A109794 a(2n) = A001906(n+1), a(2n+1) = A002878(n).

1, 1, 3, 4, 8, 11, 21, 29, 55, 76, 144, 199, 377, 521, 987, 1364, 2584, 3571, 6765, 9349, 17711, 24476, 46368, 64079, 121393, 167761, 317811, 439204, 832040, 1149851, 2178309, 3010349, 5702887, 7881196, 14930352, 20633239, 39088169

Offset: 0

Author

Creighton Dement, Aug 14 2005

Keywords

• nonn
• easy

Comments

Sequence relates bisections of Lucas and Fibonacci numbers (see also A098149).

Floretion Algebra Multiplication Program, FAMP code: 4jesleftforsumseq[ + .25'i + .25i' + .25'ii' + .25'jj' + .25'kk' + .25'jk' + .25'kj' + .25e], vesleftforsumseq = A000045, sumtype: (Y[15], *, inty*sum) (internal program code)

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-1)

Crossrefs

Cf. A001906, A002878, A098149, A000045, A189761.

Programs

GAP

a:=[1,1,3,4];; for n in [5..40] do a[n]:=3*a[n-2]-a[n-4]; od; a; # Muniru A Asiru, Aug 09 2018

Maple

a:= n-> (<<0|1>, <-1|3>>^iquo(n, 2, 'r'). <<1, 3+r>>)[1, 1]:
seq(a(n), n=0..50);  # Alois P. Heinz, May 02 2011

Mathematica

LinearRecurrence[{0, 3, 0, -1}, {1, 1, 3, 4}, 40] (* Robert G. Wilson v, Aug 06 2018 *)
CoefficientList[Series[(1+x+x^3)/((1+x-x^2)(1-x-x^2)),{x,0,40}],x] (* Harvey P. Dale, Aug 10 2021 *)

Formula

G.f.: (1+x+x^3)/((1+x-x^2)*(1-x-x^2)).

a(n) = ((3/20)*sqrt(5) + 3/4)*(1/2 + (1/2)*sqrt(5))^n + (-(3/20)*sqrt(5) + 3/4)*(1/2 - (1/2)*sqrt(5))^n + (-(3/20)*sqrt(5) - 1/4)*(-1/2 + (1/2)*sqrt(5))^n + ((3/20)*sqrt(5) - 1/4) *(-1/2 - (1/2)*sqrt(5))^n.

a(n) = 3*a(n-2) - a(n-4), n >= 4; a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4. - Daniel Forgues, May 07 2011