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A038505
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Sum of every 4th entry of row n in Pascal's triangle, starting at "n choose 2".
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14
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0, 1, 3, 6, 10, 16, 28, 56, 120, 256, 528, 1056, 2080, 4096, 8128, 16256, 32640, 65536, 131328, 262656, 524800, 1048576, 2096128, 4192256, 8386560, 16777216, 33558528, 67117056, 134225920, 268435456, 536854528, 1073709056
(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 0 and subtrace 1.
Same as number of strings over GF(2) of length n with trace 0 and subtrace 1.
Binomial transform of (0,1,1,0,0,1,1,0,...) - Paul Barry (pbarry(AT)wit.ie), Jul 07 2003
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 13 2009: (Start)
M^n * [1,0,0,0] = [A038503(n), A000749(n), a(n-1), A038504(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)=sum{k=0..n+1, (1/2)C(n+1, k)(-1)^C(k+3, 3)} - Paul Barry (pbarry(AT)wit.ie), Jul 07 2003
G.f. : x(1-x)/((1-x)^4-x^4) = x(1-x)/((1-2x)(1-2x+2x^2)); a(n)=sum{k=0..floor((n+1)/2), binomial(n+1, 2k)(1-(-1)^k)/2}. - Paul Barry (pbarry(AT)wit.ie), Nov 29 2004
Conjecture: 2*a(n+2) = A038504(n+3) + A000749(n+3) + 2*A009545(n+1) - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), May 22 2005
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.
a(n)=4a(n-1)-6a(n-2)+4a(n-3), n > 3; sequence is identical to its fourth differences. - Paul Curtz (bpcrtz(AT)free.fr), Dec 21 2007
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EXAMPLE
| a(3;0,1)=3 since the three binary strings of trace 0, subtrace 1 and length 3 are { 011, 101, 110 }.
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CROSSREFS
| Cf. A038503, A038504, A000749.
Cf. A009116.
Sequence in context: A130578 A107068 A033541 * A119971 A094272 A005045
Adjacent sequences: A038502 A038503 A038504 * A038506 A038507 A038508
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KEYWORD
| easy,nonn
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AUTHOR
| Frank Ruskey (ruskey(AT)cs.uvic.ca)
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