A154235 -id:A154235 - 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

A164532 a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 4.

Original entry on oeis.org

1, 4, 6, 24, 36, 144, 216, 864, 1296, 5184, 7776, 31104, 46656, 186624, 279936, 1119744, 1679616, 6718464, 10077696, 40310784, 60466176, 241864704, 362797056, 1451188224, 2176782336, 8707129344, 13060694016, 52242776064, 78364164096

Offset: 1

Klaus Brockhaus, Aug 15 2009

nonn

Interleaving of A000400 and A067411 without initial term 1.

Binomial transform is apparently A123011. Fourth binomial transform is A154235.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,6).

Cf. A000400 (powers of 6), A067411, A123011, A154235.

[ n le 2 select 3*n-2 else 6*Self(n-2): n in [1..29] ];

LinearRecurrence[{0,6}, {1,4}, 40] (* G. C. Greubel, Jul 16 2021 *)

[((1 - (-1)^n)*sqrt(6)/2 + 2*(1 + (-1)^n))*6^(n/2 -1) for n in (1..40)] # G. C. Greubel, Jul 16 2021

a(n) = (5 - (-1)^n)*6^(1/4*(2*n - 5 + (-1)^n)).
G.f.: x*(1+4*x)/(1-6*x^2).
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = ((1-(-1)^n)*sqrt(6)/2 + 2*(1+(-1)^n))*6^(n/2 -1). - G. C. Greubel, Jul 16 2021

A164550 a(n) = 6a(n-1) - 3a(n-2) for n > 1; a(0) = 1, a(1) = 7.

Original entry on oeis.org

1, 7, 39, 213, 1161, 6327, 34479, 187893, 1023921, 5579847, 30407319, 165704373, 903004281, 4920912567, 26816462559, 146136037653, 796366838241, 4339792916487, 23649656984199, 128878563155733, 702322407981801

Offset: 0

Klaus Brockhaus, Aug 15 2009

nonn

Binomial transform of A164549.

Inverse binomial transform of A154235.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-3).

Cf. A154235, A164549.

[ n le 2 select 6*n-5 else 6*Self(n-1)-3*Self(n-2): n in [1..21] ];

LinearRecurrence[{6,-3}, {1,7}, 31] (* G. C. Greubel, Jul 16 2021 *)

[3^((n-1)/2)*(sqrt(3)*chebyshev_U(n, sqrt(3)) + chebyshev_U(n-1, sqrt(3))) for n in (0..30)] # G. C. Greubel, Jul 16 2021

a(n) = ((3+2*sqrt(6))*(3+sqrt(6))^n + (3-2*sqrt(6))*(3-sqrt(6))^n)/6.
G.f.: (1+x)/(1-6*x+3*x^2).
a(n) = 3^((n-1)/2)*(sqrt(3)*ChebyshevU(n, sqrt(3)) + ChebyshevU(n-1, sqrt(3))). - G. C. Greubel, Jul 16 2021

A164551 a(n) = 10a(n-1)-19a(n-2) for n > 1; a(0) = 1, a(1) = 9.

Original entry on oeis.org

1, 9, 71, 539, 4041, 30169, 224911, 1675899, 12485681, 93014729, 692919351, 5161913659, 38453668921, 286460329689, 2133983587391, 15897089609819, 118425207937761, 882207376791049, 6571994817093031, 48958008011900379

Offset: 0

Klaus Brockhaus, Aug 15 2009

nonn

Binomial transform of A154235. Inverse binomial transform of A164552.

Index entries for linear recurrences with constant coefficients, signature (10, -19).

Cf. A154235, A164552.

[ n le 2 select 8*n-7 else 10*Self(n-1)-19*Self(n-2): n in [1..20] ];

LinearRecurrence[{10,-19},{1,9},30] (* Harvey P. Dale, Dec 26 2015 *)

a(n) = ((3+2*sqrt(6))*(5+sqrt(6))^n+(3-2*sqrt(6))*(5-sqrt(6))^n)/6.

G.f.: (1-x)/(1-10*x+19*x^2).

cp's OEIS Frontend

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

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A164532 a(n) = 6*a(n-2) for n > 2; a(1) = 1, a(2) = 4.

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A164550 a(n) = 6*a(n-1) - 3*a(n-2) for n > 1; a(0) = 1, a(1) = 7.

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Author

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Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A164551 a(n) = 10*a(n-1)-19*a(n-2) for n > 1; a(0) = 1, a(1) = 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

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

A164550 a(n) = 6a(n-1) - 3a(n-2) for n > 1; a(0) = 1, a(1) = 7.

A164551 a(n) = 10a(n-1)-19a(n-2) for n > 1; a(0) = 1, a(1) = 9.