A067331 -id:A067331 - cp's OEIS frontend

A002940 Arrays of dumbbells.

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

A067330 Triangle read by rows of incomplete convolutions of Fibonacci numbers F(n+1) = A000045(n+1), n>=0.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 3, 5, 7, 10, 5, 8, 12, 15, 20, 8, 13, 19, 25, 30, 38, 13, 21, 31, 40, 50, 58, 71, 21, 34, 50, 65, 80, 96, 109, 130, 34, 55, 81, 105, 130, 154, 180, 201, 235, 55, 89, 131, 170, 210, 250, 289, 331, 365, 420, 89, 144, 212, 275, 340, 404, 469, 532, 600, 655, 744, 144, 233, 343, 445

Offset: 0

Wolfdieter Lang, Feb 15 2002

The diagonals d>=0 (d=0: main diagonal) give convolutions of Fibonacci numbers F(n+1), n>=0, with those with d-shifted index: a(d+n,d)=sum(F(k+1)*F(d+n+1-k),k=0..n), n>=0.
The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(x*z)*(A(z)-x*A(x*z))/(1-x), with A(x) := 1/(1-x-x^2) (g.f. Fibonacci F(n+1), n>=0).
The diagonals give A001629(n+2), A023610, A067331-4, A067430-1, A067977-8 for d= n-m= 0..9, respectively.
A row with n terms = the dot product of vectors with n terms: (1,1,2,3,...)dot(...3,2,1,1) with carryovers; such that (3, 5, 7, 10) = (1*3=3), (1*2+3=5), (2*1+5=7), (3*1+7=10).

			{1}; {1,2}; {2,3,5}; {3,5,7,10}; ...; p(2,n)= 2+3*x+5*x^2.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A067418 (triangle with rows read backwards).

Table[Sum[Fibonacci[k + 1] Fibonacci[n - k + 1], {k, 0, m}], {n, 0, 11}, {m, 0, n}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

a(n, m)= sum(F(k+1)*F(n-k+1), k=0..m), n>=m>=0, else 0.
a(n, m)= (((3*m+5)*F(m+1)+(m+1)*F(m))*F(n-m+1)+(m*F(m+1)+2*(m+1)*F(m))*F(n-m))/5.
G.f. for diagonals d=n-m>=0: (x^d)*(F(d+1)+F(d)*x)/(1-x-x^2)^2, with F(n) := A000045(n) (Fibonacci).
a(n, m) = ((-1)^m*F(n-2*m-1)+m*L(n+2)+5*F(n)+4*F(n-1))/5, with F(-n) = (-1)^(n+1)*F(n), hence a(n, m) = (2*(m+1)*L(n+2)-A067979(n, m))/5, n>=m>=0. - Ehren Metcalfe, Apr 11 2016

A067418 Triangle A067330 with rows read backwards.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 10, 7, 5, 3, 20, 15, 12, 8, 5, 38, 30, 25, 19, 13, 8, 71, 58, 50, 40, 31, 21, 13, 130, 109, 96, 80, 65, 50, 34, 21, 235, 201, 180, 154, 130, 105, 81, 55, 34, 420, 365, 331, 289, 250, 210, 170, 131, 89, 55, 744, 655, 600, 532, 469, 404, 340, 275, 212, 144, 89, 1308, 1164, 1075, 965

Offset: 0

Wolfdieter Lang, Feb 15 2002

The column m (without leading 0's) gives the convolution of Fibonacci numbers F(n+1) := A000045(n+1), n>=0, with those with m-shifted index: a(n+m,m)=sum(F(k+1)*F(m+n+1-k),k=0..n), n>=0, m=0,1,...
The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(z)*(A(z)-x*A(x*z))/(1-x), with A(x) := 1/(1-x-x^2) (g.f. for Fibonacci F(n+1), n>=0).
The columns give A001629(n+2), A023610, A067331-4, A067430-1, A067977-8 for m= 0..9, respectively. Row sums give A067988.

			{1}; {2,1}; {5,3,2}; {10,7,5,3}; ...; p(2,n)=5+3*x+2*x^2.

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Reverse /@ Table[Sum[Fibonacci[k + 1] Fibonacci[n - k + 1], {k, 0, m}], {n, 0, 11}, {m, 0, n}] // Flatten (* Michael De Vlieger, Apr 11 2016 *)

a(n, m)=A067330(n, n-m), n>=m>=0, else 0.
a(n, m)= (((3*(n-m)+5)*F(n-m+1)+(n-m+1)*F(n-m))*F(m+1)+((n-m)*F(n-m+1)+2*(n-m+1)*F(n-m))*F(m))/5.
G.f. for column m=0, 1, ...: (x^m)*(F(m+1)+F(m)*x)/(1-x-x^2)^2, with F(m) := A000045(m) (Fibonacci).
a(n, m) = ((-1)^m*F(n-2*m+1)-m*L(n+2)+n*L(n+2)+5*F(n)+4*F(n-1))/5, with F(-n) = (-1)^(n+1)*F(n), hence a(n, m) = (2*(n-m+1)*L(n+2)-A067990(n, m))/5, n>=m>=0. - Ehren Metcalfe, Apr 11 2016

A226545 Number A(n,k) of squares in all tilings of a k X n rectangle using integer-sided square tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 5, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 25, 34, 25, 5, 0, 0, 6, 50, 98, 98, 50, 6, 0, 0, 7, 96, 256, 386, 256, 96, 7, 0, 0, 8, 180, 654, 1402, 1402, 654, 180, 8, 0, 0, 9, 331, 1625, 4938, 6940, 4938, 1625, 331, 9, 0

Offset: 0

Alois P. Heinz, Jun 10 2013

			A(3,3) = 1 + 6 + 6 + 6 + 6 + 9 = 34:
  ._____.  ._____.  ._____.  ._____.  ._____.  ._____.
  |     |  |   |_|  |_|   |  |_|_|_|  |_|_|_|  |_|_|_|
  |     |  |___|_|  |_|___|  |_|   |  |   |_|  |_|_|_|
  |_____|  |_|_|_|  |_|_|_|  |_|___|  |___|_|  |_|_|_|
Square array A(n,k) begins:
  0, 0,   0,    0,     0,      0,       0,        0, ...
  0, 1,   2,    3,     4,      5,       6,        7, ...
  0, 2,   5,   12,    25,     50,      96,      180, ...
  0, 3,  12,   34,    98,    256,     654,     1625, ...
  0, 4,  25,   98,   386,   1402,    4938,    16936, ...
  0, 5,  50,  256,  1402,   6940,   33502,   157279, ...
  0, 6,  96,  654,  4938,  33502,  221672,  1426734, ...
  0, 7, 180, 1625, 16936, 157279, 1426734, 12582472, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened

Columns (or rows) k=0-10 give: A000004, A001477, A067331(n-1) for n>0, A226546, A226547, A226548, A226549, A226550, A226551, A226552, A226553.
Main diagonal gives A226554.
Cf. A113881, A219924.

b:= proc(n, l) option remember; local i, k, s, t;
      if max(l[])>n then [0,0] elif n=0 or l=[] then [1,0]
    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))
    else for k do if l[k]=0 then break fi od; s:=[0$2];
         for i from k to nops(l) while l[i]=0 do s:=s+(h->h+[0, h[1]])
           (b(n, [l[j]$j=1..k-1, 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))
         od; s
      fi
    end:
A:= (n, k)-> `if`(n>=k, b(n, [0$k]), b(k, [0$n]))[2]:
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, l_List] := b[n, l] = Module[{i, k, s, t}, Which[Max[l] > n, {0, 0}, n == 0 || l == {}, {1, 0}, Min[l] > 0, t=Min[l]; b[n-t, l-t], True, k = Position[l, 0, 1][[1, 1]]; s={0, 0}; For[i=k, i <= Length[l] && l[[i]] == 0, i++, s = s + Function[h, h+{0, h[[1]]}][b[n, Join[l[[1 ;; k-1]], Table[1+i-k, {j, k, i}], l[[i+1 ;; -1]]]]] ]; s]]; a[n_, k_] := If[n >= k, b[n, Array[0&, k]], b[k, Array[0&, n]]][[2]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Dec 13 2013, translated from Maple *)

A178523 The path length of the Fibonacci tree of order n.

Original entry on oeis.org

0, 0, 2, 6, 16, 36, 76, 152, 294, 554, 1024, 1864, 3352, 5968, 10538, 18478, 32208, 55852, 96420, 165800, 284110, 485330, 826752, 1404816, 2381616, 4029216, 6803666, 11468502, 19300624, 32433204, 54426364, 91216184, 152691702, 255313658, 426460288, 711634648

Offset: 0

Emeric Deutsch, Jun 15 2010

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 path length of a tree is the sum of the levels of all of its nodes.

This is also the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that all but one such pair are joined by an edge; equivalently the number of "(n-1)-domino" configurations in the game of memory played on a 2 X n rectangular array, see [Young]. - Donovan Young, Oct 23 2018

			a(2)=2 because the Fibonacci tree of order 2 is /\ with path length 1 + 1. - _Emeric Deutsch_, Sep 13 2010

Ralph P. Grimaldi, Properties of Fibonacci trees, In Proceedings of the Twenty-second Southeastern Conference on Combinatorics, Graph Theory, and Computing (Baton Rouge, LA, 1991); Congressus Numerantium 84 (1991), 21-32. - Emeric Deutsch, Sep 13 2010
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

Muniru A Asiru, Table of n, a(n) for n = 0..4500
Carlos Alirio Rico Acevedo, and Ana Paula Chaves, Double-Recurrence Fibonacci Numbers and Generalizations, arXiv:1903.07490 [math.NT], 2019.
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
W. Kuszmaul, Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations, arXiv preprint arXiv:1509.08216 [cs.DM], 2015-2017.
D. Young, The Number of Domino Matchings in the Game of Memory, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.
Donovan Young, Generating Functions for Domino Matchings in the 2 * k Game of Memory, arXiv:1905.13165 [math.CO], 2019. Also in J. Int. Seq., Vol. 22 (2019), Article 19.8.7.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-3,1,1).

Cf. A000045, A006478, A178522, A002940, A067331.

a:=[0,2];;  for n in [3..35] do a[n]:=a[n-1]+a[n-2]+ 2*Fibonacci(n +1) -2; od; Concatenation([0],a); # Muniru A Asiru, Oct 23 2018

[2+(2/5)*(4*n-9)*Fibonacci(n)+(2/5)*(3*n-5)*Fibonacci(n-1): n in [0..40]]; // Vincenzo Librandi, Oct 24 2018

with(combinat): a := proc (n) options operator, arrow: 2+((8/5)*n-18/5)*fibonacci(n)+((6/5)*n-2)*fibonacci(n-1) end proc: seq(a(n), n = 0 .. 35);
G := 2*z^2/((1-z)*(1-z-z^2)^2): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 35);

Table[2 +2/5 (4n-9) Fibonacci[n] +2/5 (3n -5) Fibonacci[n-1], {n, 0, 40}] (* or *) LinearRecurrence[{3, -1, -3, 1, 1}, {0, 0, 2, 6, 16}, 40] (* Harvey P. Dale, Oct 02 2016 *)

vector(40, n, n--; (10+(8*n-18)*fibonacci(n)+(6*n-10)*fibonacci(n-1))/5) \\ G. C. Greubel, Jan 31 2019

[(10+(8*n-18)*fibonacci(n)+(6*n-10)*fibonacci(n-1))/5 for n in range(40)] # G. C. Greubel, Jan 31 2019

a(n) = 2 + (2/5)*(4n-9)*F(n) + (2/5)*(3n-5)*F(n-1), where F(n) = A000045(n) (Fibonacci numbers).
a(n) = 2*A006478(n+1).
a(n) = Sum_{k=0..n-1} k*A178522(n,k).
G.f.: 2*z^2/((1-z)*(1-z-z^2)^2).
From Emeric Deutsch, Sep 13 2010: (Start)
a(0)=a(1)=0, a(n) = a(n-1)+a(n-2)+2F(n+1)-2 if n>=2; here F(j)=A000045(j) are the Fibonacci numbers (see the Grimaldi reference, Eq. (**) on p. 23).
An explicit formula for a(n) is given in the Grimaldi reference (Theorem 2).
(End)
E.g.f.: 2*exp(x) + 2*exp(x/2)*(5*(4*x - 5)*cosh(sqrt(5)*x/2) + sqrt(5)*(10*x - 13)*sinh(sqrt(5)*x/2))/25. - Stefano Spezia, Dec 04 2023

A178522 Triangle read by rows: T(n,k) is the number of nodes at level k in the Fibonacci tree of order n (n>=0, 0<=k<=n-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 4, 6, 2, 1, 2, 4, 8, 8, 2, 1, 2, 4, 8, 14, 10, 2, 1, 2, 4, 8, 16, 22, 12, 2, 1, 2, 4, 8, 16, 30, 32, 14, 2, 1, 2, 4, 8, 16, 32, 52, 44, 16, 2, 1, 2, 4, 8, 16, 32, 62, 84, 58, 18, 2, 1, 2, 4, 8, 16, 32, 64, 114, 128, 74, 20, 2, 1, 2, 4, 8, 16, 32, 64, 126

Offset: 0

Emeric Deutsch, Jun 15 2010

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.
Sum of entries in row n is A001595(n).
Sum_{k=0..n-1} k*T(n,k) = A178523(n).

			Triangle starts:
1,
1,
1,2,
1,2,2,
1,2,4,2,
1,2,4,6,2,
1,2,4,8,8,2,
1,2,4,8,14,10,2,
1,2,4,8,16,22,12,2,
1,2,4,8,16,30,32,14,2,
...

D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.

Cf. A001595, A059214, A178523, A067331, A002940. See A059250 for another version.

G := (1-t*z+t*z^2)/((1-z)*(1-t*z-t*z^2)): Gser := simplify(series(G, z = 0, 17)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: 1; for n to 13 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form

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

T(k,n) = T(k-1,n-1)+T(k-1,n) with T(0,0)=1, T(k,0)=1 for k>0, T(0,n)=2 for n>0. - Frank M Jackson, Aug 30 2011

A166106 a(n) = a(n-1) + a(n-2) + F(n), with a(0) = 0, a(1) = 1, a(2) = a(1) + a(0), a(3) = a(2) + a(1), a(4) = a(3) + a(2) + 2.

Original entry on oeis.org

0, 1, 1, 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, 17048641, 28559450, 47786400, 79870428, 133359715, 222457608, 370747675

Offset: 0

Gerald McGarvey, Oct 06 2009

Consider the recursive call tree for Fibonacci numbers shown in the Wilson, Abelson et al., and Bloch links. This type of tree is a variant of Fibonacci trees shown/defined elsewhere. Here, let us refer it as a recursive Fibonacci tree. A Fibonacci number F(n) is the weight of the root of a weighted recursive Fibonacci tree of order n in which the leafs have a weight of 1, and the weight of a node equals the sum of the weights of its two children. If instead we weight each leaf by the number of nodes above it (considering the root as the topmost node), then for n > 2, a(n) is the weight of the root of such a weighted tree of order n. For example, a(5) = 2+2+2+3+3.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Harold Abelson and Gerald Jay Sussman with Julie Sussman, Tree Recursion from "Structure and Interpretation of Computer Programs", MIT Press, 1996, LaTeX2HTML translation by Ryan Bender.
Laurent Bloch, Analyse de l'algorithme de Fibonacci
Bill Wilson, The Prolog Dictionary - memoisation (shows Recursive call tree for Fibonacci number f_6).
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Cf. A000045.

a[n_] := a[n] = a[n-1] + a[n-2] + Fibonacci[n]; a[0] = 0; a[1] = 1; a[2] = 1; a[3] = 2; a[4] = 5; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Oct 03 2011 *)

s = 33; a = concat([0,1,1,2,5],vector(s-5)); for(n=6,s,a[n]=a[n-1]+a[n-2]+fibonacci(n)); for(n=1,s,print1(a[n],", "))

concat(0, Vec(x*(x^5+3*x^4+2*x^3-x^2-x+1)/(x^2+x-1)^2 + O(x^100))) \\ Colin Barker, May 25 2014

For n > 1, a(n) = A067331(n-2).
From Colin Barker, May 25 2014: (Start)
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) for n > 6.
G.f.: x*(x^5 + 3*x^4 + 2*x^3 - x^2 - x + 1) / (x^2+x-1)^2. (End)
a(n) = (1/25)*2^(-n-1)*(5*((1 - sqrt(5))^(n+1) + (1 + sqrt(5))^(n+1))*n - (25 + sqrt(5))*(1 + sqrt(5))^n + (sqrt(5) - 25)*(1 - sqrt(5))^n), n > 2. - Ilya Gutkovskiy, Apr 26 2016

More terms from Colin Barker, May 25 2014

A108038 Triangle read by rows: g.f. = (x+y+xy)/((1-x-x^2)(1-y-y^2)).

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 2, 4, 4, 2, 3, 7, 5, 7, 3, 5, 11, 9, 9, 11, 5, 8, 18, 14, 16, 14, 18, 8, 13, 29, 23, 25, 25, 23, 29, 13, 21, 47, 37, 41, 39, 41, 37, 47, 21, 34, 76, 60, 66, 64, 64, 66, 60, 76, 34, 55, 123, 97, 107, 103, 105, 103, 107, 97, 123, 55, 89, 199, 157, 173, 167, 169, 169, 167

Offset: 0

N. J. A. Sloane, Jun 01 2005

Start with 3 rows 0; 1 1; 1 3 1; then rule is each entry is maximum of sum of two entries diagonally above it to the left or to the right. Borders are Fibonacci numbers (A000045).

			Triangle begins:
      k=0  1  2  3  4
  n=0:  0;
  n=1:  1, 1;
  n=2:  1, 3, 1;
  n=3:  2, 4, 4, 2;
  n=4:  3, 7, 5, 7, 3;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Matthew Blair, Rigoberto Flórez, Antara Mukherjee, and José L. Ramírez, Matrices in the Determinant Hosoya Triangle, Fibonacci Quart. 58 (2020), no. 5, 34-54.
Matthew Blair, Rigoberto Flórez and Antara Mukherjee, Geometric Patterns in The Determinant Hosoya Triangle, INTEGERS, A90, 2021.
Hsin-Yun Ching, Rigoberto Flórez and Antara Mukherjee, Families of Integral Cographs within a Triangular Arrays, Special Matrices, 8 (2020), 257-273; see also arXiv preprint, arXiv:2009.02770 [math.CO], 2020.
Hsin-Yun Ching, Rigoberto Flórez, F. Luca, Antara Mukherjee, and J. C. Saunders, Primes and composites in the determinant Hosoya triangle, arXiv:2211.10788 [math.NT], 2022.

Cf. A000045, A067331 (row sums).

Block[{nn = 11, s}, s = Series[(x + y + x*y)/((1 - x - x^2)*(1 - y - y^2)), {x, 0, nn}, {y, 0, nn}]; Table[Function[m, SeriesCoefficient[s, {m, k}]][n - k], {n, 0, nn}, {k, 0, n}]] // Flatten (* Michael De Vlieger, Dec 04 2020 *)
G[n_,k_] := Fibonacci[k]*Fibonacci[n-k+1]; T[n_,k_]:= G[n+2,k+1]-G[n,k]; RowPointHosoya[n_] := Table[Inset[T[n,i+1], {1-n+2i,1-n}], {i,0,n-1}]; T[n_] := Graphics[ Flatten[Table[RowPointHosoya[i], {i,1,n}],1]]; Manipulate[T[n], Style["Determinant Hosoya Triangle",12,Red], {{n,6,"Rows"}, Range[12]}, ControlPlacement -> Up] (* Rigoberto Florez, Feb 07 2022 *)

From Rigoberto Florez, Feb 08 2022: (Start)
T(n,k) = F(k+2)*F(n-k+2) - F(k+1)*F(n-k+1), where F(n) = Fibonacci(n) = A000045(n).
T(n,k) = F(k)*F(n-k+2) + F(k+1)*F(n-k), where F(n) = Fibonacci(n).
T(n,k) = T(n-1,k) + T(n-2,k) and T(n,k) = T(n-1,k-1) + T(n-2,k-2), where T(1,1) = 0, T(2,1) = T(2,2) = 1, and T(3,2) = 3.
G.f: (x + x*y + x^2*y)/((1 - x - x^2)*(1 - x*y - x^2*y^2)). (End)

A178524 Triangle read by rows: T(n,k) is the number of leaves at level k in the Fibonacci tree of order n (n>=0, 0<=k<=n-1).

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 2, 0, 0, 3, 2, 0, 0, 1, 5, 2, 0, 0, 0, 4, 7, 2, 0, 0, 0, 1, 9, 9, 2, 0, 0, 0, 0, 5, 16, 11, 2, 0, 0, 0, 0, 1, 14, 25, 13, 2, 0, 0, 0, 0, 0, 6, 30, 36, 15, 2, 0, 0, 0, 0, 0, 1, 20, 55, 49, 17, 2, 0, 0, 0, 0, 0, 0, 7, 50, 91, 64, 19, 2, 0, 0, 0, 0, 0, 0, 1, 27, 105, 140, 81, 21, 2

Offset: 0

Emeric Deutsch, Jun 15 2010

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.
Sum of entries in row n is the Fibonacci number F(n+1) (A000045(n+1)).
Sum(k*T(n,k),k>=0)=A067331(n-2).
T(n,k) is the number of vertices in the Fibonacci cube G_{n-1} that have eccentricity k (see Klavzar and Mollard reference). - Michel Mollard, Aug 20 2014

			Triangle starts:
1,
1,
0, 2,
0, 1, 2,
0, 0, 3, 2,
0, 0, 1, 5, 2,
0, 0, 0, 4, 7, 2,
0, 0, 0, 1, 9, 9, 2,
0, 0, 0, 0, 5, 16, 11, 2,
0, 0, 0, 0, 1, 14, 25, 13, 2,
0, 0, 0, 0, 0, 6, 30, 36, 15, 2,

D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.

A. Castro and M. Mollard, The eccentricity sequences of Fibonacci and Lucas cubes, Discrete Math., 312 (2012), 1025-1037. See Table 1. [From _N. J. A. Sloane_, Mar 22 2012]
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
S. Klavzar, M. Mollard, Asymptotic Properties of Fibonacci Cubes and Lucas Cubes, Annals of Combinatorics, 18, 2014, 447-457.

Cf. A000045, A067331, A178522, A004070

G := (1+z-t*z)/(1-t*z-t*z^2): Gser := simplify(series(G, z = 0, 17)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: 1; for n to 13 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form

G.f.: G(t,z) = (1+z-t*z) / (1-t*z-t*z^2).

A212338 Sequence of coefficients of x in marked mesh pattern generating function Q_{n,132}^(0,0,2,0)(x).

Original entry on oeis.org

2, 7, 21, 53, 124, 273, 577, 1181, 2358, 4614, 8880, 16854, 31612, 58691, 108003, 197203, 357596, 644463, 1155059, 2059897, 3656988, 6465660, 11388480, 19990140, 34976870, 61019071, 106160481, 184228193, 318948124, 550962717, 949781269, 1634103701, 2806342578

Offset: 3

N. J. A. Sloane, May 09 2012

Apparently the number of Dyck n-paths that have n-2 peaks after changing each valley to a peak by the transformation DU -> UD. E.g., the Dyck 3-paths UUUDDD and UUDUDD have 1 peak after changing DU to UD so a(3) = 2. - David Scambler, Sep 03 2012

Robert P. P. McKone, Table of n, a(n) for n = 3..5000
S. Kitaev, J. Remmel and M. Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243 [math.CO], 2012-2014.
Index entries for linear recurrences with constant coefficients, signature (3,0,-5,0,3,1).

Cf. Column 2 of A091533. Partial sums of A036681.

LinearRecurrence[{3, 0, -5, 0, 3, 1}, {0, 0, 2, 7, 21, 53}, {3, 35}] (* Robert P. P. McKone, Jan 14 2022 *)
A212338[1, n2_] = 0; A212338[n1_, 1] = 0; A212338[2, n2_] = 0; A212338[n1_, 2] = 0; A212338[3, 3] = 1; A212338[n1_, n2_] := A212338[n1, n2] = A212338[n1 - 1, n2] + A212338[n1, n2 - 1] + A212338[n1 - 1, n2 - 1] + A212338[n1 - 2, n2] + A212338[n1, n2 - 2]; Table[A212338[5, y], {y, 3, 35}] (* Robert P. P. McKone, Jan 14 2022 *)
QQQ2[t,x]=2/(1 + (t*x - t)*(1 + t) +((1 + (t*x - t)*(1 + t))^2 - 4*t*x)^(1/2)); CoefficientList[Coefficient[Series[QQQ2[t, x], {t, 0, 22}], x],t] (* Robert Price, Jun 05 2012 *)

g.f. -x^3*(2+x) / (x^2+x-1)^3, i.e., a(n) = 2*A001628(n-3) + A001628(n-4). - R. J. Mathar, Jun 27 2012

a(n) = a(n-1) + a(n-2) + A067331(n-3). E.g., a(5) = 21 = 7 + 2 + 12. - David Scambler, Sep 03 2012

cp's OEIS Frontend

A002940 Arrays of dumbbells.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Sage

Formula

Extensions

A067330 Triangle read by rows of incomplete convolutions of Fibonacci numbers F(n+1) = A000045(n+1), n>=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A067418 Triangle A067330 with rows read backwards.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

Formula

A226545 Number A(n,k) of squares in all tilings of a k X n rectangle using integer-sided square tiles; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A178523 The path length of the Fibonacci tree of order n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A178522 Triangle read by rows: T(n,k) is the number of nodes at level k in the Fibonacci tree of order n (n>=0, 0<=k<=n-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Formula

A166106 a(n) = a(n-1) + a(n-2) + F(n), with a(0) = 0, a(1) = 1, a(2) = a(1) + a(0), a(3) = a(2) + a(1), a(4) = a(3) + a(2) + 2.

Views

Author

A108038 Triangle read by rows: g.f. = (x+y+xy)/((1-x-x^2)(1-y-y^2)).