A299913 a(n) = a(n-1) + 2*a(n-2) if n even, or 3*a(n-1) + 4*a(n-2) if n odd, starting with 0, 1.

0, 1, 1, 7, 9, 55, 73, 439, 585, 3511, 4681, 28087, 37449, 224695, 299593, 1797559, 2396745, 14380471, 19173961, 115043767, 153391689, 920350135, 1227133513, 7362801079, 9817068105, 58902408631, 78536544841, 471219269047, 628292358729, 3769754152375, 5026338869833

Offset: 0

Author

N. J. A. Sloane, Mar 10 2018

Keywords

• nonn
• easy

References

• Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (-1,8,8)

Crossrefs

Bisections give A023001, A083068.

Cf. A299914, A299915, A299916.

Programs

Maple

a:= n-> (<<0|1|0>, <0|0|1>, <8|8|-1>>^n. <<0, 1, 1>>)[1,1]:
seq(a(n), n=0..35);  # Alois P. Heinz, Mar 10 2018

Mathematica

Fold[Append[#1, Inner[Times, 2 Boole[OddQ@ #2] + {1, 2}, {#1[[-1]], #1[[-2]]}, Plus]] &, {0, 1}, Range[2, 30]] (* or *)
CoefficientList[Series[-x (2 x + 1)/((x + 1) (8 x^2 - 1)), {x, 0, 30}], x] (* Michael De Vlieger, Mar 10 2018 *)
nxt[{n_,a_,b_}]:={n+1,b,If[OddQ[n],b+2a,3b+4a]}; NestList[nxt,{1,0,1},30][[;;,2]] (* Harvey P. Dale, Mar 02 2025 *)

PARI

concat(0, Vec(x*(1 + 2*x) / ((1 + x)*(1 - 8*x^2)) + O(x^40))) \\ Colin Barker, Mar 11 2018

Formula

G.f.: -x*(2*x+1)/((x+1)*(8*x^2-1)). - Alois P. Heinz, Mar 10 2018

From Colin Barker, Mar 11 2018: (Start)

a(n) = (2^(3*n/2) - 1) / 7 for n even.

a(n) = 3*2^((3*(n-1))/2+1)/7 + 1/7 for n odd.

a(n) = -a(n-1) + 8*a(n-2) + 8*a(n-3) for n>2.

(End)

Extensions

More terms from Altug Alkan, Mar 10 2018