A062188 -id:A062188 - cp's OEIS frontend

A062186 a(n) = a(n-1) - a(floor(n/2)), with a(1)=1.

1, 0, -1, -1, -1, 0, 1, 2, 3, 4, 5, 5, 5, 4, 3, 1, -1, -4, -7, -11, -15, -20, -25, -30, -35, -40, -45, -49, -53, -56, -59, -60, -61, -60, -59, -55, -51, -44, -37, -26, -15, 0, 15, 35, 55, 80, 105, 135, 165, 200, 235, 275, 315, 360, 405, 454, 503, 556, 609, 665, 721, 780, 839, 899, 959, 1020, 1081, 1141, 1201, 1260

Offset: 1

Henry Bottomley, Jun 13 2001

sign

Period of oscillations above and below the axis more than doubles at each cycle.

			a(14) = a(13) - a(7) = 5 - 1 = 4.
a(15) = a(14) - a(7) = 4 - 1 = 3.

Cf. A033485, A062187, A062188.

from itertools import islice
from collections import deque
def A062186_gen(): # generator of terms
    aqueue, f, b, a = deque([0]), True, 1, 0
    yield from (1,0)
    while True:
        a -= b
        yield a
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A062186_list = list(islice(A062186_gen(),40)) # Chai Wah Wu, Jun 08 2022

G.f.: A(x) satisfies: A(x) = (x - (1 + x)*A(x^2))/(1 - x). - Ilya Gutkovskiy, May 04 2019

A062187 a(n+1) = a(n) - a(floor(n/2)), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 0, -1, -2, -3, -3, -3, -2, -1, 1, 3, 6, 9, 12, 15, 18, 21, 23, 25, 26, 27, 26, 25, 22, 19, 13, 7, -2, -11, -23, -35, -50, -65, -83, -101, -122, -143, -166, -189, -214, -239, -265, -291, -318, -345, -371, -397, -422, -447, -469, -491, -510, -529, -542, -555, -562, -569, -567, -565, -554, -543, -520, -497, -462

Offset: 0

Henry Bottomley, Jun 13 2001

sign

Period of oscillations above and below the axis more than doubles at each cycle.

			a(6) = a(5) - a(2) = -2 - 1 = -3.
a(7) = a(6) - a(3) = -3 - 0 = -3.

Cf. A033485, A062186, A062188.

from itertools import islice
from collections import deque
def A062187_gen(): # generator of terms
    aqueue, f, b, a = deque([1]), True, 0, 1
    yield from (0,1)
    while True:
        a -= b
        yield a
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A061287_list = list(islice(A062187_gen(),40)) # Chai Wah Wu, Jun 08 2022

G.f. A(x) satisfies: A(x) = x * (1 - (1 + x)*A(x^2))/(1 - x). - Ilya Gutkovskiy, May 04 2019

A179045 Triangle T(n,k), 1<=k<=n, read by rows, related to A033485.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 5, 2, 1, 1, 7, 3, 1, 1, 1, 10, 4, 2, 1, 1, 1, 13, 5, 3, 1, 1, 1, 1, 18, 7, 4, 2, 1, 1, 1, 1, 23, 9, 5, 3, 1, 1, 1, 1, 1, 30, 12, 6, 4, 2, 1, 1, 1, 1, 1, 37, 15, 7, 5, 3, 1, 1, 1, 1, 1, 1, 47, 19, 9, 6, 4, 2, 1, 1, 1, 1, 1, 1, 57, 23, 11, 7, 5, 3, 1, 1, 1, 1, 1, 1, 1, 70, 28, 14, 8, 6, 4

Offset: 1

Philippe Deléham, Jun 26 2010

Column k=1 : A033485 ; column k=2 : A062188 ; row sums : A102378.

			Triangle begins : 1 ; 2,1 ; 3,1,1 ; 5,2,1,1 ; 7,3,1,1,1 ; ...

Cf. A033485, A062188, A102378

T(n,k)=T(n-1,k)+T([n/2],k), T(n,n)=1, T(n,k)=0 if k>n.

cp's OEIS Frontend

A062186 a(n) = a(n-1) - a(floor(n/2)), with a(1)=1.

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A062187 a(n+1) = a(n) - a(floor(n/2)), with a(0)=0, a(1)=1.

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Examples

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Programs

Python

Formula

A179045 Triangle T(n,k), 1<=k<=n, read by rows, related to A033485.

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Author

Keywords

Comments

Examples

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Formula