John O. Oladokun - cp's OEIS frontend

A341929 Bisection of the numerators of the convergents of cf (1,1,6,1,6,1,...,6,1).

1, 2, 15, 118, 929, 7314, 57583, 453350, 3569217, 28100386, 221233871, 1741770582, 13712930785, 107961675698, 849980474799, 6691882122694, 52685076506753, 414788729931330, 3265624762943887, 25710209373619766, 202416050226014241, 1593618192434494162, 12546529489249939055, 98778617721565018278

Offset: 0

John O. Oladokun, Feb 23 2021

15*a(n)^2 - 11 is a square for all terms.
x = a(n) and y = a(n+1) satisfy the equation x^2 + y^2 - 8*x*y = -11.
x = a(n) and y = a(n+2) satisfy x^2 + y^2 - 62*x*y = -704.

			a(3) = 8*15 - 2 = 118.

Index entries for linear recurrences with constant coefficients, signature (8,-1).

After a(0), bisection of A237262.

Cf. A341927.

Mathematica
```
LinearRecurrence [{8, -1}, {1,2}, 15]
```

my(p=Mod('x,'x^2-8*'x+1)); a(n) = subst(lift(p^n),'x,2); \\ Kevin Ryde, Feb 27 2021

a(n) = 8*a(n-1) - a(n-2) for n >= 2.
a(n) = A237262(2*n) for n >= 1.
G.f.: (1 - 6*x)/(1 - 8*x + x^2). - Stefano Spezia, Mar 01 2021

A341927 Bisection of the numerators of the convergents of cf(1,4,1,6,1,6,...,6,1).

Original entry on oeis.org

1, 6, 47, 370, 2913, 22934, 180559, 1421538, 11191745, 88112422, 693707631, 5461548626, 42998681377, 338527902390, 2665224537743, 20983268399554, 165200922658689, 1300624112869958, 10239791980300975, 80617711729537842, 634701901856001761, 4996997503118476246, 39341278123091808207

Offset: 0

John O. Oladokun, Feb 23 2021

15*a(n)^2 - 11 is a square for all terms.
x = a(n) and y = a(n+1) satisfy x^2 + y^2 - 8*x*y = -11.
x = a(n) and y = a(n+2) satisfy x^2 + y^2 - 62*x*y = -704.

			a(3) = 8*6 - 1 = 47.

Index entries for linear recurrences with constant coefficients, signature (8,-1).

Bisection of A237262.

Cf. A341929.

Mathematica
```
LinearRecurrence[{8, -1}, {1,6},15]
```

my(p=Mod('x,'x^2-8*'x+1)); a(n) = subst(lift(p^n),'x,6); \\ Kevin Ryde, Mar 01 2021

a(0) = 1; a(1) = 6; a(n) = 8*a(n-1) - a(n-2).
G.f.: (1 - 2*x)/(1 - 8*x + x^2). - Stefano Spezia, Feb 26 2021
a(n) = A237262(2*n + 1).

A335823 Triangle read by rows: A080779 with rows reversed.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 12, 6, 0, 24, 60, 40, 0, -4, 120, 360, 300, 0, -60, 0, 720, 2520, 2520, 0, -840, 0, 120, 5040, 20160, 23520, 0, -11760, 0, 3360, 0, 40320, 181440, 241920, 0, -169344, 0, 80640, 0, -12096, 362880, 1814400, 2721600, 0, -2540160, 0, 1814400, 0, -544320, 0

Offset: 1

John O. Oladokun, Jun 25 2020

			Triangle begins:
   1;
   1,  1;
   2,  3,  1;
   6, 12,  6, 0;
  24, 60, 40, 0, -4;
  ...

Cf. A000142 (row sums).
Cf. A080779 (same triangle with rows reversed).
Cf. A027641/A027642 (Bernoulli numbers).

Table[If[k == 0, (n - 1)!, n!*Product[n - j, {j, k - 1}]*(-1)^k*BernoulliB[k]/k!], {n, 10}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Jun 27 2020 *)

T(n,k) = if (k==0, (n-1)!, n!*prod(j=1,k-1, n-j)*(-1)^k*bernfrac(k)/k!);
tabl(nn) = for(n=1, nn, for (k=0, n-1, print1(T(n,k), ", ")); print); \\ Michel Marcus, Jun 25 2020

T(n,k) = n!*(n-1)*(n-2)*...*(n-k+1)*(-1)^k*Bk/k! where Bk is a Bernoulli number and T(n,0) = (n-1)! and T(n,m) = 0 if m >= n.

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User: John O. Oladokun

A341929 Bisection of the numerators of the convergents of cf (1,1,6,1,6,1,...,6,1).

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A341927 Bisection of the numerators of the convergents of cf(1,4,1,6,1,6,...,6,1).

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A335823 Triangle read by rows: A080779 with rows reversed.

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