A177445 -id:A177445 - cp's OEIS frontend

A120562 Sum of binomial coefficients binomial(i+j, i) modulo 2 over all pairs (i,j) of positive integers satisfying 3i+j=n.

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

Offset: 0

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

a(n) is the 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.
A120562 can be generated from triangle A177444. Partial sums of A120562 = A177445. - 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 crossrefs. - Johannes W. Meijer, Jun 05 2011

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

Michael De Vlieger, Table of n, a(n) for n = 0..10000
K. Anders, Counting Non-Standard Binary Representations, JIS vol 19 (2016) #16.3.3 example 6.
Cristina Ballantine and George Beck, Partitions enumerated by self-similar sequences, arXiv:2303.11493 [math.CO], 2023.
S. Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.

Cf. A001316 (1,1), A002487 (2,1), A120562 (3,1), A112970 (4,1), A191373 (5,1).
Cf. A177444, A177445. - Gary W. Adamson, 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). - Johannes W. Meijer, Jun 05 2011
Same recurrence for odd n as A000930.

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); # Johannes W. Meijer, Jun 05 2011

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-François Alcover, Sep 29 2011 *)
Nest[Append[#1, If[EvenQ@ #2, #1[[#2/2 + 1]], Total@ #1[[#2 ;; #2 + 1]] & @@ {#1, (#2 - 1)/2}]] & @@ {#, Length@ #} &, {1, 1}, 10^4 - 1] (* Michael De Vlieger, Feb 19 2019 *)

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) = Product_{i>=0} (1 + x^(2^i) + x^(3*2^i)) = (1 + x + x^3)*A(x^2).
a(n-1) << n^x with x = log_2(phi) = 0.69424... - Charles R Greathouse IV, Dec 27 2011

Reference edited and link added by Jason G. Wurtzel, Aug 22 2010

A177444 Triangle by columns, (1, 1, 0, 1, 0, 0, 0, ...); shifted down twice for columns > 0.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 0

Gary W. Adamson, May 08 2010

Let the triangle = M. Limit_{n->infinity} M^n = A120562, the left-shifted vector

considered as a sequence.

			First few rows of the triangle:
  1;
  1, 0;
  0, 1, 0;
  1, 1, 0, 0;
  0, 0, 1, 0, 0;
  0, 1, 1, 0, 0, 0;
  0, 0, 0, 1, 0, 0, 0;
  0, 0, 1, 1, 0, 0, 0, 0;
  0, 0, 0, 0, 1, 0, 0, 0, 0;
  0, 0, 0, 1, 1, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0;
  0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0;
  ...

Cf. A120562, A177445.

As an infinite lower triangular matrix, (1, 1, 0, 1, 0, 0, 0, ...) in every

column, prefaced with k zeros.

A177446 Triangle read by rows: A000012 * A177444 * the diagonalized variant of A120562.

Original entry on oeis.org

1, 2, 0, 2, 1, 0, 3, 2, 0, 0, 3, 2, 1, 0, 0, 3, 3, 2, 0, 0, 0, 3, 3, 2, 2, 0, 0, 0, 3, 3, 3, 4, 0, 0, 0, 0, 3, 3, 3, 4, 1, 0, 0, 0, 0, 3, 3, 3, 6, 2, 0, 0, 0, 0, 0, 3, 3, 3, 6, 2, 2, 0, 0, 0, 0, 0, 3, 3, 3, 6, 3, 4, 0, 0, 0, 0, 0, 0, 3, 3, 3, 6, 3, 4, 2, 0, 0, 0, 0, 0, 0

Offset: 0

Gary W. Adamson, May 08 2010

Row sums = A177445. Rows apparently tend to 3 * A120562 = 3 * (1, 1, 1, 2, 1, 2, 2, 3, 1, 3, ...).

			First few rows of the triangle =
1;
2, 0;
2, 1, 0;
3, 2, 0, 0;
3, 2, 1, 0, 0;
3, 3, 2, 0, 0, 0;
3, 3, 2, 2, 0, 0, 0;
3, 3, 3, 4, 0, 0, 0, 0;
3, 3, 3, 4, 1, 0, 0, 0, 0;
3, 3, 3, 6, 2, 0, 0, 0, 0, 0;
3, 3, 3, 6, 2, 2, 0, 0, 0, 0, 0;
3, 3, 3, 6, 3, 4, 0, 0, 0, 0, 0, 0;
3, 3, 3, 6, 3, 4, 2, 0, 0, 0, 0, 0, 0;
3, 3, 3, 6, 3, 6, 4, 0, 0, 0, 0, 0, 0, 0;
3, 3, 3, 6, 3, 6, 4, 3, 0, 0, 0, 0, 0, 0, 0;
...

Cf. A120562, A177444, A177445.

Triangle read by rows, M*Q*R; such that M = an infinite lower triangular matrix with all 1's, Q = triangle A177444, and R = the diagonalized variant of A120562 (A120562 in the diagonal and the rest zeros).

Definition corrected by Georg Fischer, May 31 2023

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A120562 Sum of binomial coefficients binomial(i+j, i) modulo 2 over all pairs (i,j) of positive integers satisfying 3i+j=n.

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A177444 Triangle by columns, (1, 1, 0, 1, 0, 0, 0, ...); shifted down twice for columns > 0.

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A177446 Triangle read by rows: A000012 * A177444 * the diagonalized variant of A120562.

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