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A038503 Sum of every 4th entry of row n in Pascal's triangle, starting at "n choose 0". 23
1, 1, 1, 1, 2, 6, 16, 36, 72, 136, 256, 496, 992, 2016, 4096, 8256, 16512, 32896, 65536, 130816, 261632, 523776, 1048576, 2098176, 4196352, 8390656, 16777216, 33550336, 67100672, 134209536, 268435456, 536887296, 1073774592 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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

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

M^n = [1,0,0,0] = [a(n), A000749(n), A038505(n), A038504(n)]; where M = the 4x4 matrix [1,1,0,0; 0,1,1,0; 0,0,1,1; 1,0,0,1]. Sum of the 4 terms = 2^n. Example: M^6 = [16, 20, 16, 12], sum of terms = 64 = 2^6. - Gary W. Adamson, Mar 13 2009

a(n) is the number of generalized compositions of n when there are i^2/2-5i/2+3 different types of i, (i=1,2,...). [From Milan Janjic, Sep 24 2010]

REFERENCES

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, 2nd ed., Problem 38, p. 70, gives an explicit formula for the sum.

LINKS

Table of n, a(n) for n=0..32.

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

G.f.: (1-x)^3/((1-x)^4-x^4); a(n)=sum{k=0..floor(n/4), binomial(n, 4k)}; a(n)=2^(n-1)+2^((n-2)/2)(cos(pi*n/4)-sin(pi*n/4)). - Paul Barry, Mar 18 2004

Binomial transform of 1/(1-x^4). a(n)=4a(n-1)-6a(n-2)+4a(n-3); a(n)=sum{k=0..n, binomial(n, k)(sin(pi*(k+1)/2)/2+(1+(-1)^k)/4)}; a(n)=sum{k=0..floor(n/4), binomial(n, 4k) }. - Paul Barry, Jul 25 2004

a(n)=sum{k=0..n, binomial(n, 4(n-k))} - Paul Barry, Aug 30 2004

a(n)=sum{k=0..floor(n/2), binomial(n, 2k)(1+(-1)^k)/2} - Paul Barry, Nov 29 2004

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.

E.g.f.: exp(z)*(cosh(z)+cos(z))/2. - Peter Luschny, Jul 10 2012

EXAMPLE

a(3;0,0)=1 since the one binary string of trace 0, subtrace 0 and length 3 is { 000 }.

MAPLE

ZL:=[S, {a = Atom, b = Atom, S = Prod(X, Sequence(Prod(X, X, X, X))), X = Sequence(b, card >= 1)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=1..33); - Zerinvary Lajos, Mar 26 2008

A038503_list := proc(n) local i; series(exp(z)*(cosh(z)+cos(z))/2, z, n+2):

seq(i!*coeff(%, z, i), i=0..n) end: A038503_list(32); # Peter Luschny, Jul 10 2012

MATHEMATICA

nn = 18; a = Sum[x^(4 i)/(4 i)!, {i, 0, nn}]; b = Exp[x]; Range[0, nn]! CoefficientList[Series[a b, {x, 0, nn}], x]  (*Geoffrey Critzer, Dec 27 2011*)

Join[{1}, LinearRecurrence[{4, -6, 4}, {1, 1, 1}, 40]] (* Harvey P. Dale, Dec 02 2014 *)

CROSSREFS

Cf. A024493, A024494, A024495, A038505, A038504, A000749.

Row sums of A098173.

Sequence in context: A254119 A145126 A005676 * A079990 A127902 A157136

Adjacent sequences:  A038500 A038501 A038502 * A038504 A038505 A038506

KEYWORD

easy,nonn,changed

AUTHOR

Frank Ruskey

STATUS

approved

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