A023610 -id:A023610 - cp's OEIS frontend

A240847 a(n) = 2a(n-1) + a(n-2) - 2a(n-3) - a(n-4) for n>3, a(0)=a(1)=a(3)=0, a(2)=1.

0, 0, 1, 0, 1, 0, 0, -2, -5, -12, -25, -50, -96, -180, -331, -600, -1075, -1908, -3360, -5878, -10225, -17700, -30509, -52390, -89664, -153000, -260375, -442032, -748775, -1265832, -2136000, -3598250, -6052061, -10164540

Offset: 0

Paul Curtz, Apr 13 2014

F1(m, n) is the difference table of a(n):
    0,   0,   1,   0,   1,   0,   0,  -2, ...
    0,   1,  -1,   1,  -1,   0,  -2,  -3, ...
    1,  -2,   2,  -2,   1,  -2,  -1,  -4, ...
   -3,   4,  -4,   3,  -3,   1,  -3,  -2, ...
    7,  -8,   7,  -6,   4,  -4,   1,  -4, ...
  -15,  15, -13,  10,  -8,   5,  -5,   1, ...
   30, -28,  23, -18,  13, -10,   6,  -6, ...
The recurrence holds for every row and every signed column.
Main diagonal: F1(n, n) = A001477(n).
First upper diagonal: F1(n, n+1) = -A001477(n).
F1(m, n) = F1(m, n-1) + F1(m+1, n-1).
Inverse binomial transform: 0, 0, 1, -3, 7, -15, 30, ... = 0, 0, followed by (-1)^n*A023610(n). Without signs: F2(0, n) = 0, 0, 1, 3, 7, 15, 30, ... = b(n) has the same recurrence.
F1(0, n) + F2(0, n) = 0, followed by A099920(n).
a(n) and b(n) are reciprocal by their inverse binomial transform.
0, followed by A001629(n) is an autosequence.
F1(m, 1) = (-1)^n*A029907(n).
F1(1, n) = 0, 1, -1, 1, -1, followed by -A226432(n+3).
F1(m, 2) = (-1)^n*A208354(n).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000032, A000045, A001629 (main sequence for the recurrence), A067331.

List([0..40], n-> (6*Fibonacci(n-3) - (n-3)*Lucas(1,-1,n-3)[2])/5 ); # G. C. Greubel, Feb 06 2020

[(6*Fibonacci(n-3) - (n-3)*Lucas(n-3))/5: n in [0..40]]; // G. C. Greubel, Feb 06 2020

with(combinat): seq( ((n+3)*fibonacci(n-3) - 2*(n-3)*fibonacci(n-2))/5, n=0..40); # G. C. Greubel, Feb 06 2020

a[n_]:= a[n]= 2*a[n-1] +a[n-2] -2*a[n-3] -a[n-4]; a[0]= a[1]= a[3]= 0; a[2]= 1; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Apr 17 2014 *)
CoefficientList[Series[x^2*(1-2*x)/(1-x-x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, May 09 2014 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,2d+c-2b-a}; NestList[nxt,{0,0,1,0},40][[All,1]] (* Harvey P. Dale, Sep 17 2022 *)

Vec(x^2*(1-2*x)/(1-x-x^2)^2 + O(x^100)) \\ Colin Barker, Apr 13 2014

vector(41, n, my(m=n-1); ((m+3)*fibonacci(m-3) - 2*(m-3)*fibonacci(m-2) )/5 ) \\ G. C. Greubel, Feb 06 2020

[((n+3)*fibonacci(n-3) - 2*(n-3)*fibonacci(n-2))/5 for n in (0..40)] # G. C. Greubel, Feb 06 2020

a(n) = 0, 0, 1, 0, 1, 0, 0, followed by -A067331.
G.f.: x^2*(1-2*x)/(1-x-x^2)^2. - Colin Barker, Apr 13 2014
a(n) = ( (10*n + (3-5*n)*t)*(1+t)^n + (10*n-(3-5*n)*t)*(1-t)^n )/(25*2^n), where t=sqrt(5). - Bruno Berselli, Apr 17 2014
a(n) = (6*Fibonacci(n-3) - (n-3)*Lucas(n-3))/5 = ((n+3)*Fibonacci(n-3) - 2*(n-3)*Fibonacci(n-2))/5. - G. C. Greubel, Feb 06 2020

A101350 Triangle read by rows: T(n,k) = number of k-matchings in the graph obtained by a zig-zag triangulation of a convex n-gon, T(0,0)=T(1,0)=T(2,0)=T(2,1)=1 (n > 2, 0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 5, 2, 1, 7, 7, 1, 9, 16, 3, 1, 11, 29, 15, 1, 13, 46, 43, 5, 1, 15, 67, 95, 30, 1, 17, 92, 179, 104, 8, 1, 19, 121, 303, 271, 58, 1, 21, 154, 475, 591, 235, 13, 1, 23, 191, 703, 1140, 705, 109, 1, 25, 232, 995, 2010, 1746, 506, 21, 1, 27, 277, 1359, 3309, 3780

Offset: 0

Emeric Deutsch, Dec 25 2004

			T(5,2)=7 because in the triangulation of the convex pentagon ABCDEA with diagonals AD and AC we have seven 2-matchings: {AB,CD},{AB,DE},{BC,AD},{BC,DE},{BC,EA},{CD,EA} and {DE,AC}.
Triangle begins:
  1;
  1;
  1,  1;
  1,  3;
  1,  5,  2;
  1,  7,  7;
  1,  9, 16,  3;
  1, 11, 29, 15;
  1, 13, 46, 43, 5;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..649

Row sums yield A000078 (the tetranacci numbers). T(2n+1, n) = A023610(n) (n > 0). T(2n, n) = A000045(n+1) (the Fibonacci numbers).

Cf. A000078, A023610, A000045.

G:=1/(1-z-t*z^2-t*z^3-t^2*z^4):Gserz:=simplify(series(G,z=0,18)):P[0]:=1: for n from 1 to 16 do P[n]:=sort(coeff(Gserz,z^n)) od:for n from 0 to 16 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od;# yields the sequence in triangular form

s(n) = 1/(1-x-y*x^2-y*x^3-y^2*x^4) + O(x^n);
my(gf=Pol(s(20))); for(n=0, poldegree(gf), my(p=polcoeff(gf,n)); for(k=0, poldegree(p), print1(polcoeff(p,k), ", ")); print) \\ Andrew Howroyd, Nov 04 2017

G.f.: 1/(1 - z - tz^2 - tz^3 - t^2z^4).

A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 5, 3, 0, 0, 8, 7, 1, 0, 0, 13, 15, 4, 0, 0, 0, 21, 30, 12, 1, 0, 0, 0, 34, 58, 31, 5, 0, 0, 0, 0, 55, 109, 73, 18, 1, 0, 0, 0, 0, 89, 201, 162, 54, 6, 0, 0, 0, 0, 0, 144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 10 2012

A skew version of A122075.

			Triangle begins :
  1
  2, 0
  3, 1, 0
  5, 3, 0, 0
  8, 7, 1, 0, 0
  13, 15, 4, 0, 0, 0
  21, 30, 12, 1, 0, 0, 0
  34, 58, 31, 5, 0, 0, 0, 0
  55, 109, 73, 18, 1, 0, 0, 0, 0
  89, 201, 162, 54, 6, 0, 0, 0, 0, 0
  144, 365, 344, 145, 25, 1, 0, 0, 0, 0, 0
  ...

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A122075, A122950, A000045, A023610, A129707

T[0, 0] := 1; T[1, 0] := 2; T[1, 1] := 0; T[n_, k_] := T[n, k] = If[n<0, 0, If[k > n, 0, T[n - 1, k] + T[n - 2, k] + T[n - 2, k - 1]]]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Dec 19 2017 *)

G.f.: (1+x)/(1-x-(1+y)*x^2).
T(n,k) = T(n-1,k) + T(n-2,k) + T(n-2,k-1), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0, T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A040000(n), A000045(n+2), A000079(n), A006138(n), A026597(n), A133407(n), A133467(n), A133469(n), A133479(n), A133558(n), A133577(n), A063092(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively.

A236076 A skewed version of triangular array A122075.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 3, 5, 0, 0, 1, 7, 8, 0, 0, 0, 4, 15, 13, 0, 0, 0, 1, 12, 30, 21, 0, 0, 0, 0, 5, 31, 58, 34, 0, 0, 0, 0, 1, 18, 73, 109, 55, 0, 0, 0, 0, 0, 6, 54, 162, 201, 89, 0, 0, 0, 0, 0, 1, 25, 145, 344, 365, 144, 0, 0, 0, 0, 0, 0, 7, 85, 361

Offset: 0

Philippe Deléham, Jan 19 2014

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

Subtriangle of the triangle A122950.

			Triangle begins:
  1;
  0,  2;
  0,  1,  3;
  0,  0,  3,  5;
  0,  0,  1,  7,  8;
  0,  0,  0,  4, 15, 13;
  0,  0,  0,  1, 12, 30, 21;
  0,  0,  0,  0,  5, 31, 58, 34;

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
H. Fuks and J.M.G. Soto, Exponential convergence to equilibrium in cellular automata asymptotically emulating identity, arXiv:1306.1189 [nlin.CG], 2013.

Cf. variant: A055830, A122075, A122950, A208337.
Cf. A167704 (diagonal sums), A000079 (row sums).
Cf. A111006.

a236076 n k = a236076_tabl !! n !! k
a236076_row n = a236076_tabl !! n
a236076_tabl = [1] : [0, 2] : f [1] [0, 2] where
   f us vs = ws : f vs ws where
     ws = [0] ++ zipWith (+) (zipWith (+) ([0] ++ us) (us ++ [0])) vs
-- Reinhard Zumkeller, Jan 19 2014

T[n_, k_]:= If[k<0 || k>n, 0, If[n==0 && k==0, 1, If[k==0, 0, If[n==1 && k==1, 2, T[n-1, k-1] + T[n-2, k-1] + T[n-2, k-2]]]]]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 21 2019 *)

{T(n,k) = if(k<0 || k>n, 0, if(n==0 && k==0, 1, if(k==0, 0, if(n==1 && k==1, 2, T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2) ))))}; \\ G. C. Greubel, May 21 2019

def T(n, k):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return 1
    elif (k==0): return 0
    elif (n==1 and k==1): return 2
    else: return T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 21 2019

G.f.: (1+x*y)/(1 - x*y - x^2*y - x^2*y^2).
T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0)=1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k < 0 or if k > n.
Sum_{k=0..n} T(n,k) = 2^n = A000079(n).
Sum_{n>=k} T(n,k) = A078057(k) = A001333(k+1).
T(n,n) = Fibonacci(n+2) = A000045(n+2).
T(n+1,n) = A023610(n-1), n >= 1.
T(n+2,n) = A129707(n).

A323212 The Fibonacci-Catalan Hybrid. Expansion of 1 + x(2(x + 1))/(sqrt(1 - 4y) - 2x*(x + 1) + 1). Square array read by descending antidiagonals, A(n,k) for n,k >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 3, 3, 0, 5, 7, 7, 5, 0, 14, 19, 19, 15, 8, 0, 42, 56, 56, 46, 30, 13, 0, 132, 174, 174, 146, 103, 58, 21, 0, 429, 561, 561, 477, 351, 220, 109, 34, 0, 1430, 1859, 1859, 1595, 1205, 801, 453, 201, 55, 0, 4862, 6292, 6292, 5434, 4180, 2884, 1756, 908, 365, 89

Offset: 0

Peter Luschny, Feb 14 2019

			      1,   0,    0,    0,     0,      0,      0,       0,       0, ...
      1,   1,    2,    5,    14,     42,    132,     429,    1430, ... [A000108]
      2,   3,    7,   19,    56,    174,    561,    1859,    6292, ... [A005807]
      3,   7,   19,   56,   174,    561,   1859,    6292,   21658, ... [A005807]
      5,  15,   46,  146,   477,   1595,   5434,   18798,   65858, ...
      8,  30,  103,  351,  1205,   4180,  14651,   51844,  185028, ...
     13,  58,  220,  801,  2884,  10372,  37401,  135420,  492558, ...
     21, 109,  453, 1756,  6621,  24674,  91532,  339184, 1257762, ...
     34, 201,  908, 3734, 14719,  56796, 216698,  821848, 3107583, ...
     55, 365, 1781, 7746, 31872, 127245, 499164, 1937439, 7470819, ...
A000045,A023610,...
Seen as a triangle a refinement of A000958:
[0]                                1
[1]                              0, 1
[2]                            0, 1, 2
[3]                           0, 2, 3, 3
[4]                         0, 5, 7, 7, 5
[5]                      0, 14, 19, 19, 15, 8
[6]                   0, 42, 56, 56, 46, 30, 13
[7]               0, 132, 174, 174, 146, 103, 58, 21
[8]            0, 429, 561, 561, 477, 351, 220, 109, 34
[9]       0, 1430, 1859, 1859, 1595, 1205, 801, 453, 201, 55

Antidiagonal sums (or row sums of the triangle) are A000958.

Cf. A000045, A000108, A023610, A005807.

gf := 1 + x*(2*(x + 1))/(sqrt(1 - 4*y) - 2*x*(x + 1) + 1):
serx := series(gf, x, 20): sery := n -> series(coeff(serx, x, n), y, 20):
row := n -> seq(coeff(sery(n), y, j), j=0..9):
seq(lprint(row(n)), n=0..9);

m = 11; T = PadRight[CoefficientList[#+O[y]^m, y], m]& /@ CoefficientList[1 + 2x(x+1)/(Sqrt[1-4y] - 2x(x+1) + 1) + O[x]^m, x]; Table[T[[n-k+1, k]], {n, 1, m}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Mar 20 2019 *)

A113684 Expansion of x(1-x^2-x^3)/((1-x)(1-x-x^2))^2.

Original entry on oeis.org

0, 1, 4, 11, 25, 52, 102, 193, 356, 645, 1153, 2040, 3580, 6241, 10820, 18671, 32089, 54956, 93826, 159745, 271300, 459721, 777409, 1312176, 2211000, 3719617, 6248452, 10482323, 17562841, 29391460, 49132638, 82048705, 136884260

Offset: 0

Paul Barry, Nov 05 2005

Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,0,-1)

Cf. A045925, A006478, A023610.

a(n)=4a(n-1)-4a(n-2)-2a(n-3)+4a(n-4)-a(n-6); a(n)=sum{k=0..n, (n-k)*C(n-k, k+1)}; a(n)=n*(F(n+2)-1)-(1+((n-5)*F(n-1)+(3n-8)*F(n))/5).

A129709 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 011 subwords (0<=k<=floor(n/3)). A Fibonacci binary word is a binary word having no 00 subword.

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 3, 6, 7, 7, 13, 1, 8, 22, 4, 9, 34, 12, 10, 50, 28, 1, 11, 70, 58, 5, 12, 95, 108, 18, 13, 125, 188, 50, 1, 14, 161, 308, 121, 6, 15, 203, 483, 261, 25, 16, 252, 728, 520, 80, 1, 17, 308, 1064, 968, 220, 7, 18, 372, 1512, 1710, 536, 33, 19, 444, 2100

Offset: 0

Emeric Deutsch, May 12 2007

Also number of Fibonacci binary words of length n and having k 110 subwords. Row n has 1+floor(n/3) terms. Row sums are the Fibonacci numbers (A000045). T(n,0)=n+1. Sum(k*T(n,k), k>=0)=A023610(n-3).

			T(7,2)=4 because we have 1011011,0111011,0110110 and 0110111.
Triangle starts:
1;
2;
3;
4,1;
5,3;
6,7;
7,13,1;
8,22,4;
9,34,12;
10,50,28,1;

Cf. A000045, A023610.

G:=(1+z)/(1-z-z^2+z^3-t*z^3): Gser:=simplify(series(G,z=0,23)): for n from 0 to 20 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 20 do seq(coeff(P[n],t,j),j=0..floor(n/3)) od; # yields sequence in triangular form

G.f.=G(t,z)=(1+z)/(1-z-z^2+z^3-tz^3).

cp's OEIS Frontend

A240847 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n>3, a(0)=a(1)=a(3)=0, a(2)=1.

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A101350 Triangle read by rows: T(n,k) = number of k-matchings in the graph obtained by a zig-zag triangulation of a convex n-gon, T(0,0)=T(1,0)=T(2,0)=T(2,1)=1 (n > 2, 0 <= k <= floor(n/2)).

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A209599 Triangle T(n,k), read by rows, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A236076 A skewed version of triangular array A122075.

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A323212 The Fibonacci-Catalan Hybrid. Expansion of 1 + x*(2*(x + 1))/(sqrt(1 - 4*y) - 2*x*(x + 1) + 1). Square array read by descending antidiagonals, A(n,k) for n,k >= 0.

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A113684 Expansion of x(1-x^2-x^3)/((1-x)(1-x-x^2))^2.

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A129709 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 011 subwords (0<=k<=floor(n/3)). A Fibonacci binary word is a binary word having no 00 subword.

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A240847 a(n) = 2a(n-1) + a(n-2) - 2a(n-3) - a(n-4) for n>3, a(0)=a(1)=a(3)=0, a(2)=1.

A323212 The Fibonacci-Catalan Hybrid. Expansion of 1 + x(2(x + 1))/(sqrt(1 - 4y) - 2x*(x + 1) + 1). Square array read by descending antidiagonals, A(n,k) for n,k >= 0.