A054454 -id:A054454 - cp's OEIS frontend

A202453 Fibonacci self-fusion matrix, by antidiagonals.

1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 5, 5, 6, 5, 5, 8, 8, 9, 9, 8, 8, 13, 13, 15, 15, 15, 13, 13, 21, 21, 24, 24, 24, 24, 21, 21, 34, 34, 39, 39, 40, 39, 39, 34, 34, 55, 55, 63, 63, 64, 64, 63, 63, 55, 55, 89, 89, 102, 102, 104, 104, 104, 102, 102, 89, 89, 144, 144, 165, 165

Offset: 1

Clark Kimberling, Dec 19 2011

The Fibonacci self-fusion matrix, F, is the fusion P**Q, where P and Q are the lower and upper triangular Fibonacci matrices.  See A193722 for the definition of fusion of triangular arrays.
Every term F(n,k) of F is a product of two Fibonacci numbers; indeed,
F(n,k)=F(n)*F(k+1) if k is even;
F(n,k)=F(n+1)*F(k) if k is odd.
antidiagonal sums: (1,2,6,12,...), A054454
diagonal (1,2,6,15,...), A001654
diagonal (1,3,9,24,...), A064831
diagonal (2,5,15,39,..), A059840
diagonal (3,8,24,63,..), A080097
diagonal (5,13,39,102,...), A080143
diagonal (8,21,63,165,...), A080144
principal submatrix sums, A202462
All the principal submatrices are invertible, and the terms in the inverses are in {-3,-2,-1,0,1,2,3}.

			Northwest corner:
1...1....2....3....5....8....13
1...2....3....5....8...13....21
2...3....6....9...15...24....39
3...5....9...15...24...39....63
5...8...15...24...40...64...104

Clark Kimberling, Fusion, Fission, and Factors, Fib. Q., 52 (2014), 195-202.

Cf. A000045, A202451, A202452, A202503 (Fibonacci fission array).

n = 12;
Q = NestList[Most[Prepend[#, 0]] &, #, Length[#] - 1] &[Table[Fibonacci[k], {k, 1, n}]];
P = Transpose[Q]; F = P.Q;
Flatten[Table[P[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202452 as a sequence *)
Flatten[Table[Q[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202451 as a sequence *)
Flatten[Table[F[[i]][[k + 1 - i]], {k, 1, n}, {i, 1, k}]] (* A202453 as a sequence *)
TableForm[Q]  (* A202451, upper tri. Fibonacci array *)
TableForm[P]  (* A202452, lower tri. Fibonacci array *)
TableForm[F]  (* A202453, Fibonacci fusion array *)
TableForm[FactorInteger[F]]

Matrix product P*Q, where P, Q are the lower and upper triangular Fibonacci matrices, A202451 and A202452.

A002940 Arrays of dumbbells.

Original entry on oeis.org

1, 4, 11, 26, 56, 114, 223, 424, 789, 1444, 2608, 4660, 8253, 14508, 25343, 44030, 76136, 131110, 224955, 384720, 656041, 1115784, 1893216, 3205416, 5416441, 9136084, 15384563, 25866914, 43429784, 72821274, 121953943, 204002680, 340886973, 569047468, 949022608

Offset: 1

N. J. A. Sloane

Whitney transform of n. The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)). - Paul Barry, Feb 16 2005
a(n-1) is the permanent of the n X n 0-1 matrix with 1 in (i,j) position iff (i=1 and j1). For example, with n=5, a(4) = per([[1, 1, 1, 1, 0], [1, 1, 1, 1, 1], [1, 1, 1, 1, 1], [0, 1, 1, 1, 1], [0, 0, 1, 1, 1]]) = 26. - David Callan, Jun 07 2006
a(n) is the internal path length of the Fibonacci tree of order n+2. A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node. The internal path length of a tree is the sum of the levels of all of its internal (i.e. non-leaf) nodes. - Emeric Deutsch, Jun 15 2010
Partial Sums of A023610 - John Molokach, Jul 03 2013

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983,(2.3.14).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Carlos Alirio Rico Acevedo and Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 23-24.
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15 (1974), 214-216.
R. C. Grimson, Exact formulas for 2 x n arrays of dumbbells, J. Math. Phys., 15.2 (1974), 214-216. (Annotated scanned copy)
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
R. B. McQuistan and S. J. Lichtman, Exact recursion relation for 2 x N arrays of dumbbells, J. Math. Phys., 11 (1970), 3095-3099.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Cf. A002941, A002889, A055608, A062123, A062124, A062125, A062126, A062127, A046741.

Cf. A000045, A001925, A023610, A054454, A006478, A067331, A178523.

a002940 n = a002940_list !! (n-1)
a002940_list = 1 : 4 : 11 : zipWith (+)
   (zipWith (-) (map (* 2) $ drop 2 a002940_list) a002940_list)
   (drop 5 a000045_list)
-- Reinhard Zumkeller, Jan 18 2014

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)/((1-x)*(1-x-x^2)^2) )); // G. C. Greubel, Jan 31 2019

a[n_]:= a[n]= If[n<3, n^2, 2a[n-1] -a[n-3] +Fibonacci[n+1]]; Array[a, 32] (* Jean-François Alcover, Jul 31 2018 *)

my(x='x+O('x^35)); Vec((1+x)/((1-x)*(1-x-x^2)^2)) \\ G. C. Greubel, Jan 31 2019

((1+x)/((1-x)*(1-x-x^2)^2)).series(x, 35).coefficients(x, sparse=False) # G. C. Greubel, Jan 31 2019

a(n) = 2*a(n-1) - a(n-3) + A000045(n+1).
G.f.: x*(1+x)/((1-x)*(1-x-x^2)^2).
a(n) = Sum_{k=0..n} ( Sum_{i=0..n} k*C(k, i-k) ). - Paul Barry, Feb 16 2005
E.g.f.: 2*exp(x) + exp(x/2)*((55*x - 50)*cosh(sqrt(5)*x/2) + sqrt(5)*(25*x - 22)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 03 2023

More terms from Henry Bottomley, Jun 02 2000

A225084 Triangle read by rows: T(n,k) is the number of compositions of n with maximal up-step k; n>=1, 0<=k

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 5, 2, 1, 0, 7, 6, 2, 1, 0, 11, 12, 6, 2, 1, 0, 15, 26, 14, 6, 2, 1, 0, 22, 50, 33, 14, 6, 2, 1, 0, 30, 97, 72, 34, 14, 6, 2, 1, 0, 42, 180, 156, 77, 34, 14, 6, 2, 1, 0, 56, 332, 328, 173, 78, 34, 14, 6, 2, 1, 0, 77, 600, 681, 378, 177, 78, 34, 14, 6, 2, 1, 0, 101, 1078, 1393, 818, 393, 178, 78, 34, 14, 6, 2, 1, 0

Offset: 1

Joerg Arndt, Apr 27 2013

T(n,k) is the number of compositions [p(1), p(2), ..., p(k)] of n such that max(p(j) - p(j-1)) == k.
The first column is A000041 (partition numbers).
Sum of first and second column is A003116.
Sum of the first three columns is A224959.
The second columns deviates from A054454 after the term 600.
Row sums are A011782.

			Triangle starts:
01: 1,
02: 2, 0,
03: 3, 1, 0,
04: 5, 2, 1, 0,
05: 7, 6, 2, 1, 0,
06: 11, 12, 6, 2, 1, 0,
07: 15, 26, 14, 6, 2, 1, 0,
08: 22, 50, 33, 14, 6, 2, 1, 0,
09: 30, 97, 72, 34, 14, 6, 2, 1, 0,
10: 42, 180, 156, 77, 34, 14, 6, 2, 1, 0,
11: 56, 332, 328, 173, 78, 34, 14, 6, 2, 1, 0,
12: 77, 600, 681, 378, 177, 78, 34, 14, 6, 2, 1, 0,
13: 101, 1078, 1393, 818, 393, 178, 78, 34, 14, 6, 2, 1, 0,
14: 135, 1917, 2821, 1746, 863, 397, 178, 78, 34, 14, 6, 2, 1, 0,
15: 176, 3393, 5660, 3695, 1872, 877, 398, 178, 78, 34, 14, 6, 2, 1, 0,
...
The fifth row corresponds to the following statistics:
#:  M   composition
01:  0  [ 1 1 1 1 1 ]
02:  1  [ 1 1 1 2 ]
03:  1  [ 1 1 2 1 ]
04:  2  [ 1 1 3 ]
05:  1  [ 1 2 1 1 ]
06:  1  [ 1 2 2 ]
07:  2  [ 1 3 1 ]
08:  3  [ 1 4 ]
09:  0  [ 2 1 1 1 ]
10:  1  [ 2 1 2 ]
11:  0  [ 2 2 1 ]
12:  1  [ 2 3 ]
13:  0  [ 3 1 1 ]
14:  0  [ 3 2 ]
15:  0  [ 4 1 ]
16:  0  [ 5 ]
There are 7 compositions with no up-step (M=0), 6 with M=1, 2 with M=2, and 1 with M=3.

Joerg Arndt and Alois P. Heinz, Rows n = 1..141, flattened

Cf. A225085 (partial sums of rows).

T(2n,n) gives A059570.

b:= proc(n, v) option remember; `if`(n=0, 1, add((p->
      `if`(i seq(coeff(b(n, 0), x, i), i=0..n-1):
seq(T(n), n=1..14);  # Alois P. Heinz, Feb 22 2014

b[n_, v_] := b[n, v] = If[n == 0, 1, Sum[Function[{p}, If[iJean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

A054453 Triangle of partial row sums of triangle A054450(n,m), n >= m >= 0.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 5, 2, 1, 15, 10, 6, 2, 1, 28, 20, 12, 7, 2, 1, 51, 38, 26, 14, 8, 2, 1, 92, 71, 50, 33, 16, 9, 2, 1, 164, 130, 97, 64, 41, 18, 10, 2, 1, 290, 235, 180, 130, 80, 50, 20, 11, 2, 1, 509, 420, 332, 244, 171, 98, 60, 22, 12, 2, 1

Offset: 0

Wolfdieter Lang, Apr 27 2000 and May 08 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is ((1-z^2)*(Fib(z))^2)/(1-x*z/(1-z^2)) Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).
This is the second member of the family of Riordan-type matrices obtained from the unsigned convolution matrix A049310(n,m) by repeated application of the partial row sums procedure.
The column sequences are A029907, A001629, A054454 for m=0..2.

			{1}; {2,1}; {4,2,1}; {8,5,2,1};...
Fourth row polynomial (n=3): p(3,x)= 8+5*x+2*x^2+x^3

Gregg Musiker, Nick Ovenhouse, and Sylvester W. Zhang, Double Dimers and Super Ptolemy Relations, Séminaire Lotharingien de Combinatoire XX, Proc. 35th Conf. Formal Power, Series and Algebraic Combinatorics (Davis) 2023, Art. #YY. See p. 12.

Cf. A049310, A054450, A000045, A029907, A001629. Row sums: A054455(n).

a(n, m)=sum(A054450(n, k), k=m..n), n >= m >= 0, a(n, m) := 0 if n

Column m recursion: a(n, m)= sum(a(j-1, m)*|A049310(n-j, 0)|, j=m..n) + A054450(n, m), n >= m >= 0, a(n, m) := 0 if n

G.f. for column m: ((1-x^2)*(Fib(x))^2)*(x/(1-x^2))^m, m >= 0, with Fib(x) G.f. for A000045(n+1).

A191308 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n having k peaks at height >=2.

Original entry on oeis.org

1, 1, 2, 3, 5, 1, 8, 2, 13, 6, 1, 21, 12, 2, 34, 26, 9, 1, 55, 50, 19, 2, 89, 97, 52, 13, 1, 144, 180, 108, 28, 2, 233, 332, 243, 97, 18, 1, 377, 600, 488, 210, 39, 2, 610, 1076, 999, 552, 170, 24, 1, 987, 1908, 1942, 1168, 376, 52, 2, 1597, 3361, 3773, 2663, 1162, 282, 31, 1, 2584, 5878, 7140, 5466, 2543, 630, 67, 2

Offset: 0

Emeric Deutsch, May 30 2011

Also number of dispersed Dyck paths of length n having k valleys (i.e., DU's). Example: T(7,2)=2 because we have HUDUDUD and UDUDUDH, where U=(1,1), D=(1,-1), H=(1,0).
Also number of dispersed Dyck paths of length n having k doublerises (i.e., UU's). Example: T(7,2)=2 because we have HUUUDDD and UUUDDDH, where U=(1,1), D=(1,-1), H=(1,0).
Row n>=2 has floor(n/2) entries.
Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
T(n,0) = A000045(n+1) (the Fibonacci numbers).
T(n,1) = A054454(n-4).
Sum_{k>=0} k*T(n,k) = A191309(n).

			T(7,2)=2 because we have HUUDUDD and UUDUDDH, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
   1;
   1;
   2;
   3;
   5,  1;
   8,  2;
  13,  6,  1;
  21, 12,  2;

Cf. A000045, A001405, A054454, A191309.

r := ((1-z^2-t*z^2-sqrt(((1-z)^2-t*z^2)*((1+z)^2-t*z^2)))*1/2)/(t*z^2): G := 1/(1-z-z^2*(1+t*r)): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: 1; 1; for n from 2 to 17 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)-1) end do; # yields sequence in triangular form

G.f.: G=G(t,z) satisfies G = 1+z*G + z^2*G(1+t*r), where r=r(t,z) is given by r=z^2*(1+r)*(1+t*r) (the Narayana function).

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A202453 Fibonacci self-fusion matrix, by antidiagonals.

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A002940 Arrays of dumbbells.

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A225084 Triangle read by rows: T(n,k) is the number of compositions of n with maximal up-step k; n>=1, 0<=k

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A054453 Triangle of partial row sums of triangle A054450(n,m), n >= m >= 0.

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A191308 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n having k peaks at height >=2.

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