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A038503
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Sum of every 4th entry of row n in Pascal's triangle, starting at "n choose 0".
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23
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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; internal format)
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OFFSET
| 0,5
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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.
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 13 2009: (Start)
M^n = [1,0,0,0] = [a(n), A000749(n), A038505(n-1), 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. (End)
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 R. Janjic (agnus(AT)blic.net), Sep 24 2010]
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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.
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
F. Ruskey, Strings over Z_2 of given Trace and Subtrace
F. Ruskey, Strings over GF(2) of given Trace and Subtrace
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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 (pbarry(AT)wit.ie), 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 (pbarry(AT)wit.ie), Jul 25 2004
a(n)=sum{k=0..n, binomial(n, 4(n-k))} - Paul Barry (pbarry(AT)wit.ie), Aug 30 2004
a(n)=sum{k=0..floor(n/2), binomial(n, 2k)(1+(-1)^k)/2} - Paul Barry (pbarry(AT)wit.ie), 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.
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EXAMPLE
| a(3;0,0)=1 since the one binary string of trace 0, subtrace 0 and length 3 is { 000 }.
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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 (zerinvarylajos(AT)yahoo.com), Mar 26 2008
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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*)
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CROSSREFS
| Cf. A024493, A024494, A024495, A038505, A038504, A000749.
Row sums of A098173
Sequence in context: A159938 A145126 A005676 * A079990 A127902 A157136
Adjacent sequences: A038500 A038501 A038502 * A038504 A038505 A038506
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KEYWORD
| easy,nonn
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AUTHOR
| Frank Ruskey (ruskey(AT)cs.uvic.ca)
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