A101399 a(0) = 1, a(1) = 2, a(2) = 5; for n >= 3, a(n) = a(n-1) + 2*a(n-2) + a(n-3).

1, 2, 5, 10, 22, 47, 101, 217, 466, 1001, 2150, 4618, 9919, 21305, 45761, 98290, 211117, 453458, 973982, 2092015, 4493437, 9651449, 20730338, 44526673, 95638798, 205422482, 441226751, 947710513, 2035586497, 4372234274

Offset: 0

Author

Jeroen F.J. Laros, Jan 15 2005

Keywords

• nonn
• easy

Comments

Lengths of successive words (starting with a) under the substitution: {a -> ab, b -> aac, c -> a}.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Dun Qiu, Classical pattern distributions in Sn(132) and Sn(123), arXiv:1810.10099 [math.CO], 2018. See p. 11.
• Index entries for linear recurrences with constant coefficients, signature (1, 2, 1).

Crossrefs

Pairwise sums of A078007. Bisection of A003410 and A058278.

Programs

GAP

a:=[1,2,5];; for n in [4..35] do a[n]:=a[n-1]+2*a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Apr 03 2018

Magma

I:=[1,2,5]; [n le 3 select I[n] else Self(n-1) + 2*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Apr 03 2018

Magma

m:=25; R:=PowerSeriesRing( Integers(), m); Coefficients(R!((1+x+x^2)/(1-x-2*x^2-x^3))); // G. C. Greubel, Apr 03 2018

Mathematica

a[0] = 1; a[1] = 2; a[2] = 5; a[n_] := a[n] = a[n - 1] + 2a[n - 2] + a[n - 3]; Table[ a[n], {n, 0, 30}] (* Robert G. Wilson v, Jan 15 2005 *)
LinearRecurrence[{1,2,1},{1,2,5},30] (* Harvey P. Dale, Aug 29 2012 *)
CoefficientList[Series[(1+x+x^2)/(1-x-2*x^2-x^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 03 2018 *)

PARI

x='x+O('x^30); Vec((1+x+x^2)/(1-x-2*x^2-x^3)) \\ G. C. Greubel, Apr 03 2018

Formula

G.f.: (1+x+x^2)/(1-x-2*x^2-x^3). - G. C. Greubel, Apr 03 2018

Extensions

More terms from Robert G. Wilson v and Lior Manor, Jan 15 2005