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A038504
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Sum of every 4th entry of row n in Pascal's triangle, starting at "n choose 1".
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17
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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; internal format)
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OFFSET
| 0,3
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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 (qntmpkt(AT)yahoo.com), Mar 13 2009: (Start)
M^n * [1,0,0,0] = [A038503(n), A000749(n), A038505(n-1), 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)
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LINKS
| F. Ruskey, Strings over Z_2 of given Trace and Subtrace
F. Ruskey, Strings over GF(2) of given Trace and Subtrace
Index to sequences with linear recurrences with constant coefficients, signature (4,-6,4).
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FORMULA
| a(n)=4a(n-1)-6a(n-2)+4a(n-3), n > 3. Also a(n)=6a(n-1)-14a(n-2)+16a(n-3)-8a(n-4), n > 4. - Paul Curtz (bpcrtz(AT)free.fr), Mar 01 2008
G.f.: x(1-x)^2/((1-2x)(1-2x+2x^2)).
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, 4k+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. - Paul Barry (pbarry(AT)wit.ie), Jul 25 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.
(1, 2, 3, 4, 6,...) is the binomial transform of (1, 1, 0, 0, 1, 1,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 15 2007
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EXAMPLE
| a(2;1,0)=3 since the two binary strings of trace 1, subtrace 0 and length 2 are { 10, 01 }.
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CROSSREFS
| Cf. A038503, A038505, A000749.
Cf. A099855.
Sequence in context: A018369 A078495 A161701 * A018405 A018419 A095938
Adjacent sequences: A038501 A038502 A038503 * A038505 A038506 A038507
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KEYWORD
| easy,nonn
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AUTHOR
| Frank Ruskey (ruskey(AT)cs.uvic.ca)
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