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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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7
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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; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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 [A001316] 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.
A120562 can be generated from triangle A177444. Partial sums of A120562 = A177445. [From Gary W. Adamson, May 08 2010]
The Ca1 and Ca2 triangle sums, see A180662 for their definitions, of Sierpinski’s triangle A047999 equal this sequence. Some A120562(2^n-p) sequences, 0<=p<=32, lead to known sequences, see the cross-refs. [From Johannes W. Meijer, Jun 05 2011]
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LINKS
| S. Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.
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FORMULA
| Recurrence; a(0)=a(1)=1, a(2*n)=a(n) and a(2*n+1)=a(n)+a(n-1).
G.f.: A(x) = prod(i>=0, 1+x^(2^i)+x^(3*2^i) ) = (1+x+x^3)*A(x^2).
a(n-1) << n^x with x = lg(phi) = 0.69424... - Charles R. Greathouse IV, Dec 27 2011
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EXAMPLE
| a(2^n)=1 since a(2n)=a(n).
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MAPLE
| 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:
A120562:=proc(n) option remember; if n <0 then A120562(n):=0 fi: if (n=0 or n=1) then 1 elif n mod 2 = 0 then A120562(n/2) else A120562((n-1)/2) + A120562((n-3)/2); fi; end: seq(A120562(n), n=0..64); [From Johannes W. Meijer, Jun 05 2011]
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MATHEMATICA
| a[0] = a[1] = 1; a[n_?EvenQ] := a[n] = a[n/2]; a[n_?OddQ] := a[n] = a[(n-1)/2] + a[(n-1)/2 - 1]; Table[a[n], {n, 0, 64}] (* From Jean-François Alcover, Sep 29 2011 *)
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CROSSREFS
| Cf. A001316 (1,1), A002487 (2,1), A120562 (3,1), A112970 (4,1), A191373 (5,1).
Cf. A177444, A177445 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 08 2010]
Cf. A000012 (p=0), A000045 (p=1, p=2, p=4, p=8, p=16, p=32), A000071 (p=3, p=6, p=12, p=13, p=24, p=26), A001610 (p=5, p=10, p=20), A001595 (p=7, p=14, p=28), A014739 (p=11, p=22, p=29), A111314 (p=15, p=30), A027961 (p=19), A154691 (p=21), A001911 (p=23) [From Johannes W. Meijer, Jun 05 2011]
Sequence in context: A202389 A176853 A000374 * A178692 A033666 A139124
Adjacent sequences: A120559 A120560 A120561 * A120563 A120564 A120565
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KEYWORD
| easy,nonn
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AUTHOR
| Sam Northshield (samuel.northshield(AT)plattsburgh.edu), Aug 07 2006
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EXTENSIONS
| Reference edited and link added by Jason G. Wurtzel (j_seq(AT)wurtzel.com), Aug 22 2010
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