A039722 -id:A039722 - cp's OEIS frontend

A168276 a(n) = 2*n - (-1)^n - 1.

2, 2, 6, 6, 10, 10, 14, 14, 18, 18, 22, 22, 26, 26, 30, 30, 34, 34, 38, 38, 42, 42, 46, 46, 50, 50, 54, 54, 58, 58, 62, 62, 66, 66, 70, 70, 74, 74, 78, 78, 82, 82, 86, 86, 90, 90, 94, 94, 98, 98, 102, 102, 106, 106, 110, 110, 114, 114, 118, 118, 122, 122, 126, 126, 130, 130

Offset: 1

Vincenzo Librandi, Nov 22 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A039722, A168277.

Cf. A063210. - R. J. Mathar, Nov 25 2009

[2*n-1-(-1)^n: n in [1..70]]; // Vincenzo Librandi, Sep 16 2013

CoefficientList[Series[2 (1 + x^2) / ((1 + x) (1 - x)^2), {x, 0, 80}], x] (* Vincenzo Librandi, Sep 16 2013 *)
Table[2 n - 1 - (-1)^n, {n, 70}] (* Bruno Berselli, Sep 17 2013 *)
LinearRecurrence[{1,1,-1},{2,2,6},70] (* Harvey P. Dale, Oct 22 2014 *)

a(n) = 4*n - a(n-1) - 4, with n>1, a(1)=2.
from R. J. Mathar, Nov 25 2009: (Start)
a(n) = 2*n - (-1)^n - 1.
a(n) = 2*A109613(n-1).
G.f.: 2*x*(1 + x^2)/((1+x)*(1-x)^2). (End)
a(n) = a(n-1) + a(n-2) - a(n-3). - Vincenzo Librandi, Sep 16 2013
a(n) = A168277(n) + 1. - Vincenzo Librandi, Sep 17 2013
E.g.f.: (-1 + 2*exp(x) + (2*x -1)*exp(2*x))*exp(-x). - G. C. Greubel, Jul 16 2016
Sum_{n>=1} 1/a(n)^2 = Pi^2/16. - Amiram Eldar, Aug 21 2022

Previous definition replaced with closed-form expression by Bruno Berselli, Sep 17 2013

A039721 a(1) = 1, a(m+1) = 2*Sum_{k=1..floor((m+1)/2)} a(m+1-k).

Original entry on oeis.org

1, 2, 4, 12, 32, 96, 280, 840, 2496, 7488, 22400, 67200, 201408, 604224, 1812112, 5436336, 16307328, 48921984, 146760960, 440282880, 1320833664, 3962500992, 11887458176, 35662374528, 106986989184, 320960967552, 962882499840, 2888647499520, 8665941290112

Offset: 1

Leroy Quet, Dec 11 1999

			a(6)=2*(a(5)+a(4)+a(3)) = 2*(32+12+4) = 96.

Robert Israel, Table of n, a(n) for n = 1..212

Cf. A039722 (similar definition).

a[1]:= 1;
for m from 1 to 100 do
   a[m+1]:= 2*add(a[m+1-k],k=1..floor((m+1)/2));
od:
seq(a[i],i=1..100); # Robert Israel, May 18 2014

Fold[Append[#1, 2 Total[#1[[#2 - Range[Floor[#2/2] ] ]] ] ] &, {1}, Range[2, 29]] (* Michael De Vlieger, Dec 11 2017 *)

lista(nn) = {v = vector(nn); v[1] = 1; for (n=2, nn, v[n] = 2*sum(k=1, n\2, v[n-k]);); v;} \\ Michel Marcus, May 18 2014

a(1)=1, a(2)=2, a(2m+1)=3*a(2m)-2*a(m), a(2m+2)=3*a(2m+1) (m is positive integer).

More terms from James Sellers, May 04 2000

Two more terms from Michel Marcus, May 18 2014

A347027 a(1) = 1; a(n) = a(n-1) + 2 * a(floor(n/2)).

Original entry on oeis.org

1, 3, 5, 11, 17, 27, 37, 59, 81, 115, 149, 203, 257, 331, 405, 523, 641, 803, 965, 1195, 1425, 1723, 2021, 2427, 2833, 3347, 3861, 4523, 5185, 5995, 6805, 7851, 8897, 10179, 11461, 13067, 14673, 16603, 18533, 20923, 23313, 26163, 29013, 32459, 35905, 39947, 43989

Offset: 1

Ilya Gutkovskiy, Aug 11 2021

nonn

Cf. A033485, A033489, A058039.

Partial sums of A039722.

a[1] = 1; a[n_] := a[n] = a[n - 1] + 2 a[Floor[n/2]]; Table[a[n], {n, 1, 47}]
nmax = 47; A[] = 0; Do[A[x] = (x + 2 (1 + x) A[x^2])/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest

from collections import deque
from itertools import islice
def A347027_gen(): # generator of terms
    aqueue, f, b, a = deque([3]), True, 1, 3
    yield from (1, 3)
    while True:
        a += 2*b
        yield a
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A347027_list = list(islice(A347027_gen(),40)) # Chai Wah Wu, Jun 08 2022

G.f. A(x) satisfies: A(x) = (x + 2 * (1 + x) * A(x^2)) / (1 - x).

a(n) = 1 + 2 * Sum_{k=2..n} a(floor(k/2)).

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A168276 a(n) = 2*n - (-1)^n - 1.

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A039721 a(1) = 1, a(m+1) = 2*Sum_{k=1..floor((m+1)/2)} a(m+1-k).

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A347027 a(1) = 1; a(n) = a(n-1) + 2 * a(floor(n/2)).

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