A111927 -id:A111927 - cp's OEIS frontend

A024495 a(n) = C(n,2) + C(n,5) + ... + C(n, 3*floor(n/3)+2).

0, 0, 1, 3, 6, 11, 21, 42, 85, 171, 342, 683, 1365, 2730, 5461, 10923, 21846, 43691, 87381, 174762, 349525, 699051, 1398102, 2796203, 5592405, 11184810, 22369621, 44739243, 89478486, 178956971, 357913941, 715827882, 1431655765, 2863311531, 5726623062

Offset: 0

Clark Kimberling

Trisections give A082365, A132804, A132805. - Paul Curtz, Nov 18 2007
If the offset is changed to 1, this is the maximal number of closed regions bounded by straight lines after n straight line cuts in a plane: a(n) = a(n-1) + n - 3, a(1)=0; a(2)=0; a(3)=1; and so on. - Srikanth K S, Jan 23 2008
M^n * [1,0,0] = [A024493(n), a(n), A024494(n)]; where M = a 3x3 matrix [1,1,0; 0,1,1; 1,0,1]. Sum of terms = 2^n. Example: M^5 * [1,0,0] = [11, 11, 10], sum = 2^5 = 32. - Gary W. Adamson, Mar 13 2009
For n>=1, a(n-1) is the number of generalized compositions of n when there are i^2/2 - 3*i/2 + 1 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010
Let M be any endomorphism on any vector space, such that M^3 = 1 (identity). Then (1+M)^n = A024493(n) + A024494(n)*M + a(n)*M^2. - Stanislav Sykora, Jun 10 2012
{A024493, A131708, A024495} is the difference analog of the hyperbolic functions {h_1(x), h_2(x), h_3(x)} of order 3. For the definitions of {h_i(x)} and the difference analog {H_i(n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Aug 01 2017
This is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S^3; see A291000.  - Clark Kimberling, Aug 24 2017

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd. ed., Problem 38, p. 70.

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Paul Barry, A note on Krawtchouk Polynomials and Riordan Arrays, JIS 11 (2008) 08.2.2, example 9.
Antoine-Augustin Cournot, Solution d'un problème d'analyse combinatoire, Bulletin des Sciences Mathématiques, Physiques et Chimiques, item 34, volume 11, 1829, pages 93-97. Also at Google Books. Page 97 case p=3 formula y^(2) = a(n).
Christian Ramus, Solution générale d'un problème d'analyse combinatoire, Journal für die Reine und Angewandte Mathematik (Crelle's journal), volume 11, 1834, pages 353-355. Page 353 case p=3 formula y^(2) = a(n).
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Eric Weisstein's World of Mathematics, Plane division by lines
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A007283, A010892, A011655, A024493, A024494, A024495, A057079, A082365.
Cf. A083322, A139469, A111927, A113405, A131577, A131708, A132804, A132805.
Cf. A175805, A291000.
Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), this sequence (m=3), A000749 (m=4), A049016 (m=5), A192080 (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

R:=PowerSeriesRing(Integers(), 30); [0,0] cat Coefficients(R!( x^2/((1-x)^3-x^3) )); // G. C. Greubel, Apr 11 2023

a:= proc(n) option remember; `if`(n=0, 0, 2*a(n-1)+
      [-1, 0, 1, 1, 0, -1, -1][1+(n mod 6)])
    end:
seq(a(n), n=0..33); # Paul Weisenhorn, May 17 2020

LinearRecurrence[{3,-3,2},{0,0,1},40] (* Harvey P. Dale, Sep 20 2016 *)

a(n) = sum(k=0,n\3,binomial(n,3*k+2)) /* Michael Somos, Feb 14 2006 */

a(n)=if(n<0, 0, ([1,0,1;1,1,0;0,1,1]^n)[3,1]) /* Michael Somos, Feb 14 2006 */

def A024495(n): return (2^n - chebyshev_U(n, 1/2) - chebyshev_U(n-1, 1/2))/3
[A024495(n) for n in range(41)] # G. C. Greubel, Apr 11 2023

a(n) = ( 2^n + 2*cos((n-4)*Pi/3) )/3 = (2^n - A057079(n))/3.
a(n) = 2*a(n-1) + A010892(n-2) = a(n-1) + A024494(n-1). With initial zero, binomial transform of A011655 which is effectively A010892 unsigned. - Henry Bottomley, Jun 04 2001
a(2) = 1, a(3) = 3, a(n+2) = a(n+1) - a(n) + 2^n. - Benoit Cloitre, Sep 04 2002
a(n) = Sum_{k=0..n} 2^k*2*sin(Pi*(n-k)/3 + Pi/3)/sqrt(3) (offset 0). - Paul Barry, May 18 2004
G.f.: x^2/((1-x)^3 - x^3) =  x^2 / ( (1-2*x)*(1-x+x^2) ).
a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3). - Paul Curtz, Nov 18 2007
a(n) + A024493(n-1) = A131577(n). - Paul Curtz, Jan 24 2008
From Paul Curtz, May 29 2011: (Start)
a(n) + a(n+3) = 3*2^n = A007283(n).
a(n+6) - a(n) = 21*2^n = A175805(n).
a(n) + a(n+9) = 171*2^n.
a(n+12) - a(n) = 1365*2^n. (End)
a(n) = A113405(n) + A113405(n+1). - Paul Curtz, Jun 05 2011
Start with x(0)=1, y(0)=0, z(0)=0 and set x(n+1) = x(n) + z(n), y(n+1) = y(n) + x(n), z(n+1) = z(n) + y(n). Then a(n) = z(n). - Stanislav Sykora, Jun 10 2012
G.f.: -x^2/( x^3 - 1 + 3*x/Q(0) ) where Q(k) = 1 + k*(x+1) + 3*x - x*(k+1)*(k+4)/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Mar 15 2013
a(n) = 1/18*(-4*(-1)^floor((n - 1)/3) - 6*(-1)^floor(n/3) - 3*(-1)^floor((n + 1)/3) + (-1)^(1 + floor((n + 2)/3)) + 3*2^(n + 1)). - John M. Campbell, Dec 23 2016
a(n) = (1/63)*(-40 + 21*2^n - 42*floor(n/6) + 32*floor((n+3)/6) + 16*floor((n+ 4)/6) - 24*floor((n+5)/6) - 22*floor((n+7)/6) + 21*floor((n+8)/6) + 10*floor((n+9)/6) + 5*floor((n+10)/6) + 3*floor((n+11)/6) + floor((n+ 13)/6)). - John M. Campbell, Dec 24 2016
a(n+m) = a(n)*A024493(m) + A131708(n)*A131708(m) + A024493(n)*a(m). - Vladimir Shevelev, Aug 01 2017
From Kevin Ryde, Sep 24 2020: (Start)
a(n) = (1/3)*2^n - (1/3)*cos((1/3)*Pi*n) - (1/sqrt(3))*sin((1/3)*Pi*n). [Cournot]
a(n) + A111927(n) + A131708(n) = 2^n - 1. [Cournot, page 96 last formula, but misprint should be 2^x - 1 rather than 2^p - 1] (End)
E.g.f.: (exp(2*x) - exp(x/2)*(cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)))/3. - Stefano Spezia, Feb 06 2025

A131708 A024494 prefixed by a 0.

Original entry on oeis.org

0, 1, 2, 3, 5, 10, 21, 43, 86, 171, 341, 682, 1365, 2731, 5462, 10923, 21845, 43690, 87381, 174763, 349526, 699051, 1398101, 2796202, 5592405, 11184811, 22369622, 44739243, 89478485, 178956970, 357913941, 715827883, 1431655766, 2863311531, 5726623061, 11453246122

Offset: 0

Paul Curtz, Sep 14 2007, Mar 01 2008

Binomial transform of 0, 1, 0. Also A024495 = first differences.
Recurrence: a(n+1) - 2*a(n) = 1, 0, -1, -1, 0, 1, 1.
{A024493, A131708, A024495} is the difference analog of the hyperbolic functions {h_1(x), h_2(x), h_3(x)} of order 3. For the definitions of {h_i(x)} and the difference analog {H_i(n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Aug 01 2017

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Antoine-Augustin Cournot, Solution d'un problème d'analyse combinatoire, Bulletin des Sciences Mathématiques, Physiques et Chimiques, item 34, volume 11, 1829, pages 93-97. Also at Google Books. Page 97 case p=3 formula y^(1) = a(n).
Christian Ramus, Solution générale d'un problème d'analyse combinatoire, Journal für die Reine und Angewandte Mathematik (Crelle's journal), volume 11, 1834, pages 353-355. Page 353 case p=3 formula y^(1) = a(n).
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Index entries for linear recurrences with constant coefficients, signature (3,-3,2).

Cf. A024493, A024494, A024495, A111927, A113405, A135350, A137221.

[n le 3 select n-1 else 3*Self(n-1) -3*Self(n-2) +2*Self(n-3): n in [1..40]]; // G. C. Greubel, Jan 23 2023

LinearRecurrence[{3,-3,2}, {0,1,2}, 40] (* Harvey P. Dale, Nov 27 2013 *)

v=vector(99,i,i);for(i=4,#v,v[i]=3*v[i-1]-3*v[i-2]+2*v[i-3]);v \\ Charles R Greathouse IV, Jun 01 2011

def A131708(n): return (1/3)*(2^n -chebyshev_U(n,1/2) +2*chebyshev_U(n-1,1/2))
[A131708(n) for n in range(41)] # G. C. Greubel, Jan 23 2023

G.f.: x*(1-x)/((1-2*x)*(1-x+x^2)). - R. J. Mathar, Nov 14 2007
Recurrences:
a(n) = k*a(n-1) + (6-3*k)*a(n-2) + (3*k-7)*a(n-3) + (6-2*k)*a(n-4).
k = 0: a(n) = 6*a(n-2) - 7*a(n-3) + 6*a(n-4).
k = 1: a(n) = a(n-1) + 3*a(n-2) - 4*a(n-3) + 4*a(n-4).
k = 2: a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4), cf. A113405, A135350.
k = 3: a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3), this sequence.
k = 4: a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 2*a(n-4), cf. A111927.
k = 5: a(n) = 5*a(n-1) - 9*a(n-2) + 8*a(n-3) - 4*a(n-4), cf. A137221.
The sum of coefficients = 5 - k. Of the family k=3 gives the best recurrence.
a(n+m) = a(n)*A024493(m) + A024493(n)*a(m) + A024495(n)*A024495(m). - Vladimir Shevelev, Aug 01 2017
From Kevin Ryde, Sep 24 2020: (Start)
a(n) = (1/3)*2^n - (1/3)*cos((1/3)*Pi*n) + (1/sqrt(3))*sin((1/3)*Pi*n). [Cournot]
a(n) + A024495(n) + A111927(n) = 2^n - 1. [Cournot, page 96 last formula, but misprint should be 2^x - 1 rather than 2^p - 1]. (End)
a(n) = C(n,1) + C(n,4) + ... + C(n, 3*floor(n/3)+1). - Jianing Song, Oct 04 2021
E.g.f.: exp(x/2)*(exp(3*x/2) - cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Feb 06 2025

A153234 a(n) = floor(2^n/9).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 7, 14, 28, 56, 113, 227, 455, 910, 1820, 3640, 7281, 14563, 29127, 58254, 116508, 233016, 466033, 932067, 1864135, 3728270, 7456540, 14913080, 29826161, 59652323, 119304647, 238609294, 477218588, 954437176, 1908874353, 3817748707, 7635497415, 15270994830

Offset: 0

Paul Curtz, Dec 21 2008

Partial sums of A113405. - Mircea Merca, Dec 28 2010
Dubickas proves that infinitely many terms of this sequence are composite. - Charles R Greathouse IV, Feb 04 2016
Parity from a(4) onward gives A088911 (Period 6: repeat [1, 1, 1, 0, 0, 0]). - Jeremy Gardiner, Nov 04 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Artūras Dubickas, Prime and composite integers close to powers of a number, Monatsh. Math. 158:3 (2009), pp. 271-284.
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,3,-2).

Cf. A113405.

[Round((2*2^n-9)/18): n in [0..40]]; // Vincenzo Librandi, Jun 25 2011

seq(floor(2^n/9),n=0..25); # Mircea Merca, Dec 28 2010

Table[Floor[2^n/9], {n, 0, 37}] (* Michael De Vlieger, Mar 26 2016 *)

a(n)=2^n\9 \\ Charles R Greathouse IV, Oct 07 2015

x='x+O('x^44); concat(vector(4), Vec(x^4/((x-1)*(2*x-1)*(1+x)*(x^2-x+1)))) \\ Altug Alkan, Mar 25 2016

[floor(2^n/9) for n in (0..40)] # G. C. Greubel, Jun 05 2019

a(n+1) - 2*a(n) = A088911(n+3).
a(n) + a(n+3) = 2^n - 1 = A000225(n), n > 0.
From Mircea Merca, Dec 28 2010: (Start)
a(n) = round((2*2^n-9)/18) = floor((2^n-1)/9) = ceiling((2^n-8)/9).
a(n) = a(n-6) + 7*2^(n-6), n > 5. (End)
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + 3*a(n-4) - 2*a(n-5).
G.f.: x^4 / ( (1-2*x)*(1-x^2)*(1-x+x^2) ).
a(n) + a(n+1) = A111927(n). - R. J. Mathar, Apr 08 2013

More terms from Vincenzo Librandi, Jun 25 2011

A307393 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((1-x)^k-x^k).

Original entry on oeis.org

1, 1, 5, 1, 4, 16, 1, 4, 11, 42, 1, 4, 10, 26, 99, 1, 4, 10, 21, 57, 219, 1, 4, 10, 20, 42, 120, 466, 1, 4, 10, 20, 36, 84, 247, 968, 1, 4, 10, 20, 35, 64, 169, 502, 1981, 1, 4, 10, 20, 35, 57, 120, 340, 1013, 4017, 1, 4, 10, 20, 35, 56, 93, 240, 682, 2036, 8100

Offset: 0

Seiichi Manyama, Apr 07 2019

			Square array begins:
     1,   1,   1,   1,   1,   1,   1,   1, ...
     5,   4,   4,   4,   4,   4,   4,   4, ...
    16,  11,  10,  10,  10,  10,  10,  10, ...
    42,  26,  21,  20,  20,  20,  20,  20, ...
    99,  57,  42,  36,  35,  35,  35,  35, ...
   219, 120,  84,  64,  57,  56,  56,  56, ...
   466, 247, 169, 120,  93,  85,  84,  84, ...
   968, 502, 340, 240, 165, 130, 121, 120, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-5 give A002662(n+3), A125128(n+1), A111927(n+3), A000749(n+3), A139748(n+3).

Cf. A306915, A306846, A307078, A307394.

T[n_, k_] := Sum[Binomial[n+3, k*j + 3], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+3,k*j+3).

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i+1,k*j+1) * binomial(n-i+1,k*j+1).

A101825 G.f.: x*(1+x)^2/(1-x^3).

Original entry on oeis.org

0, 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, 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

Offset: 0

N. J. A. Sloane, Jun 17 2007

Period 3. Used in A042965.

First differences are essentially A049347. The binomial transform yields 0, 1, 4, 10,... i.e. A111927 with the first two zeros removed. - R. J. Mathar, May 14 2008

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

LinearRecurrence[{0,0,1},{0,1,2,1},120] (* Harvey P. Dale, Aug 01 2016 *)

x='x+O('x^50); Vec(x*(1+x)^2/(1-x^3)) \\ G. C. Greubel, May 03 2017

a(n) = A100063(n+1), n>0. - R. J. Mathar, Sep 01 2008

A173432 NW-SE diagonal sums of Riordan array A112468.

Original entry on oeis.org

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

Offset: 1

Mark Dols, Feb 18 2010

Matches Fibonacci-sequence, such that F(n) + a(n) and F(n) - a(n) = always even.

Periodic sequence with period: [1,1,2,1,1,0]. - Philippe Deléham, Oct 11 2011

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

Cf. A000045, A024495, A112468, A131531.

[2*Ceiling(n/6)-2*Floor(n/6)+Floor(n/3)-Ceiling(n/3) : n in [1..100]]; // Wesley Ivan Hurt, Sep 27 2014

A173432:=n->2*ceil(n/6)-2*floor(n/6)+floor(n/3)-ceil(n/3): seq(A173432(n), n=1..100); # Wesley Ivan Hurt, Sep 27 2014

Table[2 Ceiling[n/6] - 2 Floor[n/6] + Floor[n/3] - Ceiling[n/3], {n, 50}] (* Wesley Ivan Hurt, Sep 27 2014 *)

Vec(-x*(x^2+1) / ((x-1)*(x+1)*(x^2-x+1)) + O(x^100)) \\ Colin Barker, Sep 26 2014

a(n) = 1 + A131531(n) with inverse binomial transform: 1, 0, 1, -3, 6, -11, 21, .., a signed variant of A024495. - R. J. Mathar, Mar 04 2010
a(2n+1) = a(2n)-a(2n-1)+2, a(2n) = a(2n-1)-a(2n-2) with a(1) = a(2)=1. - Philippe Deléham, Oct 11 2011
a(n) = a(n-1)-a(n-3)+a(n-4). - Colin Barker, Sep 26 2014
G.f.: -x*(x^2+1) / ((x-1)*(x+1)*(x^2-x+1)). - Colin Barker, Sep 26 2014
a(n) = 2*ceiling(n/6)-2*floor(n/6)+floor(n/3)-ceiling(n/3). - Wesley Ivan Hurt, Sep 27 2014
a(n) = A001045(n) - A111927(n). - Paul Curtz, Dec 16 2020

Corrected and extended by Philippe Deléham, Oct 11 2011

A111926 Expansion of x^4/((1-2x)(x^2-x+1)*(x-1)^2).

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 15, 36, 78, 162, 331, 671, 1353, 2718, 5448, 10908, 21829, 43673, 87363, 174744, 349506, 699030, 1398079, 2796179, 5592381, 11184786, 22369596, 44739216, 89478457, 178956941, 357913911, 715827852, 1431655734, 2863311498

Offset: 0

Creighton Dement, Aug 21 2005

Binomial transform of sequence (0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0). Note: the binomial transform of the sequence (0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0) is A111927; the binomial transform of the sequence (0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0) is A024495 (disregarding first two terms, which are both zero).

Floretion Algebra Multiplication Program, FAMP Code: -4ibaseisumseq[ + .5'i + .5'j + .5'k + .5'ij' + .5'jk' + .5'ki' + e], sumtype: Y[8] = (int)Y[6] - (int)Y[7] + Y[8] + sum (internal program code).

Index entries for linear recurrences with constant coefficients, signature (5,-10,11,-7,2).

Cf. A000295, A111927, A024495.

CoefficientList[Series[x^4/((1-2x)(x^2-x+1)(x-1)^2),{x,0,40}],x] (* or *) LinearRecurrence[{5,-10,11,-7,2},{0,0,0,0,1},40] (* Harvey P. Dale, Feb 24 2016 *)

a(n+2) - a(n+1) + a(n) = A000295(n) = 2^n - n - 1 (Eulerian numbers).
a(n) = 1/3*2^n-n+2/3*(1/2+1/2*I*sqrt(3))^n*(-1/4-1/4*I*sqrt(3))+2/3*(1/2-1/2*I*sqrt(3))^n*(-1/4+1/4*I*sqrt(3)).
a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=1, a(n)=5*a(n-1)-10*a(n-2)+ 11*a(n-3)- 7*a(n-4)+2*a(n-5). - Harvey P. Dale, Feb 24 2016
a(n) = Sum_{k=1..floor(n/2)} binomial(n, 3*k+1). - Taras Goy, Jan 02 2025
E.g.f.: exp(x/2)*(exp(x/2)*(exp(x) - 3*x) - cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/3. - Stefano Spezia, Jan 03 2025

cp's OEIS Frontend

A024495 a(n) = C(n,2) + C(n,5) + ... + C(n, 3*floor(n/3)+2).

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A131708 A024494 prefixed by a 0.

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A153234 a(n) = floor(2^n/9).

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A307393 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((1-x)^k-x^k).

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A101825 G.f.: x*(1+x)^2/(1-x^3).

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PARI

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A173432 NW-SE diagonal sums of Riordan array A112468.

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PARI

Formula

A111926 Expansion of x^4/((1-2x)(x^2-x+1)*(x-1)^2).