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proposed

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editing

proposed

a(n) = Sum C(n-k,k-1)3^(k-1), _{k=0..floor(n/2)} binomial(n-k, k-1) * 3^(k-1).

In general a(n) =sum Sum_{k=0..floor(n/2), C} binomial(n-k, k-1)*r^(k-1)} has g.f. x^2/((1-r*x^2)*(1-x-r*x^2)) and satisfies the recurrence a(n) = a(n-1) +2r 2*r*a(n-2) - r*a(n-3) - r^2*a(n-4).

G. C. Greubel, <a href="/A099579/b099579.txt">Table of n, a(n) for n = 0..1000</a>

G.f.: x^2/((1-3x3*x^2)*(1-x-3x3*x^2)); a(n)=a(n-1)+6a(n-2)-3a(n-3)-9a(n-4).

a(n) = a(n-1) + 6*a(n-2) - 3*a(n-3) - 9*a(n-4).

From G. C. Greubel, Jul 24 2022: (Start)

a(n) = (i*sqrt(3))^(n-1)*ChebyshevU(n-1, -i/(2*sqrt(3))) - 3^((n-1)/2)*(1 - (-1)^n)/2.

E.g.f.: (1/sqrt(39))*( 2*sqrt(3)*exp(x/2)*sinh(sqrt(13)*x/2) - sqrt(13)*sinh(sqrt(3)*x) ). (End)

LinearRecurrence[{1, 6, -3, -9}, {0, 0, 1, 1}, 50] (* G. C. Greubel, Jul 24 2022 *)

(Magma) [n le 4 select Floor((n-1)/2) else Self(n-1) +6*Self(n-2) -3*Self(n-3) -9*Self(n-4): n in [1..41]]; // G. C. Greubel, Jul 24 2022

(SageMath)

@CachedFunction

def a(n): # a = A099579

if (n<4): return (n//2)

else: return a(n-1) +6*a(n-2) -3*a(n-3) -9*a(n-4)

[a(n) for n in (0..40)] # G. C. Greubel, Jul 24 2022

Cf. A097038, A099580, A140167.

approved

editing

<a href="/index/Rec#order_04">Index to sequences with entries for linear recurrences with constant coefficients</a>, signature (1,6,-3,-9).

editing

approved

<a href="/index/Rec#order_04">Index to sequences with linear recurrences with constant coefficients</a>, signature (1,6,-3,-9).

approved

editing

_Paul Barry (pbarry(AT)wit.ie), _, Oct 23 2004

In general a(n)=sum{k=0..floor(n/2), C(n-k,k-1)r^(k-1)} has g.f. x^2/((1-r*x^2)(1-x-r*x^2)), and satisfies the recurrence a(n)=a(n-1)+2r*a(n-2)-r*a(n-3)-r^2*a(n-4).

easy,nonn,new

Sum C(n-k,k-1)3^(k-1), k=0..floor(n/2).

0, 0, 1, 1, 7, 10, 40, 70, 217, 427, 1159, 2440, 6160, 13480, 32689, 73129, 173383, 392770, 919480, 2097790, 4875913, 11169283, 25856071, 59363920, 137109280, 315201040, 727060321, 1672663441, 3855438727, 8873429050, 20444528200

0,5

In general a(n)=sum{k=0..floor(n/2), C(n-k,k-1)r^(k-1)} has g.f. x^2/((1-r*x^2)(1-x-r*x^2)), and satisfies the recurrence a(n)=a(n-1)+2r*a(n-2)-r*a(n-3)-r^2*a(n-4).

G.f.: x^2/((1-3x^2)(1-x-3x^2)); a(n)=a(n-1)+6a(n-2)-3a(n-3)-9a(n-4).

Cf. A097038, A099580.

easy,nonn

Paul Barry (pbarry(AT)wit.ie), Oct 23 2004

approved