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Search: id:A120562
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| A120562 |
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Sum of binomial coefficients C(i+j,i) modulo 2 over all pairs (i,j) of positive integers satisfying 3i+j=n. |
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+0
1
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| 1, 1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 3, 2, 4, 3, 5, 1, 4, 3, 4, 2, 5, 3, 5, 2, 5, 4, 6, 3, 7, 5, 8, 1, 6, 4, 5, 3, 7, 4, 7, 2, 6, 5, 7, 3, 8, 5, 8, 2, 7, 5, 7, 4, 9, 6, 10, 3, 9, 7, 10, 5, 12, 8, 13, 1
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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a(n)=number of 'vectors' (...,e_k, e_{k-1},...,e_0) with e_k in {0,1,3} such that sum_k e_k 2^k=n. a(2^n-1)=F(n+1) a(2^{k+1}+j)+a(j)=a(2^k+j)+a(2^{k-1}+j) if 2^k>4j. This sequence corresponds to the pair (3,1) as Stern's diatomic sequence [A002487] corresponds to (2,1) and Gould's sequence [A007318] corresponds to (1,1). There are many interesting similarities to [A000119], the number of representations of n as as a sum of distinct Fibonacci numbers.
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REFERENCES
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S. Northshield, Sums across Pacal's triangle modulo 2, preprint
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FORMULA
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Recurrence; a(0)=a(1)=1, a(2n)=a(n), and a(2n+1)=a(n)+a(n-1). Gen. fn.: A(x)=(1+x+x^3)A(x^2)=\prod_{i\ge 0} (1+x^{2^i}+x^{3dot 2^i})
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EXAMPLE
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a(2^n)=1 since a(2n)=a(n).
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MAPLE
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p := product((1+x^(2^i)+x^(3*2^i)), i=0..25): s := series(p, x, 1000): for k from 0 to 250 do printf(`%d, `, coeff(s, x, k)) od:
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CROSSREFS
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Cf. A007318, A002487.
Adjacent sequences: A120559 A120560 A120561 this_sequence A120563 A120564 A120565
Sequence in context: A086421 A109400 A000374 this_sequence A033666 A139124 A024160
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KEYWORD
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easy,nonn
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AUTHOR
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Sam Northshield (samuel.northshield(AT)plattsburgh.edu), Aug 07 2006
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