login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A100308 Modulo 2 binomial transform of 5^n. 7
1, 6, 26, 156, 626, 3756, 16276, 97656, 390626, 2343756, 10156276, 60937656, 244531876, 1467191256, 6357828776, 38146972656, 152587890626, 915527343756, 3967285156276, 23803710937656, 95520019531876, 573120117191256 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

5^n be retrieved through 5^n=sum{k=0..n, (-1)^A010060(n-k)*mod(binomial(n,k),2)A100308(k)}.

LINKS

Robert Israel, Table of n, a(n) for n = 0..1429

V. Shevelev, On Stephan's conjectures concerning Pascal triangle modulo 2 and their polynomial generalization, J. of Algebra Number Theory: Advances and Appl., 7 (2012), no.1, 11-29.

FORMULA

a(n) = sum{k=0..n, mod(binomial(n, k), 2)5^k}.

From Vladimir Shevelev, Dec 26-27 2013: (Start)

sum{n>=0}1/a(n)^r = prod{k>=0}(1 + 1/(5^(2^k)+1)^r),

sum{n>=0}(-1)^A000120(n)/a(n)^r = prod{k>=0}(1 - 1/(5^(2^k)+1)^r), where r>0 is a real number.

In particular,

sum{n>=0}1/a(n) = prod{k>=0}(1 + 1/(5^(2^k)+1)) = 1.2134769...;

sum{n>=0}(-1)^A000120(n)/a(n) = 0.8.

a(2^n)=5^(2^n)+1, n>=0.

Note that analogs of Stephan's limit formulas (see Shevelev link) reduce to the relations:

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.

In particular, for t=2,3,4, we have the following formulas:

a(4*n+2) = 26 * a(4*n);

a(8*n+4) = 313/13 * a(8*n+2);

a(16*n+8)= 195313/8138 * a(16*n+6), etc.

(End)

MAPLE

f:= proc(n) local L, M;

   L:= convert(n, base, 2);

   mul(1+5^(2^(k-1)), k = select(t -> L[t]=1, [$1..nops(L)]));

end proc:

map(f, [$0..30]); # Robert Israel, Aug 26 2018

MATHEMATICA

a[n_] := Sum[Mod[Binomial[n, k], 2] 5^k, {k, 0, n}];

Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Sep 19 2018 *)

CROSSREFS

Cf. A001316, A001317, A038183, A100307, A100309, A100310, A100311.

Sequence in context: A140231 A144037 A187458 * A049040 A103649 A053946

Adjacent sequences:  A100305 A100306 A100307 * A100309 A100310 A100311

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Dec 06 2004

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 16 09:13 EDT 2020. Contains 335784 sequences. (Running on oeis4.)