A190980 -id:A190980 - 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

A164975 Triangle T(n,k) read by rows: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(n,0) = A000045(n), 0 <= k <= n-1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 3, 8, 8, 8, 5, 15, 25, 20, 16, 8, 30, 55, 70, 48, 32, 13, 56, 125, 175, 184, 112, 64, 21, 104, 262, 440, 512, 464, 256, 128, 34, 189, 539, 1014, 1401, 1416, 1136, 576, 256, 55, 340, 1075, 2270, 3501, 4170, 3760, 2720, 1280, 512

Offset: 1

Mark Dols, Sep 03 2009

A164975 is jointly generated with A209125 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=u(n-1,x)+ 2x*v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 05 2012

			Triangle T(n,k), 0 <= k < n, n >= 1, begins:
   1;
   1,   2;
   2,   3,   4;
   3,   8,   8,   8;
   5,  15,  25,  20,  16;
   8,  30,  55,  70,  48,  32;
  13,  56, 125, 175, 184, 112,  64;
  21, 104, 262, 440, 512, 464, 256, 128;
  ...
T(7,1) = 30 + 2*8 + 15 - 5 = 56.
T(6,1) = 15 + 2*5 +  8 - 3 = 30.

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

Cf. A000045, A000079, A000244 (row sums).

A164975 := proc(n,k) option remember; if n <=0 or k > n or k< 1 then 0; elif k= 1 then combinat[fibonacci](n); else procname(n-1,k)+2*procname(n-1,k-1)+procname(n-2,k)-procname(n-2,k-1) ; end if; end proc: # R. J. Mathar, Jan 27 2011

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A209125 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A164975 *)
(* Clark Kimberling, Mar 05 2012 *)
With[{nmax = 10}, Rest[CoefficientList[CoefficientList[Series[ x/(1 - 2*y*x-x-x^2+y*x^2), {x,0,nmax}, {y,0,nmax}], x], y]]//Flatten] (* G. C. Greubel, Jan 14 2018 *)

T(n,n-1) = A000079(n-1).
T(n,n-2) = A001792(n-2). - R. J. Mathar, Jan 27 2011
T(n,1) = A099920(n-1). - R. J. Mathar, Jan 27 2011
G.f.: x/(1-2*y*x-x-x^2+y*x^2). - Philippe Deléham, Mar 21 2012
Sum_{k=0..n-1, n>0} T(n,k)*x^k = A000045(n), A000244(n-1), A004254(n), A186446(n-1), A190980(n) for x = 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 21 2012

Corrected by Philippe Deléham, Mar 21 2012

A268344 a(n) = 11a(n - 1) - 3a(n - 2) for n>1, a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 11, 118, 1265, 13561, 145376, 1558453, 16706855, 179100046, 1919979941, 20582479213, 220647331520, 2365373209081, 25357163305331, 271832676731398, 2914087954129385, 31239469465229041, 334891900255131296, 3590092494410757133

Offset: 0

Ilya Gutkovskiy, Feb 02 2016

In general, the ordinary generating function for the recurrence relation b(n) = k*b(n - 1) - m*b(n - 2), with n>1 and b(0)=0, b(1)=1, is x/(1 - k*x + m*x^2). This recurrence gives the closed form b(n) = (2^(-n)*((sqrt(k^2 - 4*m) + k)^n - (k - sqrt(k^2 - 4*m))^n))/sqrt(k^2 - 4*m).

G. C. Greubel, Table of n, a(n) for n = 0..965
Index entries for linear recurrences with constant coefficients, signature (11,-3).

Cf. A190872, A190980.

I:=[0,1]; [n le 2 select I[n] else 11*Self(n-1) - 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 14 2018

LinearRecurrence[{11, -3}, {0, 1}, 20] (* or *) Table[(((11 + Sqrt[109])/2)^n - ((11 - Sqrt[109])/2)^n)/Sqrt[109], {n, 0, 20}]

x='x+O('x^30); concat([0], Vec(x/(1-11*x+3*x^2))) \\ G. C. Greubel, Jan 14 2018

G.f.: x/(1 - 11*x + 3*x^2).

a(n) = ( ((11 + sqrt(109))/2)^n - ((11 - sqrt(109))/2)^n )/sqrt(109).

cp's OEIS Frontend

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

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A164975 Triangle T(n,k) read by rows: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(n,0) = A000045(n), 0 <= k <= n-1.

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A268344 a(n) = 11a(n - 1) - 3a(n - 2) for n>1, a(0)=0, a(1)=1.

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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

A164975 Triangle T(n,k) read by rows: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - T(n-2,k-1), T(n,0) = A000045(n), 0 <= k <= n-1.

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Programs

Maple

Mathematica

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Extensions

A268344 a(n) = 11*a(n - 1) - 3*a(n - 2) for n>1, a(0)=0, a(1)=1.

Views

Author

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Comments

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

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

A268344 a(n) = 11a(n - 1) - 3a(n - 2) for n>1, a(0)=0, a(1)=1.