cp's OEIS Frontend

A100308 Modulo 2 binomial transform of 5^n.

%I A100308 #51 Aug 30 2024 21:11:54
%S A100308 1,6,26,156,626,3756,16276,97656,390626,2343756,10156276,60937656,
%T A100308 244531876,1467191256,6357828776,38146972656,152587890626,
%U A100308 915527343756,3967285156276,23803710937656,95520019531876,573120117191256
%N A100308 Modulo 2 binomial transform of 5^n.
%C A100308 5^n may be retrieved through 5^n = Sum_{k=0..n} (-1)^A010060(n-k) * (binomial(n,k) mod 2)*a(k).
%H A100308 Robert Israel, <a href="/A100308/b100308.txt">Table of n, a(n) for n = 0..1429</a>
%H A100308 Vladimir Shevelev, <a href="http://arxiv.org/abs/1011.6083">On Stephan's conjectures concerning Pascal triangle modulo 2 and their polynomial generalization</a>, arXiv:1011.6083 [math.NT], 2010-2012; J. of Algebra Number Theory: Advances and Appl., 7 (2012), no.1, 11-29.
%F A100308 a(n) = Sum_{k=0..n} (binomial(n, k) mod 2)*5^k.
%F A100308 From _Vladimir Shevelev_, Dec 26-27 2013: (Start)
%F A100308 Sum_{n>=0} 1/a(n)^r = Product_{k>=0} (1 + 1/(5^(2^k)+1)^r),
%F A100308 Sum_{n>=0} (-1)^A000120(n)/a(n)^r = Product_{k>=0} (1 - 1/(5^(2^k)+1)^r), where r > 0 is a real number.
%F A100308 In particular,
%F A100308 Sum_{n>=0} 1/a(n) = Product_{k>=0} (1 + 1/(5^(2^k)+1)) = 1.2134769...;
%F A100308 Sum_{n>=0} (-1)^A000120(n)/a(n) = 0.8.
%F A100308 a(2^n) = 5^(2^n) + 1, n >= 0.
%F A100308 Note that analogs of Stephan's limit formulas (see Shevelev link) reduce to the relations:
%F A100308 a(2^t*n+2^(t-1)) = 24*(5^(2^(t-1)+1))/(5^(2^(t-1))-1) * a(2^t*n+2^(t-1)-2), t >= 2.
%F A100308 In particular, for t=2,3,4, we have the following formulas:
%F A100308   a(4*n+2)  =            26 * a(4*n),
%F A100308   a(8*n+4)  =      (313/13) * a(8*n+2),
%F A100308   a(16*n+8) = (195313/8138) * a(16*n+6), etc. (End)
%p A100308 f:= proc(n) local L,M;
%p A100308    L:= convert(n,base,2);
%p A100308    mul(1+5^(2^(k-1)), k = select(t -> L[t]=1, [$1..nops(L)]));
%p A100308 end proc:
%p A100308 map(f, [$0..30]); # _Robert Israel_, Aug 26 2018
%t A100308 a[n_]:= Sum[Mod[Binomial[n, k], 2] 5^k, {k, 0, n}];
%t A100308 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Sep 19 2018 *)
%o A100308 (Magma) [(&+[5^k*(Binomial(n, k) mod 2): k in [0..n]]): n in [0..40]]; // _G. C. Greubel_, Feb 03 2023
%o A100308 (SageMath)
%o A100308 def A100308(n): return sum(5^k*(binomial(n, k)%2) for k in range(n+1))
%o A100308 [A100308(n) for n in range(41)] # _G. C. Greubel_, Feb 03 2023
%o A100308 (Python)
%o A100308 def A100308(n): return sum((bool(~n&n-k)^1)*5**k for k in range(n+1)) # _Chai Wah Wu_, May 02 2023
%Y A100308 Cf. A001316, A001317, A038183, A100307, A100309, A100310, A100311.
%K A100308 easy,nonn
%O A100308 0,2
%A A100308 _Paul Barry_, Dec 06 2004