A339572 If n even, a(n) = A000071(n/2+1); if n odd, a(n) = A001610((n-1)/2).

0, 0, 1, 2, 2, 3, 4, 6, 7, 10, 12, 17, 20, 28, 33, 46, 54, 75, 88, 122, 143, 198, 232, 321, 376, 520, 609, 842, 986, 1363, 1596, 2206, 2583, 3570, 4180, 5777, 6764, 9348, 10945, 15126, 17710, 24475, 28656, 39602, 46367, 64078, 75024, 103681, 121392, 167760, 196417, 271442

Offset

0,4

Comments

Sequences A000071 and A001610 look like long-lost cousins, and this entry smoothly interleaves them. Differences between successive terms are Fibonacci numbers.

Links

Stefano Spezia, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = Fibonacci(k+2) + Fibonacci(k)*(n mod 2) - 1, where k = floor(n/2). - Wesley Ivan Hurt, Dec 09 2020

G.f.: x^2*(1 + x - x^2)/((1 - x)*(1 - x^2 - x^4)). - Andrew Howroyd, Nov 18 2025

Mathematica

Block[{b = {0, 2}, a = {}}, Do[If[EvenQ[i], AppendTo[b, Total@ b[[-2 ;; -1]] + 1 ]; AppendTo[a, Fibonacci[i/2 + 1] - 1], AppendTo[a, b[[(i - 1)/2]]]], {i, 2, 53}]; a] (* Michael De Vlieger, Dec 09 2020 *)
Table[With[{k=Floor[n/2]}, Fibonacci[k+2]+Fibonacci[k]Mod[n, 2]-1], {n, 0, 60}] (* Harvey P. Dale, Jun 24 2025 *)

Prog

(PARI) a(n)=fibonacci(n\2+2) + if(n%2, fibonacci((n-1)/2)) - 1; \\ Andrew Howroyd, Nov 18 2025

Crossrefs

Cf. A000045, A000071, A001610.

The first differences are essentially A053602.

Sequence in context: A365827 A116665 A122135 * A027194 A039883 A024186

Adjacent sequences: A339569 A339570 A339571 * A339573 A339574 A339575

Keyword

nonn,easy

Author

N. J. A. Sloane, Dec 09 2020

Status

approved