A289415 -id:A289415 - cp's OEIS frontend

A289414 a(n) = ((10-sqrt(10))^n + (10+sqrt(10))^n) / 2.

1, 10, 110, 1300, 16100, 205000, 2651000, 34570000, 452810000, 5944900000, 78145100000, 1027861000000, 13524161000000, 177975730000000, 2342340110000000, 30828986500000000, 405769120100000000, 5340773617000000000, 70296251531000000000, 925255405090000000000

Offset: 0

Colin Barker, Jul 06 2017

Colin Barker, Table of n, a(n) for n = 0..850
Index entries for linear recurrences with constant coefficients, signature (20,-90).

Cf. A083881, A090139, A143079, A145301, A145302, A145303, A289415.

a:= n-> (<<0|1>, <-90|20>>^n. <<1,10>>)[1,1]:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 06 2017

LinearRecurrence[{20, -90}, {1, 10}, 20] (* Jean-François Alcover, Jan 29 2025 *)

Vec((1 - 10*x) / (1 - 20*x + 90*x^2) + O(x^25))

G.f.: (1 - 10*x) / (1 - 20*x + 90*x^2).

a(n) = 20*a(n-1) - 90*a(n-2) for n>1.

A361432 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 20, 8, 0, 1, 5, 20, 54, 68, 16, 0, 1, 6, 30, 112, 252, 232, 32, 0, 1, 7, 42, 200, 656, 1188, 792, 64, 0, 1, 8, 56, 324, 1400, 3904, 5616, 2704, 128, 0, 1, 9, 72, 490, 2628, 10000, 23360, 26568, 9232, 256, 0

Offset: 0

Seiichi Manyama, Mar 11 2023

			Square array begins:
  1,  1,   1,    1,    1,     1, ...
  0,  1,   2,    3,    4,     5, ...
  0,  2,   6,   12,   20,    30, ...
  0,  4,  20,   54,  112,   200, ...
  0,  8,  68,  252,  656,  1400, ...
  0, 16, 232, 1188, 3904, 10000, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Column k=0..11 give A000007, A011782, A006012, A083881, A081335, A090139, A145301, A145302, A145303, A143079, A289414, A289415.
Main diagonal gives A084062.
Cf. A084061, A084097.

T(n,k) = sum(j=0, n\2, k^(n-j)*binomial(n, 2*j));

T(n, k) = round(((k+sqrt(k))^n+(k-sqrt(k))^n))/2;

T(0,k) = 1, T(1,k) = k; T(n,k) = 2 * k * T(n-1,k) - (k-1) * k * T(n-2,k).
T(n,k) = ((k + sqrt(k))^n + (k - sqrt(k))^n)/2.
G.f. of column k: (1 - k * x)/(1 - 2 * k * x + (k-1) * k * x^2).
E.g.f. of column k: exp(k * x) * cosh(sqrt(k) * x).

cp's OEIS Frontend

A289414 a(n) = ((10-sqrt(10))^n + (10+sqrt(10))^n) / 2.

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A361432 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n,2*j).

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