A343259 -id:A343259 - cp's OEIS frontend

A298675 Rectangular array A: first differences of row entries of array A294099, read by antidiagonals.

1, 2, -1, 3, 2, -2, 4, 7, 2, -1, 5, 14, 18, 2, 1, 6, 23, 52, 47, 2, 2, 7, 34, 110, 194, 123, 2, 1, 8, 47, 198, 527, 724, 322, 2, -1, 9, 62, 322, 1154, 2525, 2702, 843, 2, -2, 10, 79, 488, 2207, 6726, 12098, 10084, 2207, 2, -1, 11, 98, 702, 3842, 15127, 39202, 57965, 37634, 5778, 2, 1

Offset: 1

L. Edson Jeffery, Bob Selcoe and Andrew Hone, Jan 24 2018

From a problem in A269254. For detailed theory, see [Hone].
From Charles L. Hohn, Sep 28 2024: (Start)
For rows n >= 3, values x >= 3 where (x^2-4)/(n^2-4) is a square.
For rows n >= 3, Lim_{k->oo}(T(n, k+1)/T(n, k)) = (sqrt(n^2-4)+n)/2. (End)

			Array begins:
   1 -1  -2   -1     1      2       1       -1        -2         -1
   2  2   2    2     2      2       2        2         2          2
   3  7  18   47   123    322     843     2207      5778      15127
   4 14  52  194   724   2702   10084    37634    140452     524174
   5 23 110  527  2525  12098   57965   277727   1330670    6375623
   6 34 198 1154  6726  39202  228486  1331714   7761798   45239074
   7 47 322 2207 15127 103682  710647  4870847  33385282  228826127
   8 62 488 3842 30248 238142 1874888 14760962 116212808  914941502
   9 79 702 6239 55449 492802 4379769 38925119 345946302 3074591599
  10 98 970 9602 95050 940898 9313930 92198402 912670090 9034502498

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Main diagonal gives A343259.

Cf. A285992, A299107, A299109, A088165, A117522, A299100, A299101, A113501, A269253, A269254, A294099, A298675, A298677, A298878, A299045, A299071.

t[n_, 0] := 2; t[n_, 1] := n; t[n_, k_] := n*t[n, k - 1] - t[n, k - 2]; Table[t[n, k], {n, 10}, {k, 10}] // Grid

A(n,k) = T_k(n), n >= 1, k >= 1, where T_j(x) = x*T_{j-1}(x) - T_{j-2}(x), j >= 2, T_0(x) = 2, T_1(x) = x, (dilated Chebyshev polynomials of the first kind).

A343261 a(n) = 2 * T(n,(n+2)/2) where T(n,x) is a Chebyshev polynomial of the first kind.

Original entry on oeis.org

2, 3, 14, 110, 1154, 15127, 238142, 4379769, 92198402, 2186871698, 57721023502, 1678243366813, 53301709843202, 1836220544383695, 68200709735854334, 2716906424134261502, 115561578124838522882, 5227260815326346060059, 250566480717349417632398

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 A299741.

Cf. A115066, A342167, A342206, A343259, A343260.

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

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

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

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

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

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A298675 Rectangular array A: first differences of row entries of array A294099, read by antidiagonals.

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A343261 a(n) = 2 * T(n,(n+2)/2) where T(n,x) is a Chebyshev polynomial of the first kind.

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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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