A000034 -id:A000034 - cp's OEIS frontend

A364447 Repeat [1,2,1,3].

1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1

Offset: 1

Rok Cestnik, Jul 25 2023

Continued fraction of sqrt(5) - 3/2 = 0.7360679... (without integer part); and (4*sqrt(5) + 6)/11 = 1.3585701... (with integer part).

Lexicographically earliest sequence in which n is banned for n terms after each appearance (see A364448 for n^2 and A364449 for n^3).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A131743 (repeat [0,1,0,2]).
Cf. A000034, A010882, A131534.
Cf. A364448, A364449.

PadRight[{}, 100, {1, 2, 1, 3}] (* Paolo Xausa, Jan 23 2025 *)

def A364447(n): return (3,1,2,1)[n&3] # Chai Wah Wu, Jul 29 2023

a(n) = A131743(n-1) + 1.

A385042 The number of unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 1, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 2, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Jun 16 2025

First differs from A367515 at n = 128.

The sum of these divisors is A385043(n), and the largest of them is A367168(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

The unitary analog of A353898.
Cf. A005117, A034444, A138302, A209229, A367168, A367515, A385043.
The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), this sequence (exponentially 2^n), A385044 (5-rough).

f[p_, e_] := Boole[e == 2^IntegerExponent[e, 2]] + 1; a[ 1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI
```
a(n) = vecprod(apply(x -> (x == 1<
				
```

Multiplicative with a(p^e) = A209229(e) + 1.
a(n) <= A034444(n), with equality if and only if n is in A138302.
a(n) <= A353898(n), with equality if and only if n is squarefree (A005117).

A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 4, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 4, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 4, 2, 2, 2, 1, 2, 2, 2, 4, 2, 2

Offset: 1

Amiram Eldar, Jun 16 2025

The sum of these divisors is A385045(n), and the largest of them is A065330(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

The unitary analog of A035218.
Cf. A007310, A034444, A065330, A385045.
Cf. A001620, A013661, A306016.
The number of unitary divisors of n that are: A000034 (power of 2), A055076 (exponentially odd), A056624 (square), A056671 (squarefree), A068068 (odd), A323308 (powerful), A365498 (cubefree), A365499 (biquadratefree), A368248 (cubefull), A380395 (cube), A382488 (3-smooth), A385042 (exponentially 2^n), this sequence (5-rough).

f[p_, e_] := If[p <= 3, 1, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> if(x <= 3, 1, 2), factor(n)[, 1]));

Multiplicative with a(p^e) = 1 if p <= 3, and 2 if p >= 5.
a(n) = A034444(n)/A382488(n).
a(n) <= A034444(n), with equality if and only if n is 5-rough.
a(n) <= A035218(n).
Dirichlet g.f.: (zeta(s)^2/zeta(2*s)) * (1/((1+1/2^s)*(1+1/3^s))).
Sum_{k=1..n} a(k) ~ (n / (2 * zeta(2))) *(log(n) + 2*gamma - 1 + log(2)/3 + log(3)/4 - 2*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).

A007687 Number of 4-colorings of cyclic group of order n.

Original entry on oeis.org

3, 10, 21, 44, 83, 218, 271, 692, 865, 2622, 2813, 9220, 9735, 35214, 35911, 135564, 136899, 533290, 535081

Offset: 1

N. J. A. Sloane

The number of 2-colorings of Z_n is A000034(n-1), the number of 3-colorings of Z_n is A005843(n). The number of n-colorings of Z_2 is A137928(n-1). - Andrey Zabolotskiy, Oct 02 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Haas, Three-colorings of finite groups or an algebra of nonequalities, Math. Mag., 63 (1990), 211-225.
Robert Haas, Letter to N. J. A. Sloane, Aug. 1994

Cf. A007688.

from itertools import product
def colorings(n, zp):
    result = 0
    for f in product(range(n), repeat=zp):
        for j1 in range(zp):
            for j2 in range(zp):
                if (f[j1]+f[j2])%n == f[(j1+j2)%zp]:
                    break
            else:
                continue
            break
        else:
            result += 1
    return result
print([colorings(4, k) for k in range(1, 12)])
# Andrey Zabolotskiy, Jul 12 2017

a(6)-a(11) from Andrey Zabolotskiy, Jul 12 2017
a(12)-a(17) from Andrey Zabolotskiy, Oct 02 2017
a(18)-a(19) from Lucas A. Brown, Sep 20 2024

A010713 Period 2: repeat (4,8).

Original entry on oeis.org

4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4

Offset: 0

N. J. A. Sloane

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A176218.

Table[4 + 4 Mod[n, 2], {n, 0, 50}] (* Jinyuan Wang, Feb 26 2020 *)
LinearRecurrence[{0,1},{4,8},100] (* or *) PadRight[{},100,{4,8}] (* Harvey P. Dale, Aug 29 2023 *)

a(n)=4+n%2*4 \\ Charles R Greathouse IV, Dec 21 2011

a(n) = -2*(-1)^n + 6. - Paolo P. Lava, Oct 20 2006
From R. J. Mathar, Oct 20 2008: (Start)
a(n) = 4*A000034(n).
G.f.: 4(1+2x)/((1-x)(1+x)). (End)

A062113 a(0)=1; a(1)=2; a(n) = a(n-1) + a(n-2)*(3 - (-1)^n)/2.

Original entry on oeis.org

1, 2, 3, 7, 10, 24, 34, 82, 116, 280, 396, 956, 1352, 3264, 4616, 11144, 15760, 38048, 53808, 129904, 183712, 443520, 627232, 1514272, 2141504, 5170048, 7311552, 17651648, 24963200, 60266496, 85229696, 205762688, 290992384, 702517760

Offset: 0

Olivier Gérard, Jun 05 2001

A bistable recurrence.

Harry J. Smith, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-2).

Cf. A007068, A062112.

a062113 n = a062113_list !! n
a062113_list = 1 : 2 : zipWith (+)
   (tail a062113_list) (zipWith (*) a000034_list a062113_list)
-- Reinhard Zumkeller, Jan 21 2012

I:=[1,2,3,7]; [n le 4 select I[n] else 4*Self(n-2) - 2*Self(n-4): n in [1..40]]; // G. C. Greubel, Oct 16 2018

LinearRecurrence[{0,4,0,-2}, {1,2,3,7}, 40] (* G. C. Greubel, Oct 16 2018 *)

{ for (n=0, 200, if (n>1, a=a1 + a2*(3 - (-1)^n)/2; a2=a1; a1=a, if (n==0, a=a2=1, a=a1=2)); write("b062113.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 01 2009

x='x+O('x^40); Vec((1+2*x-x^2-x^3)/(1-4*x^2+2*x^4)) \\ G. C. Greubel, Oct 16 2018

a(n) = a(n-1) + a(n-2) * A000034(n). - Reinhard Zumkeller, Jan 21 2012
From Colin Barker, Apr 20 2012: (Start)
a(n) = 4*a(n-2) - 2*a(n-4).
G.f.: (1+2*x-x^2-x^3)/(1-4*x^2+2*x^4). (End)

A089499 a(0)=0; a(1)=1; a(2n) = 4*Sum_{k=0..n} a(2k-1); a(2n+1) = a(2n) + a(2n-1).

Original entry on oeis.org

0, 1, 4, 5, 24, 29, 140, 169, 816, 985, 4756, 5741, 27720, 33461, 161564, 195025, 941664, 1136689, 5488420, 6625109, 31988856, 38613965, 186444716, 225058681, 1086679440, 1311738121, 6333631924, 7645370045, 36915112104, 44560482149

Offset: 0

Charlie Marion, Nov 11 2003

1, 4, 5, 24, 29, 140, ...= numerators in convergents to (sqrt(8) - 2) = continued fraction [0; 1, 4, 1, 4, 1, 4, ...]; where sqrt(8) - 2 = 0.828427124... = the inradius of a right triangle with hypotenuse 6, legs sqrt(32) and 2. Denominators of convergents to [0; 1, 4, 1, 4, 1, 4, ...] = A041011 starting (1, 5, 6, 29, 35, ...). - Gary W. Adamson, Dec 22 2007

This is a strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n,m)) for all natural numbers n and m. - Peter Bala, May 12 2014

J. L. Ramirez, F. Sirvent, A q-Analogue of the Bi-Periodic Fibonacci Sequence, J. Int. Seq. 19 (2016) # 16.4.6, t_n at a=4, b=1.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A041011.

Numerator[NestList[(4/(4+#))&,0,60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,6,0]^n*[0;1;4;5])[1,1] \\ Charles R Greathouse IV, Nov 13 2015

For n > 0, a(n) = A001333(n) + A084068(n-1)*(-1)^n.
a(n)*a(n+1) = A046729(n).
a(2n+1) = A001653(n); a(2n) = A005319(n).
a(1) = 1, a(2n) = 4*a(2n-1) + a(2n-2); a(2n-1) = a(2n-2) + a(2n-3). Given the 2 X 2 matrix X = [1, 4; 1, 5], [a(2n-1), a(2n)] = top row of X^n. The sequence starting (1, 4, 5, 24, 29, ...) = numerators in continued fraction [0; 1, 4, 1, 4, 1, 4, ...] = (sqrt(8) - 2) = 0.828427124... E.g., X^3 = [29, 140; 35, 169], where 29/35, 140/169 are convergents to (sqrt(8)-2). - Gary W. Adamson, Dec 22 2007
From R. J. Mathar, Jul 08 2009: (Start)
a(n) = A000129(n)*A000034(n+1).
a(n) = 6*a(n-2) - a(n-4).
G.f.: -x*(-1-4*x+x^2)/((x^2-2*x-1)*(x^2+2*x-1)). (End)
From Peter Bala, May 12 2014: (Start)
a(2*n + 1) = A041011(2*n + 1); a(2*n) = 4*A041011(2*n).
For n odd, a(n) = (alpha^n - beta^n)/(alpha - beta), and for n even, a(n) = 4*(alpha^n - beta^n)/(alpha^2 - beta^2), where alpha = 1 + sqrt(2) and beta = 1 - sqrt(2).
a(n) = Product_{j = 1..floor(n/2)} ( 4 + 4*cos^2(j*Pi/n) ) for n >= 1. (End)

Corrected by T. D. Noe, Nov 08 2006

Definition corrected by Jonathan Sondow, Jun 06 2014

A106524 Interleave A038573(n+1) and 2*A038573(n+1).

Original entry on oeis.org

1, 2, 1, 2, 3, 6, 1, 2, 3, 6, 3, 6, 7, 14, 1, 2, 3, 6, 3, 6, 7, 14, 3, 6, 7, 14, 7, 14, 15, 30, 1, 2, 3, 6, 3, 6, 7, 14, 3, 6, 7, 14, 7, 14, 15, 30, 3, 6, 7, 14, 7, 14, 15, 30, 7, 14, 15, 30, 15, 30, 31, 62, 1, 2, 3, 6, 3, 6, 7, 14, 3, 6, 7, 14, 7, 14, 15, 30, 3, 6, 7, 14, 7, 14, 15, 30, 7, 14, 15

Offset: 0

Paul Barry, May 06 2005

Row sums of number the number triangle (A106522 mod 2).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000034, A001316, A038573, A106522.

A106524:= func< n | 2^Multiplicity(Intseq(n+2, 2), 1) - 2^(n mod 2) >;
[A106524(n): n in [0..100]]; // G. C. Greubel, Aug 12 2021

a[n_]:= (2^DigitCount[Floor[(n+2)/2], 2, 1] - 1)*(3 - (-1)^n)/2;
Table[a[n], {n, 0, 100}] (* G. C. Greubel, Aug 11 2021 *)

a(n) = bitneg(n%2, hammingweight(n+2)); \\ Kevin Ryde, Aug 25 2021

def A000120(n): return sum(n.digits(2))
def A106524(n): return 2^A000120(n+2) - 2^(n%2)
[A106524(n) for n in (0..100)] # G. C. Greubel, Aug 11 2021

a(n) = (Sum_{k=0..n+2} binomial(n+2, k)) mod 2 - (3 - (-1)^n)/2.
a(n) = ( (Sum_{k=0..(n/2+1)} binomial(n/2+1, k)) mod 2 - 1 )*(1 + (-1)^n)/2 + ( (Sum_{k=0..(n+1)/2} binomial((n+1)/2, k)) mod 2 - 1)*(1 - (-1)^n)/2.
a(n) = A001316(n+2) - A000034(n).

A124038 Triangle read by rows: T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.

Original entry on oeis.org

1, -2, 1, -1, -2, 1, 2, -2, -2, 1, 1, 4, -3, -2, 1, -2, 3, 6, -4, -2, 1, -1, -6, 6, 8, -5, -2, 1, 2, -4, -12, 10, 10, -6, -2, 1, 1, 8, -10, -20, 15, 12, -7, -2, 1, -2, 5, 20, -20, -30, 21, 14, -8, -2, 1, -1, -10, 15, 40, -35, -42, 28, 16, -9, -2, 1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 03 2006

			Triangular sequence begins as:
   1;
  -2,   1;
  -1,  -2,   1;
   2,  -2,  -2,   1;
   1,   4,  -3,  -2,   1;
  -2,   3,   6,  -4,  -2,   1;
  -1,  -6,   6,   8,  -5,  -2,  1;
   2,  -4, -12,  10,  10,  -6, -2,  1;
   1,   8, -10, -20,  15,  12, -7, -2,  1;
  -2,   5,  20, -20, -30,  21, 14, -8, -2,  1;
  -1, -10,  15,  40, -35, -42, 28, 16, -9, -2, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A005809, A045721, A066170.
Row reversal of: A374439.
Columns are related to: A000034 (k=0), A029578 (k=1), A131259 (k=2).
Diagonals are related to: A113679 (k=n-1), A022958 (k=n-2), A005843 (k=n-3), A000217 (k=n-4), -A002378 (k=n-5).
Sums include: (-1)^floor((n+1)/2)*A016116 (signed diagonal), A057079 (row), A119910 (signed row), (-1)^n*A130706 (diagonal).

function T(n,k) // T = A124038
  if k lt 0 or k gt n then return 0;
  elif k ge n-2 then return k-n + (-1)^(n+k);
  else return T(n-1,k-1) - T(n-2,k);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 22 2025

(* First program *)
t[n_, m_, d_]:= If[n==m && n>1 && m>1, x, If[n==m-1 || n==m+1, -1, If[n==m== 1, x-2, 0]]];
M[d_]:= Table[t[n,m,d], {n,d}, {m,d}];
Join[{{1}}, Table[CoefficientList[Table[Det[M[d]], {d,10}][[d]], x], {d,10}]]//Flatten
(* Second program *)
T[n_, k_]:= T[n, k] = If[k<0 || k>n, 0, If[k>n-2, k-n+(-1)^(n-k), T[n-1, k- 1] -T[n-2,k]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 22 2025 *)

@CachedFunction
def A124038(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    h = 2*A124038(n-1,k) if n==1 else 0
    return A124038(n-1,k-1) - A124038(n-2,k) - h
for n in (0..9): [A124038(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

from sage.combinat.q_analogues import q_stirling_number2
def A124038(n,k): return (1 + ((n-k)%2))*q_stirling_number2(n+1, n-k+1, -1)
print(flatten([[A124038(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 22 2025

From G. C. Greubel, Jan 22 2025: (Start)
T(n, k) = T(n-1, k-1) - T(n-2, k), with T(n, n) = 1, T(n, n-1) = -2.
T(n, k) = (-1)^floor((n-k+1)/2)*(1 + (n-k mod 2))*qStirling2(n+1, n-k+1,-1).
T(2*n, n) = (1/2)*(-1)^floor(n/2)*( (1+(-1)^n)*A005809(n/2) - 2*(1-(-1)^n)* A045721((n-1)/2) ). (End)

Edited by G. C. Greubel, Jan 22 2025

A128316 Triangle read by rows: A000012 * A128315 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 1, 3, -1, 1, 2, 3, -2, 1, 4, -1, 4, -3, 1, 4, 3, -5, 7, -4, 1, 6, -3, 10, -13, 11, -5, 1, 4, 8, -14, 23, -24, 16, -6, 1, 7, -2, 15, -33, 46, -40, 22, -7, 1, 7, 4, -15, 47, -79, 86, -62, 29, -8, 1, 9, -6, 30, -73, 131, -166, 148, -91, 37, -9, 1, 7, 12, -37, 103, -204, 297, -314, 239, -128, 46, -10, 1

Offset: 1

Gary W. Adamson, Feb 25 2007

A128316 * [1,2,3...] = A000034: [1,2,1,2,...].

			First few rows of the triangle:
  1;
  1,  1;
  3, -1,   1;
  2,  3   -2,   1;
  4, -1,   4,  -3,   1;
  4,  3,  -5,   7,  -4,  1;
  6, -3,  10, -13,  11, -5,  1;
  4,  8, -14,  23, -24, 16, -6, 1;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000012, A000034, A014300, A026641, A059851, A128315.

Sums include: A000027 (row), A032766, A047215, A344817 (alternating sign).

A128316:= func< n,k | (&+[(-1)^(j+k)*Floor(n/j)*Binomial(j-1,k-1): j in [k..n]]) >;
[A128316(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 23 2024

T[n_, k_]:= Sum[(-1)^(j+k)*Floor[n/j]*Binomial[j-1,k-1], {j,k,n}];
Table[T[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 23 2024 *)

def A128316(n,k): return sum((-1)^(j+k)*int(n//j)*binomial(j-1,k-1) for j in range(k,n+1))
flatten([[A128316(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 23 2024

Sum_{k=1..n} T(n, k) = A000027(n) (row sums).
T(n, 1) = A059851(n).
From G. C. Greubel, Jun 23 2024: (Start)
T(n, k) = A010766(n,k) * AA130595(n-1, k-1) as infinite lower triangular matrices.
T(n, k) = Sum_{j=k..n} (-1)^(j+k) * floor(n/j) * binomial(j-1, k-1).
T(2*n-1, n) = (-1)^(n-1)*A026641(n).
T(2*n-2, n-1) = (-1)^n*A014300(n-1), for n >= 2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A344817(n).
Sum_{k=1..n} k*T(n, k) = A032766(n-1).
Sum_{k=1..n} (k+1)*T(n, k) = A047215(n). (End)

a(28) = 1 inserted and more terms from Georg Fischer, Jun 06 2023

cp's OEIS Frontend

A364447 Repeat [1,2,1,3].

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A385042 The number of unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

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A385044 The number of unitary divisors of n that are 5-rough numbers (A007310).

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A007687 Number of 4-colorings of cyclic group of order n.

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A010713 Period 2: repeat (4,8).

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A062113 a(0)=1; a(1)=2; a(n) = a(n-1) + a(n-2)*(3 - (-1)^n)/2.

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A089499 a(0)=0; a(1)=1; a(2n) = 4*Sum_{k=0..n} a(2k-1); a(2n+1) = a(2n) + a(2n-1).

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