A067331 -id:A067331 - cp's OEIS frontend

A213777 Rectangular array: (row n) = bc, where b(h) = F(h), c(h) = F(h+1), F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

1, 3, 2, 7, 5, 3, 15, 12, 8, 5, 30, 25, 19, 13, 8, 58, 50, 40, 31, 21, 13, 109, 96, 80, 65, 50, 34, 21, 201, 180, 154, 130, 105, 81, 55, 34, 365, 331, 289, 250, 210, 170, 131, 89, 55, 655, 600, 532, 469, 404, 340, 275, 212, 144, 89, 1164, 1075, 965, 863

Offset: 1

Clark Kimberling, Jun 21 2012

Principal diagonal:  A001870
Antidiagonal sums:  A152881
row 1,  (1,1,2,3,5,8,...)**(1,2,3,5,8,13,...): A023610(k-1)
row 2,  (1,1,2,3,5,8,...)**(2,3,5,8,13,21,...): A067331(k-1)
row 3,  (1,1,2,3,5,8,...)**(3,5,8,13,21,34,...)
For a guide to related arrays, see A213500.

			Northwest corner (the array is read by falling antidiagonals):
1....3....7....15....30....58
2....5....12...25....50....96
3....8....19...40....80....154
5....13...31...65....130...250
8....21...50...105...210...404
13...34...81...170...340...654

Clark Kimberling, Antidiagonals n=1..80, flattened

Cf. A213500.

b[n_] := Fibonacci[n]; c[n_] := Fibonacci[n + 1];
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A213777 *)
Table[t[n, n], {n, 1, 40}] (* A001870 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A152881 *)

T(n,k) = 2*T(n,k-1) + T(n,k-2) - 2*T(n,k-3) - T(n,k-4).
G.f. for row n:  f(x)/g(x), where f(x) = F(n-1) + F(n-2)*x and g(x) = (1 - x - x^2)^2.
T(n,k) = (k*Lucas(n+k+1) + Lucas(n)*Fibonacci(k))/5. - Ehren Metcalfe, Jul 10 2019

A228152 Triangle read by rows: T(n,k) = maximal external path length of AVL trees of height n with k (leaf-) nodes, n>=0, fibonacci(n+2)<=k<=2^n.

Original entry on oeis.org

0, 2, 5, 8, 12, 16, 20, 24, 25, 30, 35, 40, 44, 49, 54, 59, 64, 50, 56, 62, 68, 73, 79, 85, 91, 97, 102, 107, 113, 119, 125, 131, 136, 142, 148, 154, 160, 96, 103, 110, 117, 123, 130, 137, 144, 151, 157, 163, 170, 177, 184, 191, 197, 204, 211, 218, 225, 231

Offset: 0

Herbert Eberle, Aug 13 2013

The external path length of a tree is the sum of the levels of its external nodes (i.e. leaves).

			T(2,3) = 5 because in the (two) AVL trees of height 2 with 3 (leaf-) nodes one has depth 1 and two have depth 2:
       o       o
      / \     / \
     o   1   1   o
    / \         / \
   2   2       2   2
so that the sum of depths is 5 for both trees.
Triangle begins:
  0
  . 2
  . . 5 8
  . . . . 12 16 20 24
  . . . .  .  .  . 25 30 35 40 44 49 54 59 64
  . . . .  .  .  .  .  .  .  .  . 50 56 62 68 73 79 85 91 97 102 ...
  . . . .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 96 103 ...

D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 6.2.3 (7) and (8).

Alois P. Heinz, Rows n = 0..12, flattened
R. C. Richards, Shape distribution of height-balanced trees, Info. Proc. Lett., 17 (1983), 17-20.
Wikipedia, AVL tree
Index entries for sequences related to rooted trees

Row maxima give: n*2^n = A036289(n).
Row minima give: A067331(n-1) for n>0 or A166106(n+2).
Row lengths give: 1+A008466(n).
Number of AVL trees read by rows gives: A143897.
Triangle read by columns gives: A228153.
The infimum of all external path lengths of binary trees with k (leaf-) nodes is: A003314(k) for k>0.
Column maxima give: A228155(k).
Column heights give: A217710(k).
Number of AVL trees read by columns gives: A217298.

with(combinat): F:=fibonacci:
T:= proc(n, k) option remember; `if`(n<1, 0, max(seq([k+T(n-1,t)+
      T(n-1,k-t), k+T(n-1,t) +T(n-2,k-t)][], t=F(n+1)..k-1)))
    end:
seq(seq(T(n, k), k=F(n+2)..2^n), n=0..7);  # Alois P. Heinz, Aug 14 2013

maxNods = 100; Clear[T, A029837, A072649, A036289, A228155]; T[0, 1] = 0; A029837[1] = 0; A072649[1] = 1; A228155[1] = 0; For[k = 2, k <= maxNods, k++, A029837[k] = maxNods; A072649[k] = 0; A228155u = 0; For[kL = 1, kL <= Floor[k/2], kL++, For[hL = A029837[kL], hL <= A072649[kL] - 1, hL++, For[hR = Max[hL - 1, A029837[k - kL]], hR <= Min[hL + 1, A072649[k - kL] - 1], hR++, maxDepthSum = k + T[hL, kL] + T[hR, k - kL]; A228155u = Max[maxDepthSum, A228155u]; h = Max[hL, hR] + 1; If[ !IntegerQ[T[h, k]], T[h, k] = maxDepthSum, T[h, k] = Max[maxDepthSum, T[h, k]]]; A029837[k] = Min[h, A029837[k]]; If[ !IntegerQ[A036289[h]], A036289[h] = maxDepthSum, A036289[h] = Max[maxDepthSum, A036289[h]]]; A072649[k] = Max[h + 1, A072649[k]]; ]]]; A228155[k] = A228155u]; k =.; Table[T[n, k], {n, 0, maxNods}, {k, 1, maxNods}] // Flatten // Select[#, IntegerQ]& (* Jean-François Alcover, Aug 14 2013, translated and adapted from Herbert Eberle's MuPAD program *)

maxNods:=100: // max number of leaves (= external nodes)
// Triangle T for all AVL trees with <= maxNods leaves:
delete T:
// table T indexed [h, k] (h=height, k=number of leaves):
T[0, 1]:=0:
// A029837 indexed [k], min height of tree with k leaves:
A029837:=array(1..maxNods): A029837[1]:=0:
// A072649 indexed [k], 1+max height of AVL tree with k leaves:
A072649:=array(1..maxNods): A072649[1]:=1:
// A036289 indexed [h], max depthsum of all height h AVL trees:
A036289:=array(1..maxNods):
// A228155 indexed [k], max depthsum of all AVL trees with k leaves:
A228155:=array(1..maxNods): A228155[1]:=0:
for k from 2 to maxNods do:
  A029837[k]:=maxNods: // try infinity for the min height
  A072649[k]:=0:
  A228155u:=0:
  // Put together 2 AVL trees:
  for kL from 1 to floor(k/2) do:
    // kL leaves in the left tree
    for hL from A029837[kL] to A072649[kL]-1 do:
      for hR from max(hL-1, A029837[k-kL])
               to min(hL+1, A072649[k-kL]-1) do:
        // k-kL leaves in the right subtree
        maxDepthSum:=T[hL, kL]+T[hR, k-kL]+k:
        A228155u:=max(maxDepthSum, A228155u):
        h:=max(hL, hR)+1:
        if type(T[h, k]) <> DOM_INT then // T[h, k] uninit
          T[h, k]:=maxDepthSum:
        else
          T[h, k]:=max(maxDepthSum, T[h, k]):
        end_if:
        A029837[k]:=min(h, A029837[k]):
        if type(A036289[h]) <> DOM_INT then
          A036289[h]:=maxDepthSum:
        else
          A036289[h]:=max(maxDepthSum, A036289[h]):
        end_if:
        A072649[k]:=max(h+1, A072649[k]):
      end_for: // hR
    end_for: // hL
  end_for: // kL
  A228155[k]:=A228155u:
end_for: // k

A228153 Triangle read by columns: T(n,k) = maximal external path length of AVL trees of height n with k (leaf-) nodes, k>=1, A029837(k)<=n<A072649(k).

Original entry on oeis.org

0, 2, 5, 8, 12, 16, 20, 24, 25, 30, 35, 40, 44, 49, 50, 54, 56, 59, 62, 64, 68, 73, 79, 85, 91, 97, 96, 102, 103, 107, 110, 113, 117, 119, 123, 125, 130, 131, 137, 136, 144, 142, 151, 148, 157, 154, 163, 160, 170, 177, 184, 180, 191, 188, 197, 196, 204, 204

Offset: 1

Herbert Eberle, Aug 13 2013

The external path length of a tree is the sum of the levels of its external nodes (i.e. leaves).

			In the (two) AVL trees of height 2 the 3 external nodes (leaves) have once depth 1 and twice depth 2:
       o       o
      / \     / \
     o   1   1   o
    / \         / \
   2   2       2   2
so that the sum of depths is 5 for both trees.
Triangle begins:
  0
  . 2
  . . 5 8
  . . . . 12 16 20 24
  . . . .  .  .  . 25 30 35 40 44 49 54 59 64
  . . . .  .  .  .  .  .  .  .  . 50 56 62 68 73 79 85 91 97 102 ...
  . . . .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 96 103 ...

D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 6.2.3 (7) and (8).

Alois P. Heinz, Columns k = 1..3000, flattened
R. C. Richards, Shape distribution of height-balanced trees, Info. Proc. Lett., 17 (1983), 17-20.
Wikipedia, AVL tree
Index entries for sequences related to rooted trees

Triangle read by rows gives: A228152.
Row maxima give: n*2^n = A036289(n).
Row minima give: A067331(n-1) for n>0 or A166106(n+2).
Row lengths give: 1+A008466(n).
Number of AVL trees read by rows gives: A143897.
The infimum of all external path lengths of binary trees with k (leaf-) nodes is: A003314(k) for k>0.
Column maxima give: A228155(k).
Column heights give: A217710(k).
Number of AVL trees read by columns gives: A217298.

maxNods = 100; Clear[T, A029837, A072649, A036289, A228155]; T[0, 1] = 0; A029837[1] = 0; A072649[1] = 1; A228155[1] = 0; For[k = 2, k <= maxNods, k++, A029837[k] = maxNods; A072649[k] = 0; A228155u = 0; For[kL = 1, kL <= Floor[k/2], kL++, For[hL = A029837[kL], hL <= A072649[kL] - 1, hL++, For[hR = Max[hL - 1, A029837[k - kL]], hR <= Min[hL + 1, A072649[k - kL] - 1], hR++, maxDepthSum = k + T[hL, kL] + T[hR, k - kL]; A228155u = Max[maxDepthSum, A228155u]; h = Max[hL, hR] + 1; If[ !IntegerQ[T[h, k]], T[h, k] = maxDepthSum, T[h, k] = Max[maxDepthSum, T[h, k]]]; A029837[k] = Min[h, A029837[k]]; If[ !IntegerQ[A036289[h]], A036289[h] = maxDepthSum, A036289[h] = Max[maxDepthSum, A036289[h]]]; A072649[k] = Max[h + 1, A072649[k]]; ]]]; A228155[k] = A228155u]; k =.; Table[ Select[ Table[T[n, k], {n, A029837[k], A072649[k] - 1}], IntegerQ], {k, 1, maxNods}] // Flatten (* Jean-François Alcover, Aug 19 2013, translated and adapted from Herbert Eberle's MuPAD program *)

maxNods:=100: // max number of leaves (= external nodes)
// Triangle T for all AVL trees with <= maxNods leaves:
delete T:
// table T indexed [h, k] (h=height, k=number of leaves):
T[0, 1]:=0:
// A029837 indexed [k], min height of tree with k leaves:
A029837:=array(1..maxNods): A029837[1]:=0:
// A072649 indexed [k], 1+max height of AVL tree with k leaves:
A072649:=array(1..maxNods): A072649[1]:=1:
// A036289 indexed [h], max depthsum of all height h AVL trees:
A036289:=array(1..maxNods):
// A228155 indexed [k], max depthsum of all AVL trees with k leaves:
A228155:=array(1..maxNods): A228155[1]:=0:
for k from 2 to maxNods do:
  A029837[k]:=maxNods: // try infinity for the min height
  A072649[k]:=0:
  A228155u:=0:
  // Put together 2 AVL trees:
  for kL from 1 to floor(k/2) do:
    // kL leaves in the left tree
    for hL from A029837[kL] to A072649[kL]-1 do:
      for hR from max(hL-1, A029837[k-kL])
               to min(hL+1, A072649[k-kL]-1) do:
        // k-kL leaves in the right subtree
        maxDepthSum:=T[hL, kL]+T[hR, k-kL]+k:
        A228155u:=max(maxDepthSum, A228155u):
        h:=max(hL, hR)+1:
        if type(T[h, k]) <> DOM_INT then // T[h, k] uninit
          T[h, k]:=maxDepthSum:
        else
          T[h, k]:=max(maxDepthSum, T[h, k]):
        end_if:
        A029837[k]:=min(h, A029837[k]):
        if type(A036289[h]) <> DOM_INT then
          A036289[h]:=maxDepthSum:
        else
          A036289[h]:=max(maxDepthSum, A036289[h]):
        end_if:
        A072649[k]:=max(h+1, A072649[k]):
      end_for: // hR
    end_for: // hL
  end_for: // kL
  A228155[k]:=A228155u:
end_for: // k

A228155 Maximal external path length of AVL trees with n (leaf-) nodes.

Original entry on oeis.org

0, 2, 5, 8, 12, 16, 20, 25, 30, 35, 40, 44, 50, 56, 62, 68, 73, 79, 85, 91, 97, 103, 110, 117, 123, 130, 137, 144, 151, 157, 163, 170, 177, 184, 191, 197, 204, 211, 219, 227, 235, 243, 250, 257, 265, 273, 281, 289, 296, 304, 312, 320, 328, 335, 342, 349, 356

Offset: 1

Herbert Eberle, Aug 14 2013

nonn

The external path length of a tree is the sum of the levels of its external nodes (i.e. leaves).

			The (two) AVL trees with 3 (leaf-) nodes have one with depth 1 and two with depth 2:
       o       o
      / \     / \
     o   1   1   o
    / \         / \
   2   2       2   2
so a(3) = 5.

D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 6.2.3 (7) and (8).

Alois P. Heinz, Table of n, a(n) for n = 1..5000
R. C. Richards, Shape distribution of height-balanced trees, Info. Proc. Lett., 17 (1983), 17-20.
Wikipedia, AVL tree
Index entries for sequences related to rooted trees

Column maxima of triangles A228152, A228153.
Row maxima give: n*2^n = A036289(n).
Row minima give: A067331(n-1) for n>0 or A166106(n+2).
Row lengths give: 1+A008466(n).
Column heights give: A217710(k).
Number of AVL trees read by rows gives: A143897.
The infimum of all external path lengths of all binary trees with k (leaf-) nodes is: A003314(k) for k>0.
Number of AVL trees read by columns gives: A217298.

maxNods = 100; Clear[T, A029837, A072649, A036289, A228155]; T[0, 1] = 0; A029837[1] = 0; A072649[1] = 1; A228155[1] = 0; For[k = 2, k <= maxNods, k++, A029837[k] = maxNods; A072649[k] = 0; A228155u = 0; For[kL = 1, kL <= Floor[k/2], kL++, For[hL = A029837[kL], hL <= A072649[kL] - 1, hL++, For[hR = Max[hL - 1, A029837[k - kL]], hR <= Min[hL + 1, A072649[k - kL] - 1], hR++, maxDepthSum = k + T[hL, kL] + T[hR, k - kL]; A228155u = Max[maxDepthSum, A228155u]; h = Max[hL, hR] + 1; If[ !IntegerQ[T[h, k]], T[h, k] = maxDepthSum, T[h, k] = Max[maxDepthSum, T[h, k]]]; A029837[k] = Min[h, A029837[k]]; If[ !IntegerQ[A036289[h]], A036289[h] = maxDepthSum, A036289[h] = Max[maxDepthSum, A036289[h]]]; A072649[k] = Max[h + 1, A072649[k]]; ]]]; A228155[k] = A228155u]; k =.; Table[A228155[k], {k, 1, maxNods}] (* Jean-François Alcover, Aug 19 2013, translated and adapted from Herbert Eberle's MuPAD program *)

maxNods:=100: // max number of leaves (= external nodes)
// Triangle T for all AVL trees with <= maxNods leaves:
delete T:
// table T indexed [h, k] (h=height, k=number of leaves):
T[0, 1]:=0:
// A029837 indexed [k], min height of tree with k leaves:
A029837:=array(1..maxNods): A029837[1]:=0:
// A072649 indexed [k], 1+max height of AVL tree with k leaves:
A072649:=array(1..maxNods): A072649[1]:=1:
// A036289 indexed [h], max depthsum of all height h AVL trees:
A036289:=array(1..maxNods):
// A228155 indexed [k], max depthsum of all AVL trees with k leaves:
A228155:=array(1..maxNods): A228155[1]:=0:
for k from 2 to maxNods do:
  A029837[k]:=maxNods: // try infinity for the min height
  A072649[k]:=0:
  A228155u:=0:
  // Put together 2 AVL trees:
  for kL from 1 to floor(k/2) do:
    // kL leaves in the left tree
    for hL from A029837[kL] to A072649[kL]-1 do:
      for hR from max(hL-1, A029837[k-kL])
               to min(hL+1, A072649[k-kL]-1) do:
        // k-kL leaves in the right subtree
        maxDepthSum:=T[hL, kL]+T[hR, k-kL]+k:
        A228155u:=max(maxDepthSum, A228155u):
        h:=max(hL, hR)+1:
        if type(T[h, k]) <> DOM_INT then // T[h, k] uninit
          T[h, k]:=maxDepthSum:
        else
          T[h, k]:=max(maxDepthSum, T[h, k]):
        end_if:
        A029837[k]:=min(h, A029837[k]):
        if type(A036289[h]) <> DOM_INT then
          A036289[h]:=maxDepthSum:
        else
          A036289[h]:=max(maxDepthSum, A036289[h]):
        end_if:
        A072649[k]:=max(h+1, A072649[k]):
      end_for: // hR
    end_for: // hL
  end_for: // kL
  A228155[k]:=A228155u:
end_for: // k

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.

Original entry on oeis.org

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

A360932 Primes of the form H(m,k) = F(k+1)F(m-k+2) - F(k)F(m-k+1), where F(m) is the m-th Fibonacci number and m >= 0, 0 <= k <= m.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 37, 41, 47, 89, 97, 103, 107, 157, 167, 173, 199, 233, 443, 521, 733, 1597, 1741, 1867, 1871, 1877, 2207, 3037, 3571, 7841, 7919, 7951, 9349, 11933, 12823, 28657, 33503, 50549, 54277, 54287, 54293, 54319, 54497, 55717, 142099

Offset: 1

Rigoberto Florez, Feb 25 2023

nonn

This sequence appears in the triangle A108038 in this order (reading by rows): 3, 2, 7, 5, 11, 13, 29, 23, 47, 37, 41, 97, 107, 103, 89, 199, 157, 173, 167.
Are there infinitely many primes of the form H(m,k)?
This sequence appears within the determinant Hosoya triangle.

			29 is a term because it is prime and A108038(8,2) = H(8,2) = 29.  Also A108038(8,7) = H(8,7) = 29.

Robert Israel, Table of n, a(n) for n = 1..9313
Hsin-Yun Ching, Rigoberto Florez, F. Luca, Antara Mukherjee, and J. C. Saunders, Primes and composites in the determinant Hosoya triangle, Fibonacci Quarterly, 2023.

Cf. A000040, A000045, A005478 (subsequence), A108038, A153892, A067331.

Res:= {}:
M:= 50: # for terms <= F(M)
fmax:= combinat:-fibonacci(M):
T[1]:= [1,1]:
T[2]:= [1,3,1]:
for i from 3 to M do
  t1:= [op(T[i-1][1..i-1] + T[i-2][1..i-1]),T[i-1][i],0];
  t2:= ListTools:-Reverse(t1);
  T[i]:= zip(max,t1,t2);
  Res:= Res union convert(select(t -> t <= fmax and isprime(t), T[i][1..ceil((i+1)/2)]),set)
od:
sort(convert(Res,list)); # Robert Israel, Mar 14 2024

H[r_, k_] := Det[{{Fibonacci[r-k+2], Fibonacci[r-k+1]}, {Fibonacci[k], Fibonacci[k+1]}}];  DeterminantPrimes[t_, m_] :=  Table[If[PrimeQ[H[r,k]],H[r,k], Unevaluated[Sequence[]]], {r,t,m}, {k,1,Ceiling[r/2]}]; ListOfPrimes[t_,m_]:=  Sort[DeleteDuplicates[Flatten[DeterminantPrimes[t, m]]]]; ListOfPrimes[2, 100]
Select[Union[Flatten[Table[Fibonacci[k+1]Fibonacci[m-k+2]-Fibonacci[k]Fibonacci[m-k+1],{m,0,40},{k,0,m}]]],PrimeQ] (* Harvey P. Dale, Aug 14 2025 *)

Intersection of A000040 and A108038.

A210239 Triangle, read by rows, given by (2, -1, 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.

Original entry on oeis.org

1, 2, 2, 2, 5, 3, 2, 9, 12, 5, 2, 13, 28, 25, 8, 2, 17, 52, 74, 50, 13, 2, 21, 84, 167, 177, 96, 21, 2, 25, 124, 320, 470, 397, 180, 34, 2, 29, 172, 549, 1041, 1211, 850, 331, 55, 2, 33, 228, 870, 2034, 3042, 2928, 1758, 600, 89

Offset: 0

Philippe Deléham, Mar 19 2012

			Triangle begins :
1
2, 2
2, 5, 3
2, 9, 12, 5
2, 13, 28, 25, 8
2, 17, 52, 74, 50, 13
2, 21, 84, 167, 177, 96, 21
2, 25, 124, 320, 470, 397, 180, 34

Cf. A000045, A026150, A112087 (3rd column, n>2).

G.f.: (1+x+y*x)/(1-x-y*x-y*x^2-y^2*x^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A122803(n), A000007(n), A040000(n), A026150(n+1) for x = -2, -1, 0, 1 respectively.
T(n,n) = Fibonacci(n+2) = A000045(n+2), T(n+1,n) = A067331(n).

A365554 Number of increasing paths from the bottom to the top of the n-hypercube (as a graded poset) which first encounter a vector of isolated zeros at stage k, weighted by k.

Original entry on oeis.org

2, 10, 60, 396, 2976, 25056, 234720, 2423520, 27371520, 335819520, 4449150720, 63318931200, 963548006400, 15614378035200, 268480048435200, 4882321001779200, 93627018326016000, 1888394741194752000, 39963486306078720000, 885457095215616000000

Offset: 2

Brian Darrow, Jr. and Joe Fields, Feb 20 2024

nonn

These are the numerators in calculating an expected value. The expectation of the number of steps one takes in marking the elements of a predetermined list before reaching a state where only isolated unmarked entries remain.

			For n=5, an example vector of isolated 0's is 01011, which has k=3 1's.
For n=3, the following paths (from 000 to 111) reach isolated 0's at k=1 many 1's (010):
  000,010,011,111
  000,010,110,111
The following paths reach isolated 0's only at k=2 1's:
  000,100,110,111
  000,100,101,111
  000,001,101,111
  000,001,011,111
So 2 paths of k=1 and 4 paths of k=2 are weighted total a(3) = 2*1 + 4*2 = 10.

Cf. A067331.

a(n) = sum(k=n\2, n-1, k*(binomial(k+1,n-k)-binomial(k-1,n-k))*k!*(n-k)!) \\ Andrew Howroyd, Feb 23 2024

k, n = var('k,n')
sum((binomial(k+1,n-k)-binomial(k-1,n-k))*factorial(k)*factorial(n-k), k, floor(n/2),n-1)

a(n) = Sum_{k=floor(n/2)..n-1} k*(binomial(k+1,n-k)-binomial(k-1,n-k))*k!*(n-k)!.

cp's OEIS Frontend

A213777 Rectangular array: (row n) = b**c, where b(h) = F(h), c(h) = F(h+1), F=A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

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A228152 Triangle read by rows: T(n,k) = maximal external path length of AVL trees of height n with k (leaf-) nodes, n>=0, fibonacci(n+2)<=k<=2^n.

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A228153 Triangle read by columns: T(n,k) = maximal external path length of AVL trees of height n with k (leaf-) nodes, k>=1, A029837(k)<=n<A072649(k).

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A228155 Maximal external path length of AVL trees with n (leaf-) nodes.

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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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A360932 Primes of the form H(m,k) = F(k+1)*F(m-k+2) - F(k)*F(m-k+1), where F(m) is the m-th Fibonacci number and m >= 0, 0 <= k <= m.

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A210239 Triangle, read by rows, given by (2, -1, 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.

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A213777 Rectangular array: (row n) = bc, where b(h) = F(h), c(h) = F(h+1), F=A000045 (Fibonacci numbers), n>=1, h>=1, and = convolution.

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.

A360932 Primes of the form H(m,k) = F(k+1)F(m-k+2) - F(k)F(m-k+1), where F(m) is the m-th Fibonacci number and m >= 0, 0 <= k <= m.