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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.

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)

{A038503, A038504, A038505, A000749} is the difference analog of the hyperbolic functions {h_1(x), h_2(x), h_3(x), h_4(x)} of order 4. For the definitions of {h_i(x)} and the difference analog {H_i (n)} see [Erdelyi] and the Shevelev link respectively. - Vladimir Shevelev, Jul 31 2017

REFERENCES

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

LINKS

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

F. Ruskey, Strings over Z_2 of given Trace and Subtrace [Broken link ?]

F. Ruskey, Strings over GF(2) of given Trace and Subtrace [Broken link ?]

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.

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

From Vladimir Shevelev, Jul 31 2017: (Start)

For n >= 1, {H_i(n)} are linearly dependent sequences: a(n) = H_2(n) = H_1(n) + H_3(n) - H_4(n);

a(n+m) = a(n)*H_1(m) + H_1(n)*a(m) + H_4(n)*H_3(m) + H_3(n)*H_4(m), where H_1 = A038503, H_3 = A038505, H_4 = A000749.

For proofs, see Shevelev's link, Theorems 2, 3. (End)

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 *)

LinearRecurrence[{4, -6, 4}, {0, 1, 2, 3}, 40] (* Harvey P. Dale, Aug 23 2017 *)

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

AUTHOR

Frank Ruskey

STATUS

approved

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Last modified October 21 11:34 EDT 2017. Contains 293693 sequences.