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A100307 Modulo 2 binomial transform of 3^n. 7
1, 4, 10, 40, 82, 328, 820, 3280, 6562, 26248, 65620, 262480, 538084, 2152336, 5380840, 21523360, 43046722, 172186888, 430467220, 1721868880, 3529831204, 14119324816, 35298312040, 141193248160, 282472589764, 1129890359056 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

3^n may be retrieved through 3^n = Sum_{k=0..n}(-1)^A010060(n-k)*mod(binomial(n,k),2)*a(k).

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 0..200

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)*3^k.

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

Sum_{n>=0}1/a(n)^r = Product_{k>=0}(1 + 1/(3^(2^k)+1)^r),

Sum_{n>=0}(-1)^A000120(n)/a(n)^r = Product_{k>=0}(1 - 1/(3^(2^k)+1)^r), where r>0 is a real number.

In particular,

Sum_{n>=0}1/a(n) = Product_{k>=0}(1 + 1/(3^(2^k)+1)) = 1.391980...;

Sum_{n>=0}(-1)^A000120(n)/a(n) = 2/3.

a(2^n) = 3^(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)) = 8*(3^(2^(t-1)+1))/(3^(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) = 10 * a(4*n);

a(8*n+4) = 41/5 * a(8*n+2);

a(16*n+8)= 3281/410 * a(16*n+6), etc. (End)

From Tom Edgar, Oct 11 2015: (Start)

a(n) = Product_{b_j != 0} a(2^j) where n = Sum_{j>=0} b_j*2^j is the binary representation of n.

a(2*k+1) = 4*a(2*k).

(End)

MATHEMATICA

Table[Sum[Mod[Binomial[n, k], 2]3^k, {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Aug 28 2013 *)

PROG

(Sage) [sum((binomial(n, k)%2)*3^k for k in [0..n]) for n in [0..100]] # Tom Edgar, Oct 11 2015

(PARI) a(n) = subst(lift((Mod(1, 2)+'x)^n), 'x, 3); \\ Gheorghe Coserea, Jun 11 2016

CROSSREFS

Cf. A001316, A001317, A038183, A100308, A100309, A100310, A100311.

Sequence in context: A149203 A038168 A186337 * A238721 A077811 A136855

Adjacent sequences:  A100304 A100305 A100306 * A100308 A100309 A100310

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Dec 06 2004

STATUS

approved

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Last modified July 16 11:09 EDT 2020. Contains 335784 sequences. (Running on oeis4.)