 a(n)= -3a(n-1)-3a(n-2)-2a(n-3), n>3. a(0)=4, a(1)=4, a(2)=-5, a(3)=4.

Original entry on oeis.org

4, 4, -5, 4, -5, 13, -32, 67, -131, 256, -509, 1021, -2048, 4099, -8195, 16384, -32765, 65533, -131072, 262147, -524291, 1048576, -2097149, 4194301, -8388608, 16777219, -33554435, 67108864, -134217725, 268435453, -536870912, 1073741827, -2147483651

Offset: 0

Paul Curtz, Mar 31 2009

The third column of the array of differences described in A153130. The first two columns are in A158916 and A158987. Taking differences like in A158926 keeps the recurrence.

Also the inverse binomial transform of A153130 if the first two items of A153130 are omitted.

Index entries for linear recurrences with constant coefficients, signature (-3, -3, -2).

Join[{4},LinearRecurrence[{-3,-3,-2},{4,-5,4},50]] (* Harvey P. Dale, May 25 2011 *)

a(n)= A154589(n) + A099838(n+2).

G.f.: (4+16*x+19*x^2+9*x^3)/((2*x+1)*(1+x+x^2)). - R. J. Mathar, Apr 08 2009

Partially edited and extended by R. J. Mathar, Apr 08 2009

Edited by N. J. A. Sloane, Apr 08 2009

A349803 a(3n) = 1 + 4n^2, a(1+3n) = 2 + 4n(n+1), a(2+3n) = 5 + 4n(n+1).

Original entry on oeis.org

1, 2, 5, 5, 10, 13, 17, 26, 29, 37, 50, 53, 65, 82, 85, 101, 122, 125, 145, 170, 173, 197, 226, 229, 257, 290, 293, 325, 362, 365, 401, 442, 445, 485, 530, 533, 577, 626, 629, 677, 730, 733, 785, 842, 845, 901, 962

Offset: 0

Paul Curtz, Dec 01 2021

A261327 sorted in nondecreasing order.

Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

Cf. A261327.
Trisections: A053755, A069894, A078370.
Cf. A004396, A004523, A099838.

nterms=100;LinearRecurrence[{1,0,2,-2,0,-1,1},{1,2,5,5,10,13,17},nterms] (* Paolo Xausa, Dec 01 2021 *)

a(-n) = a(n) - A099838(n+2).
a(n) = a(n-3)  +  4*A004523(n-1) for n >=  3
     = a(n-6)  +  8*A004396(n-3) for n >=  6
     = a(n-9)  + 12*A004523(n-4) for n >=  9
     = a(n-12) + 16*A004396(n-6) for n >= 12
     ...

cp's OEIS Frontend

A099837 Expansion of (1 - x^2) / (1 + x + x^2) in powers of x.

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A240438 Greatest minimal difference between numbers of adjacent cells in a regular hexagonal honeycomb of order n with cells numbered from 1 through the total number of cells, the order n corresponding to the number of cells on one side of the honeycomb.

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A158916 Inverse binomial transform of A153130.

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A202241 Array F(n,m) read by antidiagonals: F(0,m)=1, F(n,0) = A130713(n), and column m+1 is recursively defined as the partial sums of column m.

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A158927 a(n) = -3a(n-1) - 3a(n-2) - 2a(n-3), n > 3.

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A158935 a(n)= -3a(n-1)-3a(n-2)-2a(n-3), n>3. a(0)=4, a(1)=4, a(2)=-5, a(3)=4.

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A349803 a(3*n) = 1 + 4*n^2, a(1+3*n) = 2 + 4*n*(n+1), a(2+3*n) = 5 + 4*n*(n+1).

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A349803 a(3n) = 1 + 4n^2, a(1+3n) = 2 + 4n(n+1), a(2+3n) = 5 + 4n(n+1).