Rigoberto Florez - cp's OEIS frontend

A378383 Number of subwords of the form UDDD in nondecreasing Dyck paths of length 2*n.

0, 0, 0, 1, 5, 19, 64, 202, 612, 1803, 5206, 14809, 41650, 116114, 321478, 885169, 2426462, 6627499, 18048088, 49026874, 132901176, 359625015, 971639014, 2621683741, 7065545950, 19022080034, 51163908874, 137499581917, 369235213742, 990822728623, 2657069356996

Offset: 0

Rigoberto Florez, Nov 24 2024

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170 (1997), 211-217.
Éva Czabarka, Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 15, 19.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
Index entries for linear recurrences with constant coefficients, signature (10,-39,74,-69,28,-4).

Cf. A000032, A000045, A377679, A377670, A375995, A377857, A377866, A377867.

Table[If[n < 3, 0, (1/5)((n-3)LucasL[2n-5]+LucasL[2n-3]+Fibonacci[2n+2]-5(n+5) 2^(n-4))], {n,0,26}]

a(n) =((n-3)*L(2n-5)+L(2n-3)+F(2n+2) -5*(n+5)*2^(n-4))/5 for n>=3, where F(n) = A000045(n) and L(n) = A000032(n).

G.f.: (-x^5+2 x^4-5 x^3+8 x^2-5 x+1)*x^3/(2 x^3-7 x^2+5 x-1)^2.

A377867 Number of subwords of the form DDDD in nondecreasing Dyck paths of length 2n.

Original entry on oeis.org

0, 0, 0, 0, 1, 7, 33, 131, 473, 1608, 5242, 16567, 51123, 154793, 461525, 1358646, 3957088, 11420995, 32707809, 93040751, 263113505, 740238852, 2073098086, 5782387855, 16070206191, 44516728277, 122956408493, 338707969266, 930787894348, 2552224341403, 6984100641117

Offset: 0

Rigoberto Florez, Nov 10 2024

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170 (1997), 211-217.
Éva Czabarka, Rigoberto Flórez, Leandro Junes and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 15, 19.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
Index entries for linear recurrences with constant coefficients, signature (10,-39,74,-69,28,-4).

Cf. A000032, A000045, A377679, A377670, A375995.

Table[If[n < 3, 0, (3*(n-2)*LucasL[2*n-4]-3*Fibonacci[2*n+1])/5+(n+9)*2^(n-4)], {n,0,20}]

a(n) = (3*(n-2)*L(2*n-4) - 3*F(2*n+1))/5 + (n+9)*2^(n-4) for n>=3, where F(n) = A000045(n) and L(n) = A000032(n).

G.f.: x^4*(1 - 3*x + 2*x^2 + x^4)/((1 - 2*x)^2*(1 - 3*x + x^2)^2).

A377866 Number of subwords of the form DUUD or DDUUD in nondecreasing Dyck paths of length 2n.

Original entry on oeis.org

0, 0, 0, 1, 5, 18, 59, 185, 564, 1685, 4957, 14406, 41455, 118321, 335400, 945193, 2650229, 7398330, 20573219, 57013865, 157517532, 433993661, 1192779085, 3270835566, 8950887895, 24448816993, 66665369424, 181489721425, 493361278949

Offset: 0

Rigoberto Florez, Nov 10 2024

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170 (1997), 211-217.
Éva Czabarka, Rigoberto Flórez, Leandro Junes and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 15, 17, 19.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A000032, A000045, A377679, A377670, A375995.

Table[If[n<3,0,(2*n*LucasL[2*n-5]-6*Fibonacci[2*n-6]-Fibonacci[2*n-7])/5], {n,0,20}]

a(n) = (2*n*L(2*n-5) - 6*F(2*n-6) - F(2*n-7))/5 for n>=3, where F(n)=A000045(n) and L(n)=A000032(n).
G.f.:  -x^3*(x^2+x-1)/ (x^2-3*x+1)^2.
E.g.f.: exp(3*x/2)*(5*(35 - 8x)*cosh(sqrt(5)*x/2) - sqrt(5)*(79 - 20*x)*sinh(sqrt(5)*x/2))/25 - 7 - x. - Stefano Spezia, Nov 10 2024

A377857 Number of subwords of the form UUUD in nondecreasing Dyck paths of length 2n.

Original entry on oeis.org

0, 0, 0, 1, 5, 18, 60, 191, 589, 1775, 5257, 15360, 44394, 127171, 361595, 1021693, 2871245, 8031246, 22372344, 62096135, 171797257, 473928875, 1304007889, 3579517116, 9804791910, 26804181643, 73145473655, 199276078201, 542076556949, 1472491141770, 3994615719732

Offset: 0

Rigoberto Florez, Nov 09 2024

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170 (1997), 211-217.
Éva Czabarka, Rigoberto Flórez, Leandro Junes and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 15, 19.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Cf. A000032, A000045, A377679, A377670, A375995.

Table[If[n<3,0,n Fibonacci[2n-5]-LucasL[2n-6]], {n,0,30}]

a(n) = n*F(2*n-5) - L(2*n-6) for n>=3, where F(n) = A000045(n) and L(n) = A000032(n).
G.f.: x^3*(1 - x)^2*(1 + x)/(1 - 3*x + x^2)^2.
a(n) = A317408(n-2)-A317408(n-3) = A030267(n-2)+A030267(n-3). - R. J. Mathar, Dec 16 2024

A377670 Number of subwords of the form UDD in nondecreasing Dyck paths of length 2n.

Original entry on oeis.org

0, 0, 1, 4, 14, 45, 138, 411, 1200, 3454, 9836, 27779, 77938, 217493, 604222, 1672246, 4613030, 12689265, 34817418, 95320335, 260436588, 710278318, 1933906496, 5257545599, 14273273314, 38699274665, 104799960058, 283487736166, 766045036730, 2067997219629, 5577597593466, 15030365074659, 40470488092008

Offset: 0

Rigoberto Florez, Nov 03 2024

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

a(n) also represents the number of subwords of the form UUDDD in nondecreasing Dyck paths of length 2n.

E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170 (1997), 211-217.
Éva Czabarka, Rigoberto Flórez, Leandro Junes and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 6, 17, 19.
Index entries for linear recurrences with constant coefficients, signature (8,-23,28,-13,2).

Cf. A000032, A000045, A377679, A375995.

Table[If[n<2, 0,(2*Fibonacci[2n-3] + n*LucasL[2n-3]+2 LucasL[2n-2]-5*2^(n-2))/5], {n,0,20}]

a(n) = (2*F(2*n-3) + n*L(2*n-3) + 2*L(2*n-2) - 5*2^(n-2))/5 for n>=2, where F(n) = A000045(n) and L(n) = A000032(n).
G.f.: x^2*(1-x)*(x^3-2*x^2+3*x-1)/((2*x-1)*(x^2-3*x+1)^2). - Alois P. Heinz, Nov 03 2024
E.g.f.: (4*exp(3*x/2)*(5*(10 - x)*cosh(sqrt(5)*x/2) - sqrt(5)*(18 - 5*x)*sinh(sqrt(5)*x/2)) - 25*(7 + exp(2*x) + 2*x))/100. - Stefano Spezia, Mar 04 2025

A377679 Number of subwords of the form DDD in nondecreasing Dyck paths of length 2n.

Original entry on oeis.org

0, 0, 0, 1, 6, 26, 97, 333, 1085, 3411, 10448, 31376, 92773, 270907, 783003, 2243815, 6383550, 18048494, 50755897, 142067625, 396014681, 1099863867, 3044737100, 8404071596, 23135752141, 63538808311, 174120317367, 476207551183

Offset: 0

Rigoberto Florez, Nov 03 2024

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170 (1997), 211-217.
Éva Czabarka, Rigoberto Flórez, Leandro Junes and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 6, 19.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
Index entries for linear recurrences with constant coefficients, signature (8,-23,28,-13,2).

Cf. A000032, A000045, A377670, A375995.

Table[If[n<2,0,n Fibonacci[2 n-3]-LucasL[2 n-2]+2^(n-2)],{n,0,30}]

a(n) = n*F(2*n-3) - L(2*n-2) + 2^(n-2) for n>=2, where F(n) = A000045(n) and L(n) = A000032(n).

G.f.: x^3*(1 - 2*x + x^2 - x^3)/((1 - 2*x)*(1 - 3*x + x^2)^2).

A375995 Number of subwords of the form UUUU in nondecreasing Dyck paths of length 2n.

Original entry on oeis.org

0, 0, 0, 0, 1, 7, 30, 109, 365, 1164, 3593, 10835, 32106, 93845, 271321, 777432, 2211025, 6248479, 17562870, 49132669, 136884293, 379975140, 1051356761, 2900587115, 7981564866, 21911096357, 60021530545, 164095925424, 447823729825, 1220105286199, 3319124711118

Offset: 0

Rigoberto Florez, Nov 03 2024

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170 (1997), 211-217.
Éva Czabarka, Rigoberto Flórez, Leandro Junes and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 15, 19.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
Index entries for linear recurrences with constant coefficients, signature (6, -11, 6, -1).

Cf. A000032, A000045, A377679, A377670.

Table[If[n<=2,0,(2(n-3)LucasL[2n-5]-3Fibonacci[2n-6])/5], {n,0,30}]

a(n) = (2(n-3)*L(2 n-5)-3F(2n-6))/5 for n>=3 and a(n) = 0 for n<=2, F(.) is a Fibonacci number, L(.) is a Lucas number.

G.f.: x^4*(-x^2+x+1)/(x^2-3x+1)^2.

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.

A338588 a(n)/A002939(n+1) is the Kirchhoff index of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.

Original entry on oeis.org

2, 77, 334, 881, 1826, 3277, 5342, 8129, 11746, 16301, 21902, 28657, 36674, 46061, 56926, 69377, 83522, 99469, 117326, 137201, 159202, 183437, 210014, 239041, 270626, 304877, 341902, 381809, 424706, 470701, 519902

Offset: 0

Rigoberto Florez, Nov 07 2020

Equivalently, the graph can be described as the graph on 3*n + 2 vertices with labels 0..3*n+1 and with i and j adjacent iff i+j>0 mod 3.
These graphs are cographs.
The initial term a(0) = 2 has been included to agree with the formula. For the graph, is not defined.

			The adjacency matrix of the graph associated with n = 2 is:
  [0, 1, 0, 0, 0, 1, 1, 1]
  [1, 0, 0, 0, 0, 1, 1, 1]
  [0, 0, 0, 1, 1, 1, 1, 1]
  [0, 0, 1, 0, 1, 1, 1, 1]
  [0, 0, 1, 1, 0, 1, 1, 1]
  [1, 1, 1, 1, 1, 0, 0, 0]
  [1, 1, 1, 1, 1, 0, 0, 0]
  [1, 1, 1, 1, 1, 0, 0, 0].
a(2) = 334 because the Kirchhoff index of the graph is 334/30=334/A002939(3).
The first few Kirchhoff indices (n >= 1) as reduced fractions are 77/12, 167/15, 881/56, 913/45, 3277/132, 2671/91, 8129/240, 5873/153, 16301/380, 10951/231.

H-Y. Ching, R. Florez, and A. Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Kirchhoff Index
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A338527, A338104, A338109.

Mathematica
```
Table[(18n^3+37n^2+20n+2), {n,0,30}]
```

a(n) = 18*n^3 + 37*n^2 + 20*n + 2.
G.f.: (2 + 69*x + 38*x^2 - x^3)/(x - 1)^4.
E.g.f.: exp(x)*(2 + 75*x + 91*x^2 + 18*x^3). - Stefano Spezia, Nov 08 2020
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Nov 08 2020

A338527 Number of spanning trees in the join of the disjoint union of two complete graphs each on n and n+1 vertices with the empty graph on n+1 vertices.

Original entry on oeis.org

24, 13500, 34420736, 239148450000, 3520397039081472, 94458953432730437824, 4179422085120000000000000, 283894102615246085842939590912, 28059580711858187192007680000000000, 3870669526565955444680027453177986243584

Offset: 1

Rigoberto Florez, Nov 07 2020

nonn

Equivalently, the graph can be described as the graph on 3*n + 2 vertices with labels 0..3*n+1 and with i and j adjacent iff i+j> 0 mod 3.

These graphs are cographs.

			The adjacency matrix of the graph associated with n = 2 is: (compare A204437)
 [0, 1, 0, 0, 0, 1, 1, 1]
 [1, 0, 0, 0, 0, 1, 1, 1]
 [0, 0, 0, 1, 1, 1, 1, 1]
 [0, 0, 1, 0, 1, 1, 1, 1]
 [0, 0, 1, 1, 0, 1, 1, 1]
 [1, 1, 1, 1, 1, 0, 0, 0]
 [1, 1, 1, 1, 1, 0, 0, 0]
 [1, 1, 1, 1, 1, 0, 0, 0]
a(2) = 13500 because the graph has 13500 spanning trees.

H-Y. Ching, R. Florez, and A. Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Spanning Tree

Cf. A338104, A338109.

Table[(n + 1)*(2 n + 2)^n*(2 n + 1)^(2 n - 1), {n, 1, 10}]

a(n) = (n + 1)*(2 n + 2)^n*(2 n + 1)^(2 n - 1).

Offset changed by Georg Fischer, Nov 03 2023

cp's OEIS Frontend

User: Rigoberto Florez

A378383 Number of subwords of the form UDDD in nondecreasing Dyck paths of length 2*n.

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A377867 Number of subwords of the form DDDD in nondecreasing Dyck paths of length 2n.

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A377866 Number of subwords of the form DUUD or DDUUD in nondecreasing Dyck paths of length 2n.

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A377857 Number of subwords of the form UUUD in nondecreasing Dyck paths of length 2n.

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A377670 Number of subwords of the form UDD in nondecreasing Dyck paths of length 2n.

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A377679 Number of subwords of the form DDD in nondecreasing Dyck paths of length 2n.

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A375995 Number of subwords of the form UUUU in nondecreasing Dyck paths of length 2n.

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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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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.