

A101624


SternJacobsthal numbers.


9



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
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OFFSET

0,3


COMMENTS

The Stern diatomic sequence A002487 could be called the SternFibonacci sequence, since it is given by A002487(n) = Sum_{k=0..floor(n/2)} (binomial(nk,k) mod 2), where F(n+1) = Sum_{k=0..floor(n/2)} binomial(nk,k). Now a(n) = Sum_{k=0..floor(n/2)} (binomial(nk,k) mod 2)*2^k, where J(n+1) = Sum_{k=0..floor(n/2)} binomial(nk,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 Sternpolynomials, and also A177219 for a reference to maybe a third kind.  Antti Karttunen, Nov 01 2016


LINKS

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. 77102.


FORMULA

a(n) = Sum_{k=0..floor(n/2)} (binomial(nk, k) mod 2)*2^k.
a(2^n1)=1, a(2*n) = 2*a(n1) + a(n+1) = A099902(n); a(2*n+1) = A101625(n+1).
a(n) = Sum_{k=0..n} (binomial(k, nk) mod 2)*2^(nk).  Paul Barry, May 10 2005
a(n) = Sum_{k=0..n} A106344(n,k)*2^(nk).  Philippe Deléham, Dec 18 2008
a(0)=1, a(1)=1, a(n) = a(n1) XOR (a(n2)*2), where XOR is the bitwise exclusiveOR operator.  Alex Ratushnyak, Apr 14 2012


PROG

(Python)
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
(Python)
def A101624(n): return sum(int(not k & ~(nk))*2**k for k in range(n//2+1)) # Chai Wah Wu, Jun 20 2022
(Haskell)
a101624 = sum . zipWith (*) a000079_list . map (flip mod 2) . a011973_row
 Reinhard Zumkeller, Jul 14 2015


CROSSREFS

Cf. A002487, A011973, A000079, A006921.
Cf. A125184, A260443, A177219.
Sequence in context: A138257 A283975 A071043 * A166519 A213043 A319740
Adjacent sequences: A101621 A101622 A101623 * A101625 A101626 A101627


KEYWORD

easy,nonn


AUTHOR

Paul Barry, Dec 10 2004


STATUS

approved



