A099156 -id:A099156 - 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

A102591 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).

Original entry on oeis.org

1, 6, 44, 328, 2448, 18272, 136384, 1017984, 7598336, 56714752, 423324672, 3159738368, 23584608256, 176037912576, 1313964867584, 9807567290368, 73204678852608, 546407161659392, 4078438577864704, 30441879976280064

Offset: 0

Paul Barry, Jan 22 2005

In general, Sum_{k=0..n} binomial(2n+1,2k)*r^(n-k) has g.f. (1-(r-1)x)/(1-2(r+1)+(r-1)^2x^2) and a(n) = ((sqrt(r)-1)^(2n+1) + (sqrt(r)+1)^(2n+1))/(2*sqrt(r)).

Index entries for linear recurrences with constant coefficients, signature (8,-4).

Bisection of A080879 or A002605.

Cf. A066443, A079935, A099156, A102592, A107903.

LinearRecurrence[{8,-4},{1,6},20] (* Harvey P. Dale, Sep 28 2021 *)

G.f.: (1-2x)/(1-8x+4x^2);
a(n) = 8*a(n-1) - 4*a(n-2);
a(n) = sqrt(3)*(sqrt(3)-1)^(2n+1)/6 + sqrt(3)*(sqrt(3)+1)^(2n+1)/6.
a(n) = 2^n*A079935(n). - R. J. Mathar, Sep 20 2012
a(n) = 2^(2*n+1)*Sum_{k >= n} binomial(2*k,2*n)*(1/3)^(k+1). Cf. A099156. - Peter Bala, Nov 29 2021
3*a(n)^2 = A107903(n)^2 + 2^(2*n+1). - Philippe Deléham, Mar 21 2023

A107903 Generalized NSW numbers.

Original entry on oeis.org

1, 10, 76, 568, 4240, 31648, 236224, 1763200, 13160704, 98232832, 733219840, 5472827392, 40849739776, 304906608640, 2275853910016, 16987204845568, 126794223124480, 946404965613568, 7064062832410624, 52726882796830720

Offset: 0

Paul Barry, May 27 2005

Counts total area under elevated Schroeder paths of length 2n+2, where horizontal steps can choose from three colors.
Case r=3 for family (1+(r-1)x)/(1-2(1+r)x+(1-r)^2*x^2). Case r=2 gives NSW numbers A002315 and case r=4 gives NSW numbers A096053.
Fifth binomial transform of (1+8x)/(1-16x^2), A107906.
If p is an odd prime, a((p-1)/2) == 1 mod p. - Altug Alkan, Mar 17 2016

Index entries for linear recurrences with constant coefficients, signature (8,-4).

Cf. A001834, A099156, A102591.

Table[Sum[Binomial[2 n + 1, 2 k] 3^k, {k, 0, n}], {n, 0, 20}] (* or *) CoefficientList[Series[(1 + 2 x)/(1 - 8 x + 4 x^2), {x, 0, 20}], x] (* Michael De Vlieger, Mar 17 2016 *)

Vec((1+2*x)/(1-8*x+4*x^2) + O(x^40)) \\ Michel Marcus, Mar 17 2016

G.f.: (1+2*x)/(1-8*x+4*x^2). [corrected by Ralf Stephan, Nov 30 2010]
a(n) = Sum_{k=0..n} binomial(2*n+1, 2*k)*3^k.
a(n) = ((1+sqrt(3))*(4+2*sqrt(3))^n+(1-sqrt(3))*(4-2*sqrt(3))^n)/2 = A099156(n+1)+2*A099156(n).
a(n) = 8*a(n-1) - 4*a(n-2); a(0) = 1, a(1) = 10. - Lekraj Beedassy, Apr 19 2020
a(n) = 2^n*A001834(n). - Philippe Deléham, Mar 18 2023

Typo corrected and link added by Johannes W. Meijer, Aug 07 2010

A342133 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2kx + k*x^2).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 14, 4, 0, 1, 8, 33, 48, 5, 0, 1, 10, 60, 180, 164, 6, 0, 1, 12, 95, 448, 981, 560, 7, 0, 1, 14, 138, 900, 3344, 5346, 1912, 8, 0, 1, 16, 189, 1584, 8525, 24960, 29133, 6528, 9, 0, 1, 18, 248, 2548, 18180, 80750, 186304, 158760, 22288, 10, 0

Offset: 0

Seiichi Manyama, Mar 01 2021

			Square array begins:
  1, 1,   1,    1,     1,     1, ...
  0, 2,   4,    6,     8,    10, ...
  0, 3,  14,   33,    60,    95, ...
  0, 4,  48,  180,   448,   900, ...
  0, 5, 164,  981,  3344,  8525, ...
  0, 6, 560, 5346, 24960, 80750, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to Chebyshev polynomials.

Columns 0..5 give A000007, A000027(n+1), A007070, A138395, A099156(n+1), A190987(n+1).
Rows 0..2 give A000012, A005843, A033991.
Main diagonal gives (-1) * A109520(n+1).
Cf. A342120, A342129, A342134.

T:= (n, k)-> (<<0|1>, <-k|2*k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 01 2021

T[n_, k_] := Sum[If[k == j == 0, 1, (2*k)^j] * (-2)^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 27 2021 *)

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

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

T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)));

T(0,k) = 1, T(1,k) = 2*k and T(n,k) = k*(2*T(n-1,k) - T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (2*k)^(n-j) * (-1/2)^j * binomial(n-j,j) = Sum_{j=0..n} (2*k)^j * (-1/2)^(n-j) * binomial(j,n-j).
T(n,k) = sqrt(k)^n * U(n, sqrt(k)) where U(n, x) is a Chebyshev polynomial of the second kind.

A110274 Expansion of (-16-7x+6x^2+28x^3+8x^4) / ((x-1)(x^2+x+1)(4x^2-8x+1)).

Original entry on oeis.org

16, 135, 1010, 7528, 56183, 419346, 3130024, 23362807, 174382354, 1301607592, 9715331319, 72516220178, 541268436136, 4040082608375, 30155587122450, 225084366546088, 1680052583878903, 12540083204846866, 93600455303259304

Offset: 0

Creighton Dement, Jul 18 2005

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-4,1,-8,4).

Cf. A110275.

seriestolist(series((-16-7*x+6*x^2+28*x^3+8*x^4)/((x-1)*(x^2+x+1)*(4*x^2-8*x+1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: bisection of 4tessigcyczapsumseq[A*B] with A = - 'j + 'k - 'ii' - 'ij' - 'ik' and B = + .5'i + .5'j - .5'k + .5i' - .5j' + .5k' + .5'ij' + .5'ik' - .5'ji' - .5'ki'; Sumtype is set to: sum[(Y[0], Y[1], Y[2]),mod(3)

Vec((16 + 7*x - 6*x^2 - 28*x^3 - 8*x^4) / ((1 - x)*(1 + x + x^2)*(1 - 8*x + 4*x^2)) + O(x^20)) \\ Colin Barker, May 12 2019

a(n) = 8*a(n-1) - 4*a(n-2) + a(n-3) - 8*a(n-4) + 4*a(n-5) for n>4. - Colin Barker, May 12 2019

117*a(n) = -47*A049347(n) -67*A049347(n-1) + 8*(209*A099156(n+1)+274*A099156(n)) +247. - R. J. Mathar, Sep 11 2019

A138240 Expansion of (1/4)(1-sqrt(1-12x)/sqrt(1-4x)).

Original entry on oeis.org

0, 1, 6, 40, 296, 2400, 20928, 192768, 1848960, 18277888, 184890368, 1904259072, 19898765312, 210424545280, 2247494172672, 24209586782208, 262696649785344, 2868744309571584, 31504024885002240, 347697247933169664

Offset: 0

Paul Barry, Mar 07 2008

Hankel transform of a(n) is -4^comb(n,2)*A099156(n)=-4^comb(n,2)*[x^n](x/(1-8x+4x^2)).
Hankel transform of a(n+1) is 4^comb(n+1,2)=A053763(n+1).
Hankel transform of a(n+2) is 4^comb(n+1,2)*A102591(n+1)=4^comb(n+1,2)*[x^n](6-4x)/(1-8x+4x^2).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A104498.

CoefficientList[Series[1/4*(1-Sqrt[1-12*x]/Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

Recurrence: n*a(n) = 4*(4*n-5)*a(n-1) - 48*(n-2)*a(n-2) . - Vaclav Kotesovec, Oct 20 2012

a(n) ~ 2^(2*n-7/2)*3^(n+1/2)/(sqrt(Pi)*n^(3/2)) . - Vaclav Kotesovec, Oct 20 2012

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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A102591 a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).

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A107903 Generalized NSW numbers.

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A342133 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2*k*x + k*x^2).

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A110274 Expansion of (-16-7*x+6*x^2+28*x^3+8*x^4) / ((x-1)*(x^2+x+1)*(4*x^2-8*x+1)).

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A138240 Expansion of (1/4)(1-sqrt(1-12x)/sqrt(1-4x)).

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

A342133 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2kx + k*x^2).

A110274 Expansion of (-16-7x+6x^2+28x^3+8x^4) / ((x-1)(x^2+x+1)(4x^2-8x+1)).