A129707 -id:A129707 - cp's OEIS frontend

A055243 First differences of A001628 (Fibonacci convolution).

1, 2, 6, 13, 29, 60, 122, 241, 468, 894, 1686, 3144, 5807, 10636, 19338, 34931, 62731, 112068, 199264, 352787, 622152, 1093260, 1914780, 3343440, 5821645, 10110278, 17515566, 30276073, 52221929, 89896332, 154461110, 264930661, 453654108, 775598634, 1324053522

Offset: 0

Wolfdieter Lang, May 10 2000

2*a(n) = C_{n+3} of Turban reference eq.(2.17), C_{1}= 0 = C_{2}.

Number of binary sequences of length n+3 such that the sequence has exactly two pairs (which may overlap) of consecutive 1's. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 07 2004

Alois P. Heinz, Table of n, a(n) for n = 0..1000
L. Turban, Lattice animals on a staircase and Fibonacci numbers, arXiv:cond-mat/0011038 [cond-mat.stat-mech], 2000; J.Phys. A 33 (2000) 2587-2595.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. A001628, A000045.

Cf. A010049, A076791, A094440, A129707.

a:= n -> (Matrix([[1,0$4,-1]]). Matrix(6, (i,j)-> if (i=j-1) then 1 elif j=1 then [3,0,-5,0,3,1][i] else 0 fi)^(n))[1,1]: seq(a(n), n=0..30); # Alois P. Heinz, Aug 05 2008

Differences[LinearRecurrence[{3,0,-5,0,3,1},{0,1,3,9,22,51,111},40]] (* Harvey P. Dale, Jun 12 2019 *)

G.f.: (1-x)/(1-x-x^2)^3. (from Turban reference eq.(2.15)).
a(n)= ((5*n^2+37*n+50)*F(n+1)+4*(n+1)*F(n))/50 with F(n)=A000045(n) (Fibonacci numbers) (from Turban reference eq. (2.17)).
From Peter Bala, Oct 25 2007 (Start):
Since F(-n) = (-1)^(n+1)*F(n), we can use the previous formula to extend the sequence to negative values of n; we find a(-n) = (-1)^n* A129707(n-3).
Recurrence relations: a(n+4) = 2*a(n+3) + a(n+2) - 2*a(n+1) - a(n) + F(n+3), with a(0) = 1, a(1) = 2, a(2) = 6 and a(3) = 13;
a(n+2) = a(n+1) + a(n) + A010049(n+3), with a(0) = 1 and a(1) = 2.
a(n-3) = Sum_{k = 2..floor((n+1)/2)} C(k,2)*C(n-k,k-1) = (1/2)*G''(n,1), where the polynomial G(n,x) := Sum_{k = 1..floor((n+1)/2)} C(n-k,k-1)*x^k = x^((n+1)/2) * F(n, 1/sqrt(x)) and where F(n,x) denotes the n-th Fibonacci polynomial. Since G(n,1) yields the Fibonacci numbers A000045 and G'(n,1) yields the second-order Fibonacci numbers A010049, a(n) may be considered as the sequence of third-order Fibonacci numbers.
For n >= 4, the polynomials Sum_{k = 0..n} C(n,k) * G''(n-k,1)*(-x)^k appear to satisfy a Riemann hypothesis; their zeros appear to lie on the vertical line Re x = 1/2 in the complex plane. Compare with the remarks in A094440 and A010049. (End)
a(n) = A076791(n+3, 2). - Michael Somos, Sep 24 2024
E.g.f.: exp(x/2)*(5*(25 + 23*x + 5*x^2)*cosh(sqrt(5)*x/2) + sqrt(5)*(29 + 65*x + 10*x^2)*sinh(sqrt(5)*x/2))/125. - Stefano Spezia, Sep 26 2024

A123585 Triangle T(n,k), 0<=k<=n, given by [1, -1, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 0, 2, 2, -1, 1, 5, 3, -1, -2, 4, 10, 5, 0, -4, -4, 12, 20, 8, 1, -2, -13, -4, 31, 38, 13, 1, 3, -11, -33, 3, 73, 71, 21, 0, 6, 6, -42, -74, 34, 162, 130, 34, -1, 3, 24, 0, -130, -146, 128, 344, 235, 55, -1, -4, 21, 72, -50, -352

Offset: 0

Philippe Deléham, Nov 13 2006

			Triangle begins:
1;
1, 1;
0, 2, 2;
-1, 1, 5, 3;
-1, -2, 4, 10, 5;
0, -4, -4, 12, 20, 8;
1, -2, -13, -4, 31, 38, 13;
1, 3, -11, -33, 3, 73, 71, 21;
0, 6, 6, -42, -74, 34, 162, 130, 34;

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

Cf. A010892, A099254, A000045, A000079 (row sums), A001629, A129707.

CoefficientList[CoefficientList[Series[1/(1 - (1 + y)*x + (1 - y^2)*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Oct 16 2017 *)

Sum_{k,0<=k<=n} T(n,k) = 2^n = A000079(n).
T(n,0) = A010892(n).
T(n,n) = Fibonacci(n+1) = A000045(n+1).
T(n+1,1) = A099254(n).
T(n+1,n) = A001629(n+2).
Sum_{k, 0<=k<=[n/2]} T(n-k,k) = A003269(n).
T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k-2) - T(n-2,k), n>0.
Sum_{k, 0<=k<=n} x^k*T(n,k) = (-1)^n*A003683(n+1), (-1)^n*A006130(n), A000007(n), A010892(n), A000079(n), A030195(n+1) for x=-3, -2, -1, 0, 1, 2 respectively . - Philippe Deléham, Dec 01 2006
T(n+2,n) = A129707(n+1).- Philippe Deléham, Dec 18 2011
G.f.: 1/(1-(1+y)*x+(1-y^2)*x^2). - Philippe Deléham, Dec 18 2011

A129706 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k inversions (n>=0, 0<=k<=floor(n(n+1)/6)). A Fibonacci binary word is a binary word having no 00 subword.

Original entry on oeis.org

1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 4, 4, 4, 2, 1, 2, 2, 2, 4, 4, 6, 6, 4, 2, 2, 2, 2, 2, 4, 4, 6, 8, 8, 6, 6, 4, 2, 1, 2, 2, 2, 4, 4, 6, 8, 10, 10, 10, 10, 8, 6, 4, 2, 1, 2, 2, 2, 4, 4, 6, 8, 10, 12, 14, 14, 14, 14, 12, 10, 8, 4, 2, 2, 2, 2, 2, 4, 4, 6, 8, 10, 12, 16, 18, 18, 20

Offset: 0

Emeric Deutsch, May 12 2007

Row n has 1+floor(n(n+1)/6) terms. Row sums are the Fibonacci numbers (A000045). Sum(k*T(n,k), k>=0)=A129707(n).

			T(5,3)=4 because we have 11101, 10101, 01110 and 01010.
Triangle starts:
1;
2;
2,1;
2,2,1;
2,2,2,2;
2,2,2,4,2,1;
2,2,2,4,4,4,2,1;

Cf. A000045, A129707.

Q[0]:=1: Q[1]:=1+x: for n from 2 to 12 do Q[n]:=expand(x*Q[n-1]+t*x*subs(x=t*x,Q[n-2])) od: for n from 0 to 15 do P[n]:=sort(subs(x=1,Q[n])) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..floor(n*(n+1)/6)) od; # yields sequence in triangular form

G.f.=G(t,z)=H(t,1,z), where H(t,x,z)=1+z+xzH(t,x,z)+txz^2*H(t,tx,z). Row generating polynomials P[n] are given by P[n](t)=Q[n](t,1), where Q[0]=1, Q[1]=1+x, Q[n](t,x)=xQ[n-1](t,x)+txQ[n-2](t,tx) for n>=2.

A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 2, 0, 3, 5, 1, -1, 5, 10, 4, -2, -1, 8, 20, 12, -4, -4, 0, 13, 38, 31, -4, -13, -2, 1, 21, 71, 73, 3, -33, -11, 3, 1, 34, 130, 162, 34, -74, -42, 6, 6, 0, 55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Offset: 0

Philippe Deléham, Dec 06 2011

Row-reversed variant of A123585. Row sums: 2^n.

			Triangle begins:
1
1, 1
2, 2, 0
3, 5, 1, -1
5, 10, 4, -2, -1
8, 20, 12, -4, -4, 0
13, 38, 31, -4, -13, -2, 1
21, 71, 73, 3, -33, -11, 3, 1
34, 130, 162, 34, -74, -42, 6, 6, 0
55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Cf. Columns: A000045, A001629, A129707.

Diagonals: A010892, A099254, Antidiagonal sums: A158943.

G.f.: 1/(1-(1+y)*x+(y+1)*(y-1)*x^2).
T(n,0) = A000045(n+1).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0)= 1 and T(n,k)= 0 if nSum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A090591(n), (-1)^n*A106852(n), A000007(n), A000045(n+1), A000079(n), A057083(n), A190966(n+1) for n = -3, -2, -1, 0, 1, 2, 3 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A010892(n), A000079(n), A030195(n+1), A180222(n+2) for x = 0, 1, 2, 3 respectively.

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).

A112973 Riordan array (1/(1-x-x^2), x(1+x)/(1-x-x^2)^2).

Original entry on oeis.org

1, 1, 1, 2, 4, 1, 3, 12, 7, 1, 5, 31, 31, 10, 1, 8, 73, 110, 59, 13, 1, 13, 162, 340, 267, 96, 16, 1, 21, 344, 956, 1022, 529, 142, 19, 1, 34, 707, 2507, 3479, 2416, 923, 197, 22, 1, 55, 1416, 6231, 10850, 9657, 4900, 1476, 261, 25, 1, 89, 2778, 14840, 31606, 34905

Offset: 0

Paul Barry, Oct 07 2005

Row sums are A091702. Diagonal sums are A052960. First column is A000045(n+1).

Second column is A129707. - Ralf Stephan, Dec 31 2013

			Rows begin
1;
1,1;
2,4,1;
3,12,7,1;
5,31,31,10,1;
8,73,110,59,13,1;

T(n,k):=sum(binomial(m,n-m)*binomial(m+k,2*k),m,floor(n/2),n); /* Vladimir Kruchinin, Apr 21 2015 */

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) - 2*T(n-3,k) - T(n-4,k), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = 2, T(2,1) = 4, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 21 2014
G.f.: (x^2+x-1)/((x^2+x)*y-x^4-2*x^3+x^2+2*x-1). - Vladimir Kruchinin, Apr 21 2015
T(n,k) = Sum_{m=floor(n/2)..n} C(m,n-m)*C(m+k,2*k). - Vladimir Kruchinin, Apr 21 2015

Definition corrected by Ralf Stephan, Dec 31 2013

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A055243 First differences of A001628 (Fibonacci convolution).

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A123585 Triangle T(n,k), 0<=k<=n, given by [1, -1, 1, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.

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A129706 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k inversions (n>=0, 0<=k<=floor(n(n+1)/6)). A Fibonacci binary word is a binary word having no 00 subword.

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A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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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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A112973 Riordan array (1/(1-x-x^2), x(1+x)/(1-x-x^2)^2).

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