A006921 -id:A006921 - cp's OEIS frontend

A257971 First differences of A006921.

0, 2, -1, 5, -2, 8, -5, 21, -8, 34, -21, 81, -34, 128, -81, 337, -128, 546, -337, 1301, -546, 2056, -1301, 5381, -2056, 8706, -5381, 20737, -8706, 32768, -20737, 86273, -32768, 139778, -86273, 333061, -139778, 526344, -333061, 1377557, -526344, 2228770

Offset: 0

Reinhard Zumkeller, Jul 14 2015

sign

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A006911, A168081.

a257971 n = a257971_list !! n
a257971_list = zipWith (-) (tail a006921_list) a006921_list

def A257971(n): return sum(int(not r & ~(n+2-r))*2**(n//2+1-r) for r in range(n//2+2)) if n & 1 else -sum(int(not r & ~(n-1-r))*2**(n//2-1-r) for r in range(n//2)) # Chai Wah Wu, Jun 20 2022

a(2*n) = - A168081(n), a(2*n+1) = A168081(n+2);

a(2*n+4) = - a(2*n+1).

A260022 A bisection of A006921.

Original entry on oeis.org

1, 3, 7, 13, 29, 55, 115, 209, 465, 883, 1847, 3357, 7437, 14087, 29443, 53505, 119041, 226051, 472839, 859405, 1903901, 3606327, 7537523, 13697489, 30474449, 57868403, 121045047, 220004381, 487391245, 923205639, 1929576451, 3506503681, 7801470977, 14814478339, 30987976711, 56321966093

Offset: 0

N. J. A. Sloane, Jul 14 2015

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A006921, A168081.

a260022 = a006921 . (* 2)  -- Reinhard Zumkeller, Jul 14 2015

def A260022(n): return sum(int(not r & ~(2*n-r))*2**(n-r) for r in range(n+1)) # Chai Wah Wu, Jun 20 2022

a(n) = A006921(2*n).

A101624 Stern-Jacobsthal numbers.

Original entry on oeis.org

1, 1, 3, 1, 7, 5, 11, 1, 23, 21, 59, 17, 103, 69, 139, 1, 279, 277, 827, 273, 1895, 1349, 2955, 257, 5655, 5141, 14395, 4113, 24679, 16453, 32907, 1, 65815, 65813, 197435, 65809, 460647, 329029, 723851, 65793, 1512983, 1381397, 3881019, 1118225

Offset: 0

Paul Barry, Dec 10 2004

The Stern diatomic sequence A002487 could be called the Stern-Fibonacci sequence, since it is given by A002487(n) = Sum_{k=0..floor(n/2)} (binomial(n-k,k) mod 2), where F(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k,k). Now a(n) = Sum_{k=0..floor(n/2)} (binomial(n-k,k) mod 2)*2^k, where J(n+1) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*2^k, with J(n) = A001045(n), the Jacobsthal numbers. - Paul Barry, Sep 16 2015

These numbers seem to encode Stern (0, 1)-polynomials in their binary expansion. See Dilcher & Ericksen paper, especially Table 1 on page 79, page 5 in PDF. See A125184 (A260443) for another kind of Stern-polynomials, and also A177219 for a reference to maybe a third kind. - Antti Karttunen, Nov 01 2016

Ivan Panchenko, Table of n, a(n) for n = 0..1000
K. Dilcher and L. Ericksen, Reducibility and irreducibility of Stern (0, 1)-polynomials, Communications in Mathematics, Volume 22/2014 , pp. 77-102.

Cf. A002487, A011973, A000079, A006921.

Cf. A125184, A260443, A177219.

a101624 = sum . zipWith (*) a000079_list . map (flip mod 2) . a011973_row
-- Reinhard Zumkeller, Jul 14 2015

prpr = 1
prev = 1
print("1, 1", end=", ")
for i in range(99):
    current = (prev)^(prpr*2)
    print(current, end=", ")
    prpr = prev
    prev = current
# Alex Ratushnyak, Apr 14 2012

def A101624(n): return sum(int(not k & ~(n-k))*2**k for k in range(n//2+1)) # Chai Wah Wu, Jun 20 2022

a(n) = Sum_{k=0..floor(n/2)} (binomial(n-k, k) mod 2)*2^k.
a(2^n-1)=1, a(2*n) = 2*a(n-1) + a(n+1) = A099902(n); a(2*n+1) = A101625(n+1).
a(n) = Sum_{k=0..n} (binomial(k, n-k) mod 2)*2^(n-k). - Paul Barry, May 10 2005
a(n) = Sum_{k=0..n} A106344(n,k)*2^(n-k). - Philippe Deléham, Dec 18 2008
a(0)=1, a(1)=1, a(n) = a(n-1) XOR (a(n-2)*2), where XOR is the bitwise exclusive-OR operator. - Alex Ratushnyak, Apr 14 2012
A000120(a(n-1)) = A002487(n). - Karl-Heinz Hofmann, Jun 18 2025

A168081 Lucas sequence U_n(x,1) over the field GF(2)[x].

Original entry on oeis.org

0, 1, 2, 5, 8, 21, 34, 81, 128, 337, 546, 1301, 2056, 5381, 8706, 20737, 32768, 86273, 139778, 333061, 526344, 1377557, 2228770, 5308753, 8388736, 22085713, 35782690, 85262357, 134742024, 352649221, 570556418, 1359020033, 2147483648, 5653987329, 9160491010

Offset: 0

Max Alekseyev, Nov 18 2009

The Lucas sequence U_n(x,1) over the field GF(2)={0,1} is: 0, 1, x, x^2+1, x^3, x^4+x^2+1, x^5+x, ... Numerical values are obtained evaluating these 01-polynomials at x=2 over the integers.
The counterpart sequence is V_n(x,1) = x*U_n(x,1) that implies identities like U_{2n}(x,1) = x*U_n(x,1)^2. - Max Alekseyev, Nov 19 2009
Also, Chebyshev polynomials of the second kind evaluated at x=1/2 (A049310) with the resulting coefficients taken modulo 2, and then evaluated at x=2. - Max Alekseyev, Jun 20 2025

G. C. Greubel, Table of n, a(n) for n = 0..1000
Karl-Heinz Hofmann and Frederik P.J. Vandecasteele, Terms in binary, including a visualization.

A bisection of A006921. Cf. A260022. - N. J. A. Sloane, Jul 14 2015
See also A257971, first differences of A006921. - Reinhard Zumkeller, Jul 14 2015
Cf. A000120, A000129, A206296, A248663, A002487.

a:= proc(n) option remember;
      `if`(n<2, n, Bits[Xor](2*a(n-1), a(n-2)))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jun 16 2025

a[0] = 0; a[1] = 1; a[n_] := a[n] = BitXor[2 a[n - 1], a[n - 2]]; Table[a@ n, {n, 0, 32}] (* Michael De Vlieger, Dec 11 2015 *)

{ a=0; b=1; for(n=1,50, c=bitxor(2*b,a); a=b; b=c; print1(c,", "); ); }

{ a168081(n) = subst(lift(polchebyshev(n-1,2,x/2)*Mod(1,2)),x,2); } \\ Max Alekseyev, Jun 20 2025

def A168081(n): return sum(int(not r & ~(2*n-1-r))*2**(n-1-r) for r in range(n)) # Chai Wah Wu, Jun 20 2022

For n>1, a(n) = (2*a(n-1)) XOR a(n-2).
a(n) = A248663(A206296(n)). - Antti Karttunen, Dec 11 2015
A000120(a(n)) = A002487(n). - Karl-Heinz Hofmann, Jun 16 2025
a(n) = Sum_{k=0..n} (A049310(n,k) mod 2) * 2^k. - Max Alekseyev, Jun 20 2025

cp's OEIS Frontend

A257971 First differences of A006921.

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A260022 A bisection of A006921.

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A101624 Stern-Jacobsthal numbers.

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A168081 Lucas sequence U_n(x,1) over the field GF(2)[x].

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