A254308 a(n) = a(n-1) + (if a(n-1) is odd a(n-2) else a(n-3)) with a(0) = 0, a(1) = 1.

0, 1, 1, 2, 3, 5, 8, 11, 19, 30, 41, 71, 112, 153, 265, 418, 571, 989, 1560, 2131, 3691, 5822, 7953, 13775, 21728, 29681, 51409, 81090, 110771, 191861, 302632, 413403, 716035, 1129438, 1542841, 2672279, 4215120, 5757961, 9973081, 15731042, 21489003, 37220045

Offset: 0

Author

Russell Walsmith, Feb 23 2015

Keywords

• nonn

Comments

Every third iteration is a tribonacci-type recursion: a(n) = a(n-1) + a(n-3) otherwise it is Fibonacci-type a(n) = a(n-1) + a(n-2).

Examples

			For n = 7, a(n-1) is even so 8 + 3 = 11.
G.f. = x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 11*x^7 + 19*x^8 + 30*x^9 + ...
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• Russell Walsmith, Fibonacci-Tribonacci Fusion
• Index entries for linear recurrences with constant coefficients, signature (0,0,4,0,0,-1).

Crossrefs

Cf. A001834, A001835, A052530.

Programs

Haskell

a254308 n = a254308_list !! n
a254308_list = 0 : 1 : 1 : zipWith3 (\u v w -> u + if odd u then v else w)
               (drop 2 a254308_list) (tail a254308_list) a254308_list
-- Reinhard Zumkeller, Feb 24 2015

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x+2*x^2-x^3+x^4)/(1-4*x^3+x^6))); // G. C. Greubel, Aug 03 2018

Mathematica

CoefficientList[Series[x*(1+x+2*x^2-x^3+x^4)/(1-4*x^3+x^6), {x, 0, 60}], x] (* G. C. Greubel, Aug 03 2018 *)
nxt[{a_,b_,c_}]:={b,c,If[OddQ[c],c+b,c+a]}; NestList[nxt,{0,1,1},50][[All,1]] (* or *) LinearRecurrence[{0,0,4,0,0,-1},{0,1,1,2,3,5},50] (* Harvey P. Dale, May 12 2022 *)

PARI

{a(n) = polcoeff( x * if( n<0, n=-n; -(1 - x + 2*x^2 + x^3 + x^4), (1 + x + 2*x^2 - x^3 + x^4)) / (1 - 4*x^3 + x^6) + x * O(x^n), n)}; /* Michael Somos, Feb 23 2015 */

Formula

Two identities: a(3n)/2 + a(3n-3)/2 = a(3n-1); a(3n)/2 - a(3n-3)/2 = a(3n-2).

G.f.: x * (1 + x + 2*x^2 - x^3 + x^4) / (1 - 4*x^3 + x^6). - Michael Somos, Feb 23 2015

0 = a(n) - 4*a(n+3) + a(n+6) for all n in Z. - Michael Somos, Feb 23 2015

a(3*n) = A052530(n). a(3*n-2) = A001835(n). a(3*n+2) = A001834(n). - Michael Somos, Feb 23 2015