A091870 -id:A091870 - cp's OEIS frontend

A093145 Third binomial transform of Fibonacci(3n)/Fibonacci(3).

0, 1, 10, 80, 600, 4400, 32000, 232000, 1680000, 12160000, 88000000, 636800000, 4608000000, 33344000000, 241280000000, 1745920000000, 12633600000000, 91417600000000, 661504000000000, 4786688000000000, 34636800000000000

Offset: 0

Paul Barry, Mar 26 2004

Fifth binomial transform of 1,5,5,25,25,125. - Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..200
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
Index entries for linear recurrences with constant coefficients, signature (10,-20).

Cf. A000045, A091870.

[n le 2 select n - 1 else 10*Self(n-1)-20*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Dec 30 2012

Join[{a=0,b=1},Table[c=10*b-20*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)
f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(5 + s)^n + (1 - s)(5 - s)^n)/2]]; Array[f, 21, 0] (* Or *)
a[n_] := 10 a[n - 1] - 20 a[n - 2]; a[0] = 0; a[1] = 1; Array[a, 22, 0] (* Or *)
CoefficientList[Series[x/(1 - 10 x + 20 x^2), {x, 0, 21}], x] (* Robert G. Wilson v, Mar 07 2011 *)
LinearRecurrence[{10,-20},{0,1},30] (* Harvey P. Dale, Jan 23 2019 *)

[lucas_number1(n,10,20) for n in range(0, 21)] # Zerinvary Lajos, Apr 26 2009

G.f.: x/(1 - 10*x + 20*x^2).
a(n) = ((5+sqrt(5))^n - (5-sqrt(5))^n)/(2*sqrt(5)).
a(n) = Sum_{k=0..n} binomial(n, 2*k+ 1)*5^(n-k-1).
a(n) = 10*a(n-1) - 20*a(n-2), n > 1; a(0)=0, a(1)=1. - Zerinvary Lajos, Apr 26 2009
G.f.: A(x) = x*G(0)/(1-5*x) where G(k) = 1 + 5*x/(1-5*x - x*(1-5*x)/(x + (1-5*x)/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 30 2012

A108404 Expansion of (1-4x)/(1-8x+11x^2).

Original entry on oeis.org

1, 4, 21, 124, 761, 4724, 29421, 183404, 1143601, 7131364, 44471301, 277325404, 1729418921, 10784771924, 67254567261, 419404046924, 2615432135521, 16310012568004, 101710347053301, 634272638178364, 3955367287840601

Offset: 0

Philippe Deléham, Jul 04 2005

Binomial transform of A098648. Second binomial transform of A001077. Third binomial transform of A084057. 4th binomial transform of (1, 0, 5, 0, 25, 0, 125, 0, 625, 0, 3125, ...).

Seiichi Manyama, Table of n, a(n) for n = 0..1258
Index entries for linear recurrences with constant coefficients, signature (8,-11).

Cf. A001077, A084057, A098648.

CoefficientList[Series[(1-4x)/(1-8x+11x^2),{x,0,30}],x] (* or *) LinearRecurrence[{8,-11},{1,4},30] (* Harvey P. Dale, Jan 03 2012 *)

E.g.f.: exp(4x)cosh(sqrt(5)x).
a(n) = 8a(n-1) - 11a(n-2), a(0) = 1, a(1) = 4.
a(n) = ((4+sqrt(5))^n + (4-sqrt(5))^n)/2.
a(n+1)/a(n) converges to 4 + sqrt(5) = 6.2360679774997896964... = 4+A002163.
a(n) = A091870(n+1)-4*A091870(n). - R. J. Mathar, Nov 10 2013

A292466 Triangle read by rows: T(n,k) = 4T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = 5^m.

Original entry on oeis.org

0, 1, 1, 0, 4, 8, 5, 5, 21, 53, 0, 20, 40, 124, 336, 25, 25, 105, 265, 761, 2105, 0, 100, 200, 620, 1680, 4724, 13144, 125, 125, 525, 1325, 3805, 10525, 29421, 81997, 0, 500, 1000, 3100, 8400, 23620, 65720, 183404, 511392, 625, 625, 2625, 6625, 19025, 52625

Offset: 0

Seiichi Manyama, Sep 22 2017

			First few rows are:
    0;
    1,   1;
    0,   4,   8;
    5,   5,  21,   53;
    0,  20,  40,  124,  336;
   25,  25, 105,  265,  761,  2105;
    0, 100, 200,  620, 1680,  4724, 13144;
  125, 125, 525, 1325, 3805, 10525, 29421, 81997.
--------------------------------------------------------------
The diagonal is      {0, 1,  8,  53, 336, 2105, ...} and
the next diagonal is {1, 4, 21, 124, 761, 4724, ...}.
Two sequences have the following property:
     1^2 - 5*   0^2 = 1      (= 11^0),
     4^2 - 5*   1^2 = 11     (= 11^1),
    21^2 - 5*   8^2 = 121    (= 11^2),
   124^2 - 5*  53^2 = 1331   (= 11^3),
   761^2 - 5* 336^2 = 14641  (= 11^4),
  4724^2 - 5*2105^2 = 161051 (= 11^5),
  ...

Seiichi Manyama, Rows n = 0..139, flattened

The diagonal of the triangle is A091870.
The next diagonal of the triangle is A108404.
T(n,k) = b*T(n-1,k-1) + T(n,k-1): A292789 (b=-3), A292495 (b=-2), A117918 and A228405 (b=1), A227418 (b=2), this sequence (b=4).

T(n+1,n)^2 - 5*T(n,n)^2 = 11^n.

A162769 a(n) = ((1+sqrt(5))(4+sqrt(5))^n + (1-sqrt(5))(4-sqrt(5))^n)/2.

Original entry on oeis.org

1, 9, 61, 389, 2441, 15249, 95141, 593389, 3700561, 23077209, 143911501, 897442709, 5596515161, 34900251489, 217640345141, 1357219994749, 8463716161441, 52780309349289, 329141597018461, 2052549373305509

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Jul 13 2009

Binomial transform of A082762. Fourth binomial transform of A162962. Inverse binomial transform of A093145 without initial 0.

Index entries for linear recurrences with constant coefficients, signature (8,-11).

Cf. A082762, A162962, A093145.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((1+r)*(4+r)^n+(1-r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 19 2009

f[n_] := Block[{s = Sqrt@ 5}, Simplify[((1 + s)(4 + s)^n + (1 - s)(4 - s)^n)/2]]; Array[f, 21, 0] (* Or *)
a[n_] := 8 a[n - 1] - 11 a[n - 2]; a[0] = 1; a[1] = 9; Array[a, 22, 0] (* Or *)
CoefficientList[Series[(1 + x)/(1 - 8 x + 11 x^2), {x, 0, 21}], x] (* Robert G. Wilson v *)

a(n) = 8*a(n-1) - 11*a(n-2) for n > 1; a(0) = 1, a(1) = 9.
G.f.: (1+x)/(1-8*x+11*x^2).
a(n) = A091870(n)+A091870(n+1). - R. J. Mathar, Feb 04 2021

Edited and extended beyond a(5) by Klaus Brockhaus, Jul 19 2009

cp's OEIS Frontend

A093145 Third binomial transform of Fibonacci(3n)/Fibonacci(3).

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A108404 Expansion of (1-4x)/(1-8x+11x^2).

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A292466 Triangle read by rows: T(n,k) = 4*T(n-1,k-1) + T(n,k-1) with T(2*m,0) = 0 and T(2*m+1,0) = 5^m.

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A162769 a(n) = ((1+sqrt(5))*(4+sqrt(5))^n + (1-sqrt(5))*(4-sqrt(5))^n)/2.

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A292466 Triangle read by rows: T(n,k) = 4T(n-1,k-1) + T(n,k-1) with T(2m,0) = 0 and T(2*m+1,0) = 5^m.

A162769 a(n) = ((1+sqrt(5))(4+sqrt(5))^n + (1-sqrt(5))(4-sqrt(5))^n)/2.