A085480 -id:A085480 - cp's OEIS frontend

A030195 a(n) = 3a(n-1) + 3a(n-2), a(0)=0, a(1)=1.

0, 1, 3, 12, 45, 171, 648, 2457, 9315, 35316, 133893, 507627, 1924560, 7296561, 27663363, 104879772, 397629405, 1507527531, 5715470808, 21668995017, 82153397475, 311467177476, 1180861724853, 4476986706987, 16973545295520

Offset: 0

Wolfdieter Lang

Scaled Chebyshev U-polynomials evaluated at I*sqrt(3)/2.
Number of zeros in the substitution system {0 -> 1111100, 1 -> 10} at step n from initial string "1" (1 -> 10 -> 101111100 -> ...). - Ilya Gutkovskiy, Apr 10 2017
a(n+1) is the number of compositions of n having parts 1 and 2, both of three kinds. - Gregory L. Simay, Sep 21 2017
More generally, define a(n) = k*a(n-1) + k*a(n-2), a(0) = 0 and a(1) = 1. Then g.f. a(n) = 1/(1 - k*x - k*x^2) and a(n+1) is the number of compositions of n having parts 1 and 2, both of k kinds. - Gregory L. Simay, Sep 22 2017

			G.f. = x + 3*x^2 + 12*x^3 + 45*x^4 + 171*x^5 + 648*x^6 + 2457*x^7 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=q=3.
Tanya Khovanova, Recursive Sequences
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs. (39), (41) and (45), rhs, m=3.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (3,3).

Equals round(A085480(n)/sqrt(21)).
Cf. A175290 (Pisano periods), A000045, A002605, A172010, A057088, A057089, A057090, A057091, A057092, A057093.
Cf. A026150, A028859, A028860, A080040, A083337, A106435, A108898, A125145.

a030195 n = a030195_list !! n
a030195_list =
   0 : 1 : map (* 3) (zipWith (+) a030195_list (tail a030195_list))
-- Reinhard Zumkeller, Oct 14 2011

I:=[0,1]; [n le 2 select I[n] else 3*Self(n-1) + 3*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

CoefficientList[Series[1/(1-3x-3x^2), {x, 0, 25}], x] (* Zerinvary Lajos, Mar 22 2007 *)
LinearRecurrence[{3, 3}, {0, 1}, 24] (* Or *)
RecurrenceTable[{a[n] == 3 a[n - 1] + 3 a[n - 2], a[0] == 0, a[1] == 1}, a, {n, 0, 23}] (* Robert G. Wilson v, Aug 18 2012 *)

{a(n) = n--; polchebyshev(n, 2, I*sqrt(3)/2) * (-I*sqrt(3))^n};

[lucas_number1(n,3,-3) for n in range(0, 25)] # Zerinvary Lajos, Apr 22 2009

a(n+1) = (-I*sqrt(3))^n*U(n, I*sqrt(3)/2).
G.f.: x / (1 - 3*x - 3*x^2).
a(n+1) = Sum_{k=0..floor(n/2)} 3^(n-k)*binomial(n-k, k). - Emeric Deutsch, Nov 14 2001
a(n) = (p^n - q^n)/sqrt(21); p = (3 + sqrt 21)/2, q = (3 - sqrt 21)/2. - Gary W. Adamson, Jul 02 2003
For n > 0, a(n) = Sum_{k=0..n-1} (2^k)*A063967(n-1,k). - Gerald McGarvey, Jul 23 2006
a(n+1) = Sum_{k=0..n} 2^k*A063967(n,k). - Philippe Deléham, Nov 03 2006

Edited by Ralf Stephan, Aug 02 2004
I simplified the definition. As a result the offsets in some of the formulas may need to shifted by 1. - N. J. A. Sloane, Apr 01 2006
Formulas shifted to match offset. - Charles R Greathouse IV, Jan 31 2011

A170931 Extended Lucas L(n,i) = n(L(n,i-1) + L(n,i-2)) = a^i + b^i where d = sqrt(n(n+4)); a=(n+d)/2; b=(n-d)/2.

Original entry on oeis.org

2, 4, 24, 112, 544, 2624, 12672, 61184, 295424, 1426432, 6887424, 33255424, 160571392, 775307264, 3743514624, 18075287552, 87275208704, 421401985024, 2034708774912, 9824443039744, 47436607258624, 229044201193472

Offset: 0

Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Feb 04 2010

Sequence gives the rational part of the radii of the circles in nested circles and squares inspired by Vitruvian Man, starting with a square whose sides are of length 4 (in some units). The radius of the circle is an integer in the real quadratic number field Q(sqrt(2)), namely R(n) = A(n-1) + B(n)*sqrt(2) with A(-1)=1, for n >= 1, A(n-1) = A170931(n-1)*-1^(n-1); and B(n) = A094013(n)*-1^n. See illustrations in the links. - Kival Ngaokrajang, Feb 15 2015

			L(n,0)=2, L(n,1)=n.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Kival Ngaokrajang, Illustration of initial terms, Vitruvian Man
Index entries for linear recurrences with constant coefficients, signature (4, 4).

Cf. similar sequences with d=sqrt(n*(n+k)): A000032 (k=1, classic Lucas), A080040 (k=2), A085480 (k=3).

Cf. A094013, A174968.

I:=[2,4]; [n le 5 select I[n] else 4*Self(n-1)+4*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 21 2017

CoefficientList[Series[2 (1 - 2 x) / (1 - 4 x - 4 x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 16 2015 *)
LinearRecurrence[{4,4},{2,4},30] (* Harvey P. Dale, Sep 03 2016 *)

x='x+O('x^30); Vec(2*(1-2*x)/(1 - 4*x - 4*x^2)) \\ G. C. Greubel, Dec 21 2017

From R. J. Mathar, Feb 05 2010: (Start)
a(n) = 2*A084128(n) = 4*a(n-1) + 4*a(n-2).
G.f.: 2*(1-2*x)/(1 - 4*x - 4*x^2). (End)

A172012 Expansion of (2-3x)/(1-3x-3*x^2) .

Original entry on oeis.org

2, 3, 15, 54, 207, 783, 2970, 11259, 42687, 161838, 613575, 2326239, 8819442, 33437043, 126769455, 480619494, 1822166847, 6908359023, 26191577610, 99299809899, 376474162527, 1427321917278, 5411388239415, 20516130470079, 77782556128482, 294896059795683

Offset: 0

Claudio Peruzzi (claudio.peruzzi(AT)gmail.com), Jan 22 2010

The case k=3 in a family of sequences a(n) = L(k,n), L(k,n)=k*(L(k,n-1)+L(k,n-2)), L(k,0)=2 and L(k,1)=k.

The case k=1 is A000032 (classic Lucas sequence), k=2 is A080040, this here is essentially A085480.

Wikipedia, Lucas sequence: Specific names.
Index entries for linear recurrences with constant coefficients, signature (3,3).

CoefficientList[Series[(2-3x)/(1-3x-3x^2),{x,0,30}],x] (* or *) LinearRecurrence[{3,3},{2,3},31] (* Harvey P. Dale, Aug 24 2011 *)

a(n) = 3*( a(n-1)+a(n-2) ) = 2*A030195(n+1)-3*A030195(n).
L(k,n) = c^n+b^n where c=(k+d)/2 ; b=(k-d)/2; d=sqrt(k*(k+4)) (Binet formula).
a(0)=2, a(1)=3, a(n) = 3*a(n-1)+3*a(n-2). [Harvey P. Dale, Aug 24 2011]
a(n) = [x^n] ( (1 + 3*x + sqrt(1 + 6*x + 21*x^2))/2 )^n for n >= 1. - Peter Bala, Jun 23 2015

Edited and extended by R. J. Mathar, Jan 23 2010

cp's OEIS Frontend

A030195 a(n) = 3a(n-1) + 3a(n-2), a(0)=0, a(1)=1.

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A170931 Extended Lucas L(n,i) = n(L(n,i-1) + L(n,i-2)) = a^i + b^i where d = sqrt(n(n+4)); a=(n+d)/2; b=(n-d)/2.

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A172012 Expansion of (2-3x)/(1-3x-3*x^2) .

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cp's OEIS Frontend

A030195 a(n) = 3*a(n-1) + 3*a(n-2), a(0)=0, a(1)=1.

Views

Author

Keywords

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Examples

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Crossrefs

Programs

Haskell

Magma

Mathematica

PARI

Sage

Formula

Extensions

A170931 Extended Lucas L(n,i) = n*(L(n,i-1) + L(n,i-2)) = a^i + b^i where d = sqrt(n*(n+4)); a=(n+d)/2; b=(n-d)/2.

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Programs

Magma

Mathematica

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Formula

A172012 Expansion of (2-3*x)/(1-3*x-3*x^2) .

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Mathematica

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Extensions

A030195 a(n) = 3a(n-1) + 3a(n-2), a(0)=0, a(1)=1.

A170931 Extended Lucas L(n,i) = n(L(n,i-1) + L(n,i-2)) = a^i + b^i where d = sqrt(n(n+4)); a=(n+d)/2; b=(n-d)/2.

A172012 Expansion of (2-3x)/(1-3x-3*x^2) .