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A112970 A generalized Stern sequence. 7
1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 2, 2, 1, 5, 3, 3, 2, 5, 2, 3, 1, 6, 4, 3, 2, 6, 2, 3, 1, 7, 5, 4, 3, 8, 3, 5, 2, 8, 5, 4, 2, 8, 3, 3, 1, 9, 6, 5, 4, 9, 3, 6, 2, 9, 6, 4, 2, 9, 3, 3, 1, 10, 7, 6, 5, 11, 4, 8, 3, 12, 8, 6, 3, 13, 5, 5, 2, 13, 8, 7, 5, 12, 4, 7, 2, 12, 8, 5, 3, 11, 3, 4, 1, 12, 9, 7, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Conjectures: a(2^n)=a(2^(n+1)+1)=A033638(n); a(2^n-1)=a(3*2^n-1)=1.

The Gi1 and Gi2 triangle sums, see A180662 for their definitions, of Sierpinski's triangle A047999 equal this sequence. The Gi1 and Gi2 sums can also be interpreted as (i + 4*j = n) and (4*i + j = n) sums, see the Northshield reference. Some A112970(2^n-p) sequences, 0<=p<=32, lead to known sequences, see the crossrefs. - Johannes W. Meijer, Jun 05 2011

LINKS

Table of n, a(n) for n=0..99.

Sam Northshield, Sums across Pascal’s triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010. [Johannes W. Meijer, Jun 05 2011]

FORMULA

a(n) = Sum_{k=0..n} mod(sum{j=0..n, (-1)^(n-k)*C(j, n-j)*C(k, j-k)}, 2).

From Johannes W. Meijer, Jun 05 2011: (Start)

a(2*n+1) = a(n) and a(2*n) = a(n) + a(n-2) with a(0) = 1, a(1) = 1 and a(n)=0 for n<=-1.

G.f.: Product_{n>=0} ((1 + x^(2^n) + x^(4*2^n)). (End)

G.f. A(x) satisfies: A(x) = (1 + x + x^4) * A(x^2). - Ilya Gutkovskiy, Jul 09 2019

MAPLE

A112970:=proc(n) option remember; if n <0 then A112970(n):=0 fi: if (n=0 or n=1) then 1 elif n mod 2 = 0 then A112970(n/2) + A112970((n/2)-2) else A112970((n-1)/2); fi; end: seq(A112970(n), n=0..99); # Johannes W. Meijer, Jun 05 2011

CROSSREFS

Cf. A002487, A112971.

Cf. A120562 (Northshield).

Cf. A033638 (p=0), A000012 (p=1), A004526 (p=2, p=3, p=5, p=9, p=17), A002620 (p=4, p=7, p=13, p=25), A000027 (p=6, p=11, p=21), A004116 (p=8, p=15, p=29), A035106 (p=10, p=19), A024206 (p=14, p=27), A007494 (p=18), A014616 (p=22), A179207 (p=26). - Johannes W. Meijer, Jun 05 2011

Sequence in context: A320266 A277692 A129138 * A112971 A299201 A050379

Adjacent sequences:  A112967 A112968 A112969 * A112971 A112972 A112973

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Oct 07 2005

STATUS

approved

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Last modified February 23 15:52 EST 2020. Contains 332168 sequences. (Running on oeis4.)