A097690 -id:A097690 - 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

A097691 Denominators of the continued fraction n-1/(n-1/...) [n times].

Original entry on oeis.org

1, 2, 8, 56, 551, 6930, 105937, 1905632, 39424240, 922080050, 24057287759, 692686638072, 21817946138353, 746243766783074, 27543862067299424, 1091228270370045824, 46187969968474139807, 2080128468827570457762, 99318726126650358502921, 5011361251329169946919800

Offset: 1

T. D. Noe, Aug 19 2004

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.

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

Alois P. Heinz, Table of n, a(n) for n = 1..387
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 8, 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

Cf. A084844, A084845, A097690 (numerators).

Table[s=n; Do[s=n-1/s, {n-1}]; Denominator[s], {n, 20}]
Table[Abs[Fibonacci[n, I n]], {n, 20}] (* Vladimir Reshetnikov, Oct 16 2018 *)

[lucas_number1(n,n,1) for n in range(1,19)] # Zerinvary Lajos, Jul 16 2008

a(n) = ChebyshevU(n-1,n/2). - Gary Detlefs, Oct 15 2011
a(n) = abs((2^(-n) * (sqrt(4 - n^2) + i*n)^n - 2^n*(-sqrt(4 - n^2) - i*n)^(-n))/sqrt(4 - n^2)), where i is the imaginary unit, for n > 2. - Daniel Suteu, May 31 2017
a(n) ~ n^(n-1). - Vaclav Kotesovec, Jun 03 2017

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

A107995 Chebyshev polynomial of the second kind U[n,x] evaluated at x=n+2.

Original entry on oeis.org

1, 6, 63, 980, 20305, 526890, 16451071, 600940872, 25154396001, 1187422368110, 62418042417599, 3616337930622300, 228977061309711793, 15731733543660288210, 1165677769357309014015, 92665403695822344828176

Offset: 0

Roger L. Bagula, Mar 01 2006

nonn

			a(3)=980 because U[3,x]=8x^3-4x and U[3,5]=8*5^3-4*5=980.

Rosenblum and Rovnyak, Hardy Classes and Operator Theory,Dover, New York,1985, page 18.
G. Szego, Orthogonal polynomials, Amer. Math. Soc., Providence, 1939, p. 29.
G. Freud, Orthogonal Polynomials, Pergamon Press, Oxford, 1966, p. 35.

Seiichi Manyama, Table of n, a(n) for n = 0..351
Wikipedia, Chebyshev polynomials.

Cf. A097690, A323118, A342167.

Maple
```
with(orthopoly): seq(U(n,n+2),n=0..17);
```

Table[ChebyshevU[n, n + 2], {n, 0, 15}] (* Amiram Eldar, Mar 05 2021 *)

a(n) = polchebyshev(n, 2, n+2); \\ Seiichi Manyama, Mar 05 2021

a(n) = sum(k=0, n, (2*n+2)^k*binomial(n+1+k, 2*k+1)); \\ Seiichi Manyama, Mar 05 2021

a(n) = Sum_{k=0..n} (2*n+2)^(n-k) * binomial(2*n+1-k,k) = Sum_{k=0..n} (2*n+2)^k * binomial(n+1+k,2*k+1). - Seiichi Manyama, Mar 05 2021

a(n) ~ exp(2) * 2^n * n^n. - Vaclav Kotesovec, Mar 05 2021

Edited by N. J. A. Sloane, Apr 05 2006

A343259 a(n) = 2 * T(n,n/2) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

2, 1, 2, 18, 194, 2525, 39202, 710647, 14760962, 345946302, 9034502498, 260219353691, 8195978831042, 280256592535933, 10340256951198914, 409468947059131650, 17322711762013765634, 779742677038695037937, 37210469265847998489922, 1876572071974094803391179

Offset: 0

Seiichi Manyama, Apr 09 2021

nonn

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

Main diagonal of A298675.

Cf. A097690, A115066, A342205, A343260, A343261.

Table[2*ChebyshevT[n, n/2], {n, 1, 20}] (* Amiram Eldar, Apr 09 2021 *)

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

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

a(n) = 2 * cos(n*arccos(n/2)).
a(n) = 2 * n * Sum_{k=0..n} (n-2)^k * binomial(n+k,2*k)/(n+k) for n > 0.
a(n) ~ n^n. - Vaclav Kotesovec, Apr 09 2021

A372816 Table read by antidiagonals: T(t,n) = number of t-metered parking functions of length n.

Original entry on oeis.org

1, 1, 3, 1, 3, 21, 1, 3, 16, 209, 1, 3, 16, 163, 2640, 1, 3, 16, 125, 2142, 40391, 1, 3, 16, 125, 1686, 33961, 726103, 1, 3, 16, 125, 1296, 27629, 626569, 15003009, 1, 3, 16, 125, 1296, 21858, 525594, 13198604, 350382231

Offset: 1

Spencer Daugherty, May 13 2024

			Table begins:
  1, 3, 21, 209, 2640, 40391, 726103, ...
  1, 3, 16, 163, 2142, 33961, 626569, ...
  1, 3, 16, 125, 1686, 27629, 525594, ...
  1, 3, 16, 125, 1296, 21858, 430062, ...
  1, 3, 16, 125, 1296, 16807, 341192, ...
  1, 3, 16, 125, 1296, 16807, 262144, ...
  ...

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

First row is A097690 and main diagonal of A372817.
Second, third and fourth rows are main diagonals of A372818, A372819, and A372820.
Cf. A372821, A000272.

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.

A357892 T(n,k) are the values of a variant of the Chebyshev polynomials P(n,x) of order n evaluated at x = k, where T(n,k), n >= 0, k <= n is a triangle read by rows. P(0,x) = 1, P(1,x) = x, P(n,x) = x*P(n-1,x) - P(n-2,x).

Original entry on oeis.org

1, 0, 1, -1, 0, 3, 0, -1, 4, 21, 1, -1, 5, 55, 209, 0, 0, 6, 144, 780, 2640, -1, 1, 7, 377, 2911, 12649, 40391, 0, 1, 8, 987, 10864, 60605, 235416, 726103, 1, 0, 9, 2584, 40545, 290376, 1372105, 4976784, 15003009, 0, -1, 10, 6765, 151316, 1391275, 7997214, 34111385, 118118440, 350382231

Offset: 0

Hugo Pfoertner, Oct 18 2022

			The triangle begins:
   1;
   0,  1;
  -1,  0, 3;
   0, -1, 4,  21;
   1, -1, 5,  55,   209;
   0,  0, 6, 144,   780,  2640;
  -1,  1, 7, 377,  2911, 12649,  40391;
   0,  1, 8, 987, 10864, 60605, 235416, 726103

Cf. A001353 (column 4), A001906 (column 3), A097690 (diagonal).

chp(k,x) = if(k==0, 1, if(k==1, x, x*chp(k-1,x) - chp(k-2,x)));
for (k=0, 9, for(x=0, k, print1(ch(k,x),", ")); print())

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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A097691 Denominators 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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A107995 Chebyshev polynomial of the second kind U[n,x] evaluated at x=n+2.

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A343259 a(n) = 2 * T(n,n/2) where T(n,x) is a Chebyshev polynomial of the first kind.