A179943 -id:A179943 - cp's OEIS frontend

A179944 Row sums of triangle A179943.

1, 3, 7, 17, 47, 148, 518, 1977, 8138, 35879, 168500, 838944, 4409957, 24385913, 141412615, 857611641, 5426144191, 35739397738, 244573978098, 1735854397529, 12757309001222, 96941738970957, 760649367654460, 6155205917196408, 51308394497243469, 440110582561558831

Offset: 0

Gary W. Adamson, Aug 07 2010

nonn

			a(4) = 47 since row 4 of triangle A179943 = (1, 5, 15, 21, 5).

Andrew Howroyd, Table of n, a(n) for n = 0..500
Robert G. Donnelly, Molly W. Dunkum, Sasha V. Malone, and Alexandra Nance, Symmetric Fibonaccian distributive lattices and representations of the special linear Lie algebras, arXiv:2012.14991 [math.CO], 2020.

Row sums of A179943.

Cf. A341576.

seq(n)={Vec(sum(k=0, n, x^k/(1-(k+2)*x+x^2) + O(x*x^n)))} \\ Andrew Howroyd, Apr 13 2021

G.f.: Sum_{k>=0} x^k/(1 - (k+2)*x + x^2). - Andrew Howroyd, Apr 13 2021

Terms a(12) and beyond from Andrew Howroyd, Apr 13 2021

A323182 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.

Original entry on oeis.org

1, 1, 0, 1, 2, -1, 1, 4, 3, 0, 1, 6, 15, 4, 1, 1, 8, 35, 56, 5, 0, 1, 10, 63, 204, 209, 6, -1, 1, 12, 99, 496, 1189, 780, 7, 0, 1, 14, 143, 980, 3905, 6930, 2911, 8, 1, 1, 16, 195, 1704, 9701, 30744, 40391, 10864, 9, 0, 1, 18, 255, 2716, 20305, 96030, 242047, 235416, 40545, 10, -1

Offset: 0

Seiichi Manyama, Jan 06 2019

			Square array begins:
   1, 1,    1,     1,      1,      1,       1, ...
   0, 2,    4,     6,      8,     10,      12, ...
  -1, 3,   15,    35,     63,     99,     143, ...
   0, 4,   56,   204,    496,    980,    1704, ...
   1, 5,  209,  1189,   3905,   9701,   20305, ...
   0, 6,  780,  6930,  30744,  96030,  241956, ...
  -1, 7, 2911, 40391, 242047, 950599, 2883167, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Wikipedia, Chebyshev polynomials.
Index entries for sequences related to Chebyshev polynomials.

Mirror of A228161.
Columns 0-19 give A056594, A000027(n+1), A001353(n+1), A001109(n+1), A001090(n+1), A004189(n+1), A004191, A007655(n+2), A077412, A049660(n+1), A075843(n+1), A077421, A077423, A097309, A097311, A097313, A029548, A029547, A144128(n+1), A078987.
Rows 0-10 give A000012, A005843, A000466, A144138, A144139, A242850, A242851, A242852, A242853, A242854, A243130.
Main diagonal gives A323118.
Cf. A179943, A322836 (Chebyshev polynomial of the first kind).

T(n,k)  = polchebyshev(n, 2, k);
matrix(7, 7, n, k, T(n-1,k-1)) \\ Michel Marcus, Jan 07 2019

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

T(0,k) = 1, T(1,k) = 2 * k and T(n,k) = 2 * k * T(n-1,k) - T(n-2,k) for n > 1.

T(n, k) = Sum_{j=0..n} (2*k-2)^j * binomial(n+1+j,2*j+1). - Seiichi Manyama, Mar 03 2021

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)

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A179944 Row sums of triangle A179943.

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A323182 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is Chebyshev polynomial of the second kind U_{n}(x), evaluated at x=k.

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

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