This site is supported by donations to The OEIS Foundation.



Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A120562 Sum of binomial coefficients C(i+j,i) modulo 2 over all pairs (i,j) of positive integers satisfying 3i+j=n. 9


%S 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,

%T 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

%N Sum of binomial coefficients C(i+j,i) modulo 2 over all pairs (i,j) of positive integers satisfying 3i+j=n.

%C 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 a sum of distinct Fibonacci numbers.

%C A120562 can be generated from triangle A177444. Partial sums of A120562 = A177445. [From Gary W. Adamson, May 08 2010]

%C 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]

%H S. Northshield, <a href="http://faculty.plattsburgh.edu/sam.northshield/PasTriMod2v3F.pdf">Sums across Pascal's triangle modulo 2</a>, Congressus Numerantium, 200, pp. 35-52, 2010.

%F Recurrence; a(0)=a(1)=1, a(2*n)=a(n) and a(2*n+1)=a(n)+a(n-1).

%F G.f.: A(x) = prod(i>=0, 1+x^(2^i)+x^(3*2^i) ) = (1+x+x^3)*A(x^2).

%F a(n-1) << n^x with x = lg(phi) = 0.69424... - _Charles R Greathouse IV_, Dec 27 2011

%e a(2^n)=1 since a(2n)=a(n).

%p 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:

%p 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]

%t 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}] (* _Jean-Fran├žois Alcover_, Sep 29 2011 *)

%Y Cf. A001316 (1,1), A002487 (2,1), A120562 (3,1), A112970 (4,1), A191373 (5,1).

%Y Cf. A177444, A177445 [From _Gary W. Adamson_, May 08 2010]

%Y 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]

%K easy,nonn

%O 0,4

%A Sam Northshield (samuel.northshield(AT)plattsburgh.edu), Aug 07 2006

%E Reference edited and link added by _Jason G. Wurtzel_, Aug 22 2010

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 20 01:21 EST 2014. Contains 252240 sequences.