|
%I #47 Jun 20 2022 13:16:06
%S 1,1,3,1,7,5,11,1,23,21,59,17,103,69,139,1,279,277,827,273,1895,1349,
%T 2955,257,5655,5141,14395,4113,24679,16453,32907,1,65815,65813,197435,
%U 65809,460647,329029,723851,65793,1512983,1381397,3881019,1118225
%N Stern-Jacobsthal numbers.
%C 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
%C 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
%H Ivan Panchenko, <a href="/A101624/b101624.txt">Table of n, a(n) for n = 0..1000</a>
%H K. Dilcher and L. Ericksen, <a href="http://dml.cz/handle/10338.dmlcz/143908">Reducibility and irreducibility of Stern (0, 1)-polynomials</a>, Communications in Mathematics, Volume 22/2014 , pp. 77-102.
%F a(n) = Sum_{k=0..floor(n/2)} (binomial(n-k, k) mod 2)*2^k.
%F a(2^n-1)=1, a(2*n) = 2*a(n-1) + a(n+1) = A099902(n); a(2*n+1) = A101625(n+1).
%F a(n) = Sum_{k=0..n} (binomial(k, n-k) mod 2)*2^(n-k). - _Paul Barry_, May 10 2005
%F a(n) = Sum_{k=0..n} A106344(n,k)*2^(n-k). - _Philippe Deléham_, Dec 18 2008
%F 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
%o (Python)
%o prpr = 1
%o prev = 1
%o print("1, 1", end=", ")
%o for i in range(99):
%o current = (prev)^(prpr*2)
%o print(current, end=", ")
%o prpr = prev
%o prev = current
%o # _Alex Ratushnyak_, Apr 14 2012
%o (Python)
%o 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
%o (Haskell)
%o a101624 = sum . zipWith (*) a000079_list . map (flip mod 2) . a011973_row
%o -- _Reinhard Zumkeller_, Jul 14 2015
%Y Cf. A002487, A011973, A000079, A006921.
%Y Cf. A125184, A260443, A177219.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Dec 10 2004
|
