A289414 -id:A289414 - cp's OEIS frontend

A289415 a(n) = ((11-sqrt(11))^n + (11+sqrt(11))^n) / 2.

1, 11, 132, 1694, 22748, 314116, 4408272, 62429224, 888533008, 12680511536, 181232622912, 2592261435104, 37094163051968, 530922829281856, 7599944308484352, 108797263565651584, 1557545924511056128, 22298311347021560576, 319232797938258158592

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 (22,-110).

Cf. A083881, A090139, A143079, A145301, A145302, A145303, A289414.

a:= n-> (<<0|1>, <-110|22>>^n. <<1,11>>)[1,1]:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 06 2017

Vec((1 - 11*x) / (1 - 22*x + 110*x^2) + O(x^25))

G.f.: (1 - 11*x) / (1 - 22*x + 110*x^2).

a(n) = 22*a(n-1) - 110*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).

A361293 a(n) = 20 * a(n-1) - 90 * a(n-2) for n>1, with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 20, 310, 4400, 60100, 806000, 10711000, 141680000, 1869610000, 24641000000, 324555100000, 4273412000000, 56258281000000, 740558540000000, 9747925510000000, 128308241600000000, 1688851536100000000, 22229288978000000000, 292589141311000000000

Offset: 0

Seiichi Manyama, Mar 12 2023

Index entries for linear recurrences with constant coefficients, signature (20,-90).

Column k=10 of A361290.

Cf. A289414.

LinearRecurrence[{20,-90},{0,1},20] (* Harvey P. Dale, Dec 16 2023 *)

a(n) = polcoef(lift(Mod('x, ('x-10)^2-10)^n), 1);

my(N=20, x='x+O('x^N)); concat (0,  Vec(x/(1-20*x+90*x^2)))

my(N=20, x='x+O('x^N)); concat (0, apply(round, Vec(serlaplace(exp(10*x)*sinh(sqrt(10)*x)/sqrt(10)))))

a(n) = ( (10 + sqrt(10))^n - (10 - sqrt(10))^n )/(2 * sqrt(10)).
a(n) = Sum_{k=0..floor((n-1)/2)} 10^(n-1-k) * binomial(n,2*k+1).
G.f.: x/(1 - 20 * x + 90 * x^2).
E.g.f.: exp(10 * x) * sinh(sqrt(10) * x) / sqrt(10).

cp's OEIS Frontend

A289415 a(n) = ((11-sqrt(11))^n + (11+sqrt(11))^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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A361293 a(n) = 20 * a(n-1) - 90 * a(n-2) for n>1, with a(0)=0, a(1)=1.

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