A097691 -id:A097691 - cp's OEIS frontend

A084844 Denominators of the continued fraction n + 1/(n + 1/...) [n times].

1, 2, 10, 72, 701, 8658, 129949, 2298912, 46866034, 1082120050, 27916772489, 795910114440, 24851643870041, 843458630403298, 30918112619119426, 1217359297034666112, 51240457936070359069, 2296067756927144738850, 109127748348241605689981

Offset: 1

Hollie L. Buchanan II, Jun 08 2003

The (n-1)-th term of the Lucas sequence U(n,-1). The numerator is the n-th term. Adjacent terms of the sequence U(n,-1) are relatively prime. - T. D. Noe, Aug 19 2004
From Flávio V. Fernandes, Mar 05 2021: (Start)
Also, the n-th term of the n-th metallic sequence (the diagonal through the array A073133, and its equivalents, which is rows formed by sequences beginning with A000045, A000129, A006190, A001076, A052918) as shown below (for n>=1):
   0    1    0    1     0      1  ... A000035
   0   [1]   1    2     3      5  ... A000045
   0    1   [2]   5    12     29  ... A000129
   0    1    3  [10]   33    109  ... A006190
   0    1    4   17   [72]   305  ... A001076
   0    1    5   26   135   [701] ... A052918. (End)

			a(4) = 72 since 4 + 1/(4 + 1/(4 + 1/4)) = 305/72.

Alois P. Heinz, Table of n, a(n) for n = 1..387
Eric Weisstein's World of Mathematics, Lucas Sequence

Cf. A084845 (numerators).

Cf. A000045, A097690, A097691, A117715, A290864 (primes in this sequence).

A084844 :=proc(n) combinat[fibonacci](n, n) end:
seq(A084844(n), n=1..30); # Zerinvary Lajos, Jan 03 2007

myList[n_] := Module[{ex = {n}}, Do[ex = {ex, n}, {n - 1}]; Flatten[ex]] Table[Denominator[FromContinuedFraction[myList[n]]], {n, 1, 20}]
Table[s=n; Do[s=n+1/s, {n-1}]; Denominator[s], {n, 20}] (* T. D. Noe, Aug 19 2004 *)
Table[Fibonacci[n, n], {n, 1, 20}] (* Vladimir Reshetnikov, May 07 2016 *)
Table[DifferenceRoot[Function[{y,m},{y[2+m]==n*y[1+m]+y[m],y[0]==0,y[1]==1}]][n],{n,1,20}] (* Benedict W. J. Irwin, Nov 03 2016 *)

from sympy import fibonacci
def a(n):
    return fibonacci(n, n)
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Aug 12 2017

a(n) = (s^n - (-s)^(-n))/(2*s - n), where s = (n + sqrt(n^2 + 4))/2. - Vladimir Reshetnikov, May 07 2016
a(n) = y(n,n), where y(m+2,n) = n*y(m+1,n) + y(m,n), with y(0,n)=0, y(1,n)=1 for all n. - Benedict W. J. Irwin, Nov 03 2016
a(n) ~ n^(n-1). - Vaclav Kotesovec, Jun 03 2017
a(n) = A117715(n,n). - Bobby Jacobs, Aug 12 2017
a(n) = [x^n] x/(1 - n*x - x^2). - Ilya Gutkovskiy, Oct 10 2017
a(n) == 0 (mod n) for even n and 1 (mod n) for odd n. - Flávio V. Fernandes, Dec 08 2020
a(n) == 0 (mod n) for even n and 1 (mod n^2) for odd n; see A065599. - Flávio V. Fernandes, Dec 25 2020
a(n) == 0 (mod 2*(n/2)^2) for even n and 1 (mod n^2) for odd n; see A129194. - Flávio V. Fernandes, Feb 06 2021

A084845 Numerators of the continued fraction n+1/(n+1/...) [n times].

Original entry on oeis.org

1, 5, 33, 305, 3640, 53353, 927843, 18674305, 426938895, 10928351501, 309601751184, 9616792908241, 324971855514293, 11868363584907985, 465823816409224245, 19553538801258341377, 874091571490181406680

Offset: 1

Hollie L. Buchanan II, Jun 08 2003

The n-th term of the Lucas sequence U(n,-1). The denominator is the (n-1)-th term. Adjacent terms of the sequence U(n,-1) are relatively prime. - T. D. Noe, Aug 19 2004

			a(4) = 305 since 4+1/(4+1/(4+1/4)) = 305/72.

Alois P. Heinz, Table of n, a(n) for n = 1..386
Eric Weisstein's World of Mathematics, Lucas Sequence

Cf. A084844 (denominators).

Cf. A097690, A097691, A117715.

A084845 := proc(n)
    fibonacci(n+1,n) ;
end proc:
seq(A084845(n),n=1..20) ; # Zerinvary Lajos, Dec 01 2006

myList[n_] := Module[{ex = {n}}, Do[ex = {ex, n}, {n - 1}]; Flatten[ex]] Table[Numerator[FromContinuedFraction[myList[n]]], {n, 1, 20}]
Table[s=n; Do[s=n+1/s, {n-1}]; Numerator[s], {n, 20}] (* T. D. Noe, Aug 19 2004 *)

{a(n)=polcoeff(1/(1-n*x-x^2+x*O(x^n)),n)} \\ Paul D. Hanna, Dec 27 2012

from sympy import fibonacci
def a117715(n, m): return 0 if n==0 else fibonacci(n, m)
def a(n): return a117715(n + 1, n)
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Aug 12 2017

a(n) = Sum_{k=0..floor(n/2)}* binomial(n-k, k)*n^(n-2k). - Michel Lagneau
a(n) = [x^n] 1/(1 - n*x - x^2). - Paul D. Hanna, Dec 27 2012
a(n) = (s^(n+1) - (-s)^(-n-1))/(2*s - n), where s = (n + sqrt(n^2 + 4))/2. - Vladimir Reshetnikov, May 07 2016
a(n) = A117715(n+1,n). - Alois P. Heinz, Aug 12 2017

A097690 Numerators of the continued fraction n-1/(n-1/...) [n times].

Original entry on oeis.org

1, 3, 21, 209, 2640, 40391, 726103, 15003009, 350382231, 9127651499, 262424759520, 8254109243953, 281944946167261, 10393834843080975, 411313439034311505, 17391182043967249409, 782469083251377707328

Offset: 1

T. D. Noe, Aug 19 2004

The n-th term of the Lucas sequence U(n,1). The denominator is the (n-1)-th term. Adjacent terms of the sequence U(n,1) are relatively prime.

			a(4) = 209 because 4-1/(4-1/(4-1/4)) = 209/56.

Alois P. Heinz, Table of n, a(n) for n = 1..386
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 3, 8, 10, 22.
Pascual Jara and Miguel L. Rodríguez, Solving quadratic congruences, Arhimede Math. J. (2020) Vol. 7, No. 2, 105-120.
Eric Weisstein's World of Mathematics, Lucas Sequence
Wikipedia, Chebyshev polynomials.

Cf. A084844, A084845, A097691 (denominators), A179943, A323118.

Table[s=n; Do[s=n-1/s, {n-1}]; Numerator[s], {n, 20}]
Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == n*y[m] - y[m - 1], y[0] == 1, y[1] == n}]][n], {n, 1, 20}] (* Benedict W. J. Irwin, Nov 05 2016 *)

{a(n)=polcoeff(1/(1-n*x+x^2+x*O(x^n)), n)} \\ Paul D. Hanna, Dec 27 2012

a(n) = polchebyshev(n, 2, n/2); \\ Seiichi Manyama, Mar 03 2021

a(n) = sum(k=0, n, (n-2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 03 2021

[lucas_number1(n,n-1,1) for n in range(19)] # Zerinvary Lajos, Jun 25 2008

a(n) = [x^n] 1/(1 - n*x + x^2). - Paul D. Hanna, Dec 27 2012
a(n) = y(n,n), where y(m+1,n) = n*y(m,n) - y(m-1,n) with y(0,n)=1, y(1,n)=n. - Benedict W. J. Irwin, Nov 05 2016
From Seiichi Manyama, Mar 03 2021: (Start)
a(n) = U(n,n/2) where U(n,x) is a Chebyshev polynomial of the second kind.
a(n) = Sum_{k=0..n} (n-2)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (n-2)^k * binomial(n+1+k,2*k+1). (End)

A342167 a(n) = U(n, (n+2)/2) where U(n, x) is a Chebyshev polynomial of the 2nd kind.

Original entry on oeis.org

1, 3, 15, 115, 1189, 15456, 242047, 4435929, 93149001, 2205405829, 58130412911, 1688353631328, 53577891882061, 1844491975179855, 68470281953483775, 2726406212682669391, 115921586524134874897, 5241862216131004082160, 251197634537351883217999

Offset: 0

Seiichi Manyama, Mar 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..386
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 11, 22.
Wikipedia, Chebyshev polynomials.

Cf. A097690, A097691, A323118, A342168.

Table[ChebyshevU[n, (n + 2)/2], {n, 0, 18}] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
a(n) = polchebyshev(n, 2, (n+2)/2);
```

a(n) = sum(k=0, n, n^(n-k)*binomial(2*n+1-k, k));

a(n) = sum(k=0, n, n^k*binomial(n+1+k, 2*k+1));

a(n) = Sum_{k=0..n} n^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} n^k * binomial(n+1+k,2*k+1).

a(n) ~ exp(2) * n^n. - Vaclav Kotesovec, May 06 2021

A342168 a(n) = U(n, (n+3)/2) where U(n, x) is a Chebyshev polynomial of the 2nd kind.

Original entry on oeis.org

1, 4, 24, 204, 2255, 30744, 499121, 9409960, 202176360, 4878316860, 130651068911, 3846719565780, 123517560398401, 4296240885694576, 160935647131239840, 6460088606857290384, 276655979838719058119, 12591439417867717440180, 606947064800948702246681

Offset: 0

Seiichi Manyama, Mar 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..386
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 11, 22.
Wikipedia, Chebyshev polynomials.

Cf. A097690, A097691, A323118, A342167.

Table[ChebyshevU[n, (n + 3)/2], {n, 0, 18}] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
a(n) = polchebyshev(n, 2, (n+3)/2);
```

a(n) = sum(k=0, n, (n+1)^(n-k)*binomial(2*n+1-k, k));

a(n) = sum(k=0, n, (n+1)^k*binomial(n+1+k, 2*k+1));

a(n) = Sum_{k=0..n} (n+1)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (n+1)^k * binomial(n+1+k,2*k+1).

a(n) ~ exp(3) * n^n. - Vaclav Kotesovec, May 06 2021

A372817 Table read by antidiagonals: T(m,n) = number of 1-metered (m,n)-parking functions.

Original entry on oeis.org

1, 0, 2, 0, 3, 3, 0, 4, 8, 4, 0, 6, 21, 15, 5, 0, 8, 55, 56, 24, 6, 0, 12, 145, 209, 115, 35, 7, 0, 16, 380, 780, 551, 204, 48, 8, 0, 24, 1000, 2912, 2640, 1189, 329, 63, 9, 0, 32, 2625, 10868, 12649, 6930, 2255, 496, 80, 10, 0, 48, 6900, 40569, 60606, 40391, 15456, 3905, 711, 99, 11

Offset: 1

Spencer Daugherty, May 13 2024

			For T(3,2) the 1-metered (3,2)-parking functions are 111, 121, 211, 212.
Table begins:
  1,  2,    3,     4,     5,      6,      7, ...
  0,  3,    8,    15,    24,     35,     48, ...
  0,  4,   21,    56,   115,    204,    329, ...
  0,  6,   55,   209,   551,   1189,   2255, ...
  0,  8,  145,   780,  2640,   6930,  15456, ...
  0, 12,  380,  2912, 12649,  40391, 105937, ...
  0, 16, 1000, 10868, 60606, 235416, 726103, ...
  ...

Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024.

Main diagonal is A097690 and first row of A372816.
First, second, and third diagonals above main are A097691, A342167, A342168.
Second column A029744. Second row A005563. Third row A242135.
Cf. A001353, A004254, A001109, A004187, A372818, A372821.

T(m,n) = (n*(n+sqrt(n^2 - 4))-2)/(n*(n+sqrt(n^2 - 4))-4)*((n+sqrt(n^2-4))/2)^m + (n*(n-sqrt(n^2 - 4))-2)/(n*(n-sqrt(n^2 - 4))-4)*((n-sqrt(n^2-4))/2)^m.

T(m,n) = n*T(m-1,n) - T(m-2,n) with T(0,n) = 1.

A343260 a(n) = 2 * T(n,(n+1)/2) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

2, 2, 7, 52, 527, 6726, 103682, 1874888, 38925119, 912670090, 23855111399, 687808321212, 21687295069442, 742397047217294, 27420344506901023, 1086932029484351248, 46027034321342899967, 2073668380220713167378, 99042070146811639444802

Offset: 0

Seiichi Manyama, Apr 09 2021

Seiichi Manyama, Table of n, a(n) for n = 0..386
Wikipedia, Chebyshev polynomials.

Cf. A097691, A115066, A342205, A343259, A343261.

Table[2*ChebyshevT[n, (n+1)/2], {n, 0, 18}] (* Amiram Eldar, Apr 09 2021 *)

PARI
```
a(n) = 2*polchebyshev(n, 1, (n+1)/2);
```
PARI
```
a(n) = round(2*cos(n*acos((n+1)/2)));
```

a(n) = if(n==0, 2, 2*n*sum(k=0, n, (n-1)^k*binomial(n+k, 2*k)/(n+k)));

a(n) = 2 * cos(n*arccos((n+1)/2)).
a(n) = 2 * n * Sum_{k=0..n} (n-1)^k * binomial(n+k,2*k)/(n+k) for n > 0.
a(n) ~ exp(1) * n^n. - Vaclav Kotesovec, Apr 09 2021
Conjecture: a(p^r) == 1 (mod p^(2*r)) for positive integer r and all primes p >= 5. - Peter Bala, Mar 11 2024

cp's OEIS Frontend

A084844 Denominators of the continued fraction n + 1/(n + 1/...) [n times].

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A084845 Numerators of the continued fraction n+1/(n+1/...) [n times].

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A097690 Numerators of the continued fraction n-1/(n-1/...) [n times].

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A342167 a(n) = U(n, (n+2)/2) where U(n, x) is a Chebyshev polynomial of the 2nd kind.

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A342168 a(n) = U(n, (n+3)/2) where U(n, x) is a Chebyshev polynomial of the 2nd kind.

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A372817 Table read by antidiagonals: T(m,n) = number of 1-metered (m,n)-parking functions.

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A343260 a(n) = 2 * T(n,(n+1)/2) where T(n,x) is a Chebyshev polynomial of the first kind.

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