A099214 -id:A099214 - cp's OEIS frontend

A099213 a(n) = a(n-1)+a(n-2)+3a(n-3), with a(0)=a(1)=a(2)=1.

1, 1, 1, 5, 9, 17, 41, 85, 177, 385, 817, 1733, 3705, 7889, 16793, 35797, 76257, 162433, 346081, 737285, 1570665, 3346193, 7128713, 15186901, 32354193, 68927233, 146842129, 312831941, 666455769, 1419814097, 3024765689, 6443947093

Offset: 0

Paul Barry, Oct 06 2004

Binomial transform is A099214. Binomial transform of A099212.

Index entries for linear recurrences with constant coefficients, signature (1,1,3).

Cf. A099212, A099214.

Vec((1-x^2)/(1-x-x^2-3*x^3) + O(x^30)) \\ Michel Marcus, Aug 31 2022

from sage.combinat.sloane_functions import recur_gen3; it = recur_gen3(1,1,1,1,1,3); [next(it) for i in range(32)] # Zerinvary Lajos, Jun 25 2008

G.f.: (1-x^2)/(1-x-x^2-3*x^3). [corrected by Michel Marcus, Aug 31 2022]

A099215 a(n)=4a(n-1)-4a(n-2)+3a(n-3).

Original entry on oeis.org

1, 2, 4, 11, 34, 104, 313, 938, 2812, 8435, 25306, 75920, 227761, 683282, 2049844, 6149531, 18448594, 55345784, 166037353, 498112058, 1494336172, 4483008515, 13449025546, 40347076640, 121041229921, 363123689762, 1089371069284

Offset: 0

Paul Barry, Oct 06 2004

Index entries for linear recurrences with constant coefficients, signature (4,-4,3).

Cf. A099214.

LinearRecurrence[{4,-4,3},{1,2,4},30] (* Harvey P. Dale, Mar 02 2013 *)

G.f.: (1-2x)/((1-2x)^2-3x^3); a(n)=sum{k=0..floor(n/3), binomial(n-k, 2k)3^k*2^(n-3k)}.

a{x}=1/7(-Cos[(Pi*x)/3]+Cosh[x*Log[3]]+3*Sqrt[3]*Sin[(Pi*x)/3]+Sinh[x*Log[3]]) - Harvey P. Dale, Mar 02 2013

A099216 a(n)=4a(n-1)-4a(n-2)+2a(n-3).

Original entry on oeis.org

1, 2, 4, 10, 28, 80, 228, 648, 1840, 5224, 14832, 42112, 119568, 339488, 963904, 2736800, 7770560, 22062848, 62642752, 177860736, 504997632, 1433833088, 4071063296, 11558916096, 32819077376, 93182771712, 264572609536

Offset: 0

Paul Barry, Oct 06 2004

Binomial transform of tribonacci numbers A000213.

Index entries for linear recurrences with constant coefficients, signature (4,-4,2).

Cf. A099214, A099215, A055588.

G.f.: (1-2x)/((1-2x)^2-2x^3); a(n)=sum{k=0..floor(n/3), binomial(n-k, 2k)2^k*2^(n-3k)}.

cp's OEIS Frontend

A099213 a(n) = a(n-1)+a(n-2)+3a(n-3), with a(0)=a(1)=a(2)=1.

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A099215 a(n)=4a(n-1)-4a(n-2)+3a(n-3).

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A099216 a(n)=4a(n-1)-4a(n-2)+2a(n-3).

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