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A358369 Euler transform of 2^floor(n/2), (A016116). 6
1, 1, 3, 5, 12, 20, 43, 73, 146, 250, 475, 813, 1499, 2555, 4592, 7800, 13761, 23253, 40421, 67963, 116723, 195291, 332026, 552882, 932023, 1544943, 2585243, 4267081, 7094593, 11662769, 19281018, 31575874, 51937608, 84753396, 138772038, 225693778, 368017636 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..36.

MAPLE

BinaryRecurrenceSequence := proc(b, c, u0:=0, u1:=1) local u;

u := proc(n) option remember; if n < 2 then return [u0, u1][n + 1] fi;

b*u(n - 1) + c*u(n - 2) end; u end:

EulerTransform := proc(a) local b;

b := proc(n) option remember; if n = 0 then return 1 fi; add(add(d * a(d),

d = NumberTheory:-Divisors(j)) * b(n-j), j = 1..n) / n end; b end:

a := EulerTransform(BinaryRecurrenceSequence(0, 2, 1)): seq(a(n), n=0..36);

PROG

(Sage) # uses[EulerTransform from A166861]

b = BinaryRecurrenceSequence(0, 2, 1)

a = EulerTransform(b)

print([a(n) for n in range(37)])

(Python)

from typing import Callable

from functools import cache

from sympy import divisors

def BinaryRecurrenceSequence(b:int, c:int, u0:int=0, u1:int=1) -> Callable:

@cache

def u(n: int) -> int:

if n < 2:

return [u0, u1][n]

return b * u(n - 1) + c * u(n - 2)

return u

def EulerTransform(a: Callable) -> Callable:

@cache

def b(n: int) -> int:

if n == 0:

return 1

s = sum(sum(d * a(d) for d in divisors(j)) * b(n - j)

for j in range(1, n + 1))

return s // n

return b

b = BinaryRecurrenceSequence(0, 2, 1)

a = EulerTransform(b)

print([a(n) for n in range(37)])

CROSSREFS

Cf. A002513, A016116.

Sequences that can be represented as a EulerTransform(BinaryRecurrenceSequence()) include A000009, A000041, A000712, A001970, A002513, A010054, A015128, A022567, A034691, A111317, A111335, A117410, A156224, A166861, A200544, A261031, A261329, A358449.

Sequence in context: A245939 A089292 A309702 * A143360 A234005 A263346

Adjacent sequences: A358366 A358367 A358368 * A358370 A358371 A358372

KEYWORD

nonn

AUTHOR

Peter Luschny, Nov 17 2022

STATUS

approved

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Last modified January 28 17:08 EST 2023. Contains 359895 sequences. (Running on oeis4.)