A190966 -id:A190966 - cp's OEIS frontend

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

0, 1, 2, -6, -32, -4, 312, 664, -1792, -10224, -2528, 97184, 219648, -532544, -3261568, -1197696, 30220288, 72417536, -157367808, -1038910976, -504143872, 9380822016, 23803082752, -46202054656, -330434936832, -198849327104, 2906650714112, 7801794699264

Offset: 0

Vladimir Joseph Stephan Orlovsky, May 24 2011

For the difference equation a(n) = c*a(n-1) - d*a(n-2), with a(0) = 0, a(1) = 1, the solution is a(n) = d^((n-1)/2) * ChebyshevU(n-1, c/(2*sqrt(d))) and has the alternate form a(n) = ( ((c + sqrt(c^2 - 4*d))/2)^n - ((c - sqrt(c^2 - 4*d))/2)^n )/sqrt(c^2 - 4*d). In the case c^2 = 4*d then the solution is a(n) = n*d^((n-1)/2). The generating function is x/(1 - c*x + d^2) and the exponential generating function takes the form (2/sqrt(c^2 - 4*d))*exp(c*x/2)*sinh(sqrt(c^2 - 4*d)*x/2) for c^2 > 4*d, (2/sqrt(4*d - c^2))*exp(c*x/2)*sin(sqrt(4*d - c^2)*x/2) for 4*d > c^2, and x*exp(sqrt(d)*x) if c^2 = 4*d. - G. C. Greubel, Jun 10 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Index entries for linear recurrences with constant coefficients, signature (2,-10).

Sequences of the form a(n) = c*a(n-1) - d*a(n-2), with a(0)=0, a(1)=1:
c/d...1.......2.......3.......4.......5.......6.......7.......8.......9......10
.A128834,A107920,A106852,A106853,A106854,A145934,A145976,A145978,A146078,A146080
.A000027,A009545,A088137,A088138,A045873,A088139,A089181,A090591,A127357,A190958
.A001906,A000225,A057083,A049072,A190959,A190960,A190961,A190962,A190963,A190964
.A001353,A007070,A003462,A001787,A099456,A190965,A168175,A190966,A190967,A190968
.A004254,A107839,A116415,A002450,A030191,A001047,A099450,A190969,A190970,A190971
.A001109,A154244,A138395,A084326,A003463,A030192,A081179,A006516,A027471,A106392
.A004187,A186446,A190972,A190973,A190974,A003464,A030240,A152268,A099459,A016127
.A001090,A190975,A190976,A099156,A190977,A190978,A023000,A057084,A154245,A154235
.A018913,A190979,A190980,A190981,A190982,A190983,A190984,A023001,A057085,A178869
A004189,A190869,A190985,A190986,A190987,A190988,A190989,A190990,A002452,A057086

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1)-10*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 17 2011

Mathematica
```
LinearRecurrence[{2,-10}, {0,1}, 50]
```

a(n)=([0,1; -10,2]^n*[0;1])[1,1] \\ Charles R Greathouse IV, Apr 08 2016

[lucas_number1(n,2,10) for n in (0..50)] # G. C. Greubel, Jun 10 2022

G.f.: x / ( 1 - 2*x + 10*x^2 ). - R. J. Mathar, Jun 01 2011
E.g.f.: (1/3)*exp(x)*sin(3*x). - Franck Maminirina Ramaharo, Nov 13 2018
a(n) = 10^((n-1)/2) * ChebyshevU(n-1, 1/sqrt(10)). - G. C. Greubel, Jun 10 2022
a(n) = (1/3)*10^(n/2)*sin(n*arctan(3)) = Sum_{k=0..floor(n/2)} (-1)^k*3^(2*k)*binomial(n,2*k+1). - Gerry Martens, Oct 15 2022

A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 1, 2, 2, 0, 3, 5, 1, -1, 5, 10, 4, -2, -1, 8, 20, 12, -4, -4, 0, 13, 38, 31, -4, -13, -2, 1, 21, 71, 73, 3, -33, -11, 3, 1, 34, 130, 162, 34, -74, -42, 6, 6, 0, 55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Offset: 0

Philippe Deléham, Dec 06 2011

Row-reversed variant of A123585. Row sums: 2^n.

			Triangle begins:
1
1, 1
2, 2, 0
3, 5, 1, -1
5, 10, 4, -2, -1
8, 20, 12, -4, -4, 0
13, 38, 31, -4, -13, -2, 1
21, 71, 73, 3, -33, -11, 3, 1
34, 130, 162, 34, -74, -42, 6, 6, 0
55, 235, 344, 128, -146, -130, 0, 24, 3, -1

Cf. Columns: A000045, A001629, A129707.

Diagonals: A010892, A099254, Antidiagonal sums: A158943.

G.f.: 1/(1-(1+y)*x+(y+1)*(y-1)*x^2).
T(n,0) = A000045(n+1).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) - T(n-2,k-2) with T(0,0)= 1 and T(n,k)= 0 if nSum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A090591(n), (-1)^n*A106852(n), A000007(n), A000045(n+1), A000079(n), A057083(n), A190966(n+1) for n = -3, -2, -1, 0, 1, 2, 3 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A010892(n), A000079(n), A030195(n+1), A180222(n+2) for x = 0, 1, 2, 3 respectively.

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cp's OEIS Frontend

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.

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A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

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cp's OEIS Frontend

A190958 a(n) = 2*a(n-1) - 10*a(n-2), with a(0) = 0, a(1) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A201947 Triangle T(n,k), read by rows, given by (1,1,-1,0,0,0,0,0,0,0,...) DELTA (1,-1,1,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A190958 a(n) = 2a(n-1) - 10a(n-2), with a(0) = 0, a(1) = 1.