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A038504 Sum of every 4th entry of row n in Pascal's triangle, starting at "n choose 1". 21
0, 1, 2, 3, 4, 6, 12, 28, 64, 136, 272, 528, 1024, 2016, 4032, 8128, 16384, 32896, 65792, 131328, 262144, 523776, 1047552, 2096128, 4194304, 8390656, 16781312, 33558528, 67108864, 134209536, 268419072, 536854528, 1073741824 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of strings over Z_2 of length n with trace 1 and subtrace 0.

Same as number of strings over GF(2) of length n with trace 1 and subtrace 0.

Contribution from Gary W. Adamson, Mar 13 2009: (Start)

M^n * [1,0,0,0] = [A038503(n), A000749(n), A038505(n), a(n)]; where

M = a 4x4 matrix [1,1,0,0; 0,1,1,0; 0,0,1,1; 1,0,0,1]. Sum of terms = 2^n

Example: M^6 * [1,0,0,0] = [16, 20, 16, 12], sum = 2^6 = 64. (End)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

F. Ruskey, Strings over Z_2 of given Trace and Subtrace

F. Ruskey, Strings over GF(2) of given Trace and Subtrace

Index entries for linear recurrences with constant coefficients, signature (4,-6,4).

FORMULA

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3), n > 3. - Paul Curtz, Mar 01 2008

G.f.: x*(1-x)^2/((1-2*x)*(1-2*x+2*x^2)).

From Paul Barry, Jul 25 2004: (Start)

Binomial transform of x/(1-x^4).

G.f.: x(1-x)^2/((1-x)^4-x^4) = x/(1-2x)-x^3/((1-x)^4-x^4).

a(n) = sum{k=0..floor(n/4), binomial(n, 4*k+1)}.

a(n) = sum{k=0..n, binomial(n, k)*(sin(pi*k/2)/2+(1-(-1)^k)/4)}.

a(n) = 2^(n-2)+2^((n-2)/2)*sin(pi*n/4)-0^n/4. (End)

a(n; t, s) = a(n-1; t, s) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.

(1, 2, 3, 4, 6,...) is the binomial transform of (1, 1, 0, 0, 1, 1,...). - Gary W. Adamson, May 15 2007

EXAMPLE

a(2;1,0)=3 since the two binary strings of trace 1, subtrace 0 and length 2 are { 10, 01 }.

MATHEMATICA

CoefficientList[Series[x (1-x)^2/((1-2x)(1-2x+2x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 22 2012 *)

PROG

(MAGMA) I:=[0, 1, 2, 3, 4, 6, 12]; [n le 7 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jun 22 2012

CROSSREFS

Cf. A038503, A038505, A000749.

Cf. A099855.

Sequence in context: A214570 A078495 A161701 * A275448 A018405 A018419

Adjacent sequences:  A038501 A038502 A038503 * A038505 A038506 A038507

KEYWORD

easy,nonn,changed

AUTHOR

Frank Ruskey

STATUS

approved

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Last modified June 25 20:26 EDT 2017. Contains 288730 sequences.