A007729 -id:A007729 - cp's OEIS frontend

A177443 Triangle, row sums = A007729; derived from the generator for A002487, Stern's diatomic series.

1, 2, 0, 3, 1, 0, 3, 2, 0, 0, 3, 3, 2, 0, 0, 3, 3, 4, 0, 0, 0, 3, 3, 6, 1, 0, 0, 0, 3, 3, 6, 2, 0, 0, 0, 0, 3, 3, 6, 3, 3, 0, 0, 0, 0, 3, 3, 6, 3, 6, 0, 0, 0, 0, 0, 3, 3, 6, 3, 9, 2, 0, 0, 0, 0, 0, 3, 3, 6, 3, 9, 4, 0, 0, 0, 0, 0, 0, 3, 3, 6, 3, 9, 6, 3

Offset: 0

Gary W. Adamson, May 08 2010

Rows apparently tend to 3 * nonzero terms of Stern's diatomic series; i.e.,
3 * (1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5,...) = (3, 3, 6, 3, 9, 6, 9, 3, 12,...)
Row sums = A007729: (1, 2, 4, 5, 8, 10, 13, 14, ...)

			First few rows of the triangle =
1;
2, 0;
3, 1, 0;
3, 2, 0, 0;
3, 3, 2, 0, 0;
3, 3, 4, 0, 0, 0;
3, 3, 6, 1, 0, 0, 0;
3, 3, 6, 2, 0, 0, 0, 0;
3, 3, 6, 3, 3, 0, 0, 0, 0;
3, 3, 6, 3, 6, 0, 0, 0, 0, 0;
3, 3, 6, 3, 9, 2, 0, 0, 0, 0, 0;
3, 3, 6, 3, 9, 4, 0, 0, 0, 0, 0, 0;
3, 3, 6, 3, 9, 6, 3, 0, 0, 0, 0, 0, 0;
3, 3, 6, 3, 9, 6, 6, 0, 0, 0, 0, 0, 0, 0;
3, 3, 6, 3, 9, 6, 9, 1, 0, 0, 0, 0, 0, 0, 0;
3, 3, 6, 3, 9, 6, 9, 2, 0, 0, 0, 0, 0, 0, 0, 0;
3, 3, 6, 3, 9, 6, 9, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0;
3, 3, 6, 3, 9, 6, 9, 3, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0;
3, 3, 6, 3, 9, 6, 9, 3, 12, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0;
...

Cf. A002487, A007729.

Triangle read by rows, Q*R*S; where Q = an infinite lower triangular matrix with all 1's, R = the generator for A002487, and S = a diagonalized variant of A002487 (nonzero terms of A002487 as the right diagonal and the rest zeros). R, the generator for A002487 is an irregular lower triangular matrix with (1, 1, 1, 0, 0, 0,...) in each column; but each successive column for k>0 is shifted down twice from the previous column.

A072170 Square array T(n,k) (n >= 0, k >= 2) read by antidiagonals, where T(n,k) is the number of ways of writing n as Sum_{i>=0} c_i 2^i, c_i in {0,1,...,k-1}.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 2, 3, 2, 2, 1, 1, 1, 3, 3, 4, 2, 2, 1, 1, 1, 1, 4, 3, 4, 2, 2, 1, 1, 1, 4, 4, 5, 4, 4, 2, 2, 1, 1, 1, 3, 5, 4, 5, 4, 4, 2, 2, 1, 1, 1, 5, 5, 8, 5, 6, 4, 4, 2, 2, 1, 1, 1, 2, 6, 6, 8, 5, 6, 4, 4, 2, 2, 1, 1, 1, 5, 6, 9, 8, 9, 6, 6, 4, 4, 2, 2, 1, 1

Offset: 0

N. J. A. Sloane, Jun 29 2002

k-th column is k-th binary partition function.

The sequence data corresponds (via the table link) to the transpose of the array shown in example and given by the definition. - M. F. Hasler, Feb 14 2019

			Array begins: (rows n >= 0, columns k >= 2)
1 1 1 1 1 1 1 1 ...
1 1 1 1 1 1 1 1 ...
1 2 2 2 2 2 2 2 ...
1 1 2 2 2 2 2 2 ...
1 3 3 4 4 4 4 4 ...
1 2 3 3 4 4 4 4 ...
1 3 4 5 5 6 6 6 ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
B. Reznick, Some binary partition functions, in "Analytic number theory" (Conf. in honor P. T. Bateman, Allerton Park, IL, 1989), 451-477, Progr. Math., 85, Birkhäuser Boston, Boston, MA, 1990.

k=3 column is A002487, k=4 is A008619 (positive integers repeated), k = 5, 6, 7 are A007728, A007729, A007730, limiting (infinity) column is A000123 doubled up.

b:= proc(n, i, k) option remember;
      `if`(n=0, 1, `if`(i<0, 0, add(`if`(n-j*2^i<0, 0,
         b(n-j*2^i, i-1, k)), j=0..k-1)))
    end:
T:= (n, k)-> b(n, ilog2(n), k):
seq(seq(T(d+2-k, k), k=2..d+2), d=0..14); # Alois P. Heinz, Jun 21 2012

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 0, 0, Sum[If[n-j*2^i < 0, 0, b[n-j*2^i, i-1, k]], {j, 0, k-1}]]];
t[n_, k_] := b[n, Length[IntegerDigits[n, 2]] - 1, k];
Table[Table[t[d+2-k, k], {k, 2, d+2}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *)

M72170=[[]]; A072170(n,k,i=logint(n+!n,2),r=1)={if( !i, k>n, r&&(k<5||k>=n),if(k>4, A000123(n\2)-(k==n), k<3, 1, k<4, A002487(n), n\2+1), M72170[r=setsearch(M72170,[n,k,i,""],1)-1][^-1]==[n,k,i], M72170[r][4], M72170=setunion(M72170,[[n,k,i,r=sum(j=0,min(k-1,n>>i),A072170(n-j*2^i,k,i-1,0))]]);r)} \\ Code for k<5 (using A002487 for k=3) and k>=n (using A000123) is optional but makes it about 3x faster. - M. F. Hasler, Feb 14 2019

T(n,k) = T(n,n+1) = T(n,n)+1 = A000123(floor(n/2)) for all k >= n+1. - M. F. Hasler, Feb 14 2019

A174868 Partial sums of Stern's diatomic series A002487.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 10, 13, 14, 18, 21, 26, 28, 33, 36, 40, 41, 46, 50, 57, 60, 68, 73, 80, 82, 89, 94, 102, 105, 112, 116, 121, 122, 128, 133, 142, 146, 157, 164, 174, 177, 188, 196, 209, 214, 226, 233, 242, 244, 253, 260, 272, 277, 290, 298, 309, 312, 322, 329, 340, 344, 353, 358, 364, 365, 372, 378, 389, 394, 408, 417, 430, 434, 449, 460, 478, 485, 502, 512, 525, 528, 542, 553, 572, 580, 601, 614, 632, 637, 654, 666, 685

Offset: 0

Jonathan Vos Post, Dec 01 2010

After the initial 0, identical to A007729.

			a(16) = 0 + 1 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 4 + 3 + 5 + 2 + 5 + 3 + 4 + 1 = 41.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Michael J. Collins and David Wilson, Equivalence of OEIS A007729 and A174868, arXiv:1812.11174 [math.CO], 2018.
Clemens Heuberger, Daniel Krenn and Gabriel F. Lipnik, Asymptotic Analysis of q-Recursive Sequences, Algorithmica, Vol. 84 (2022), pp. 2480-2532; arXiv preprint, arXiv:2105.04334 [math.CO], 2021-2022.

Cf. A002487, A007729, A084091, A049456, A020946, A020950, A046815, A070871, A070872, A071883, A064881-A064886, A072170, A001045, A002083, A073459, A000123, A126606, A049455.

Cf. A001622, A242208.

a[n_] := a[n] = If[EvenQ[n], 2*a[n/2] + a[n/2 - 1], 2*a[(n - 1)/2] + a[(n + 1)/2]]; a[0] = 0; a[1] = 1; Array[a, 100, 0] (* Amiram Eldar, May 18 2023 *)

from itertools import accumulate, count, islice
from functools import reduce
def A174868_gen(): # generator of terms
    return accumulate((sum(reduce(lambda x,y:(x[0],x[0]+x[1]) if int(y) else (x[0]+x[1],x[1]),bin(n)[-1:2:-1],(1,0))) for n in count(1)),initial=0)
A174868_list = list(islice(A174868_gen(),30)) # Chai Wah Wu, May 07 2023

a(n) = Sum_{i=0..n} A002487(i).
G.f.: (x/(1 - x))*Product_{k>=0} (1 + x^(2^k) + x^(2^(k+1))). - Ilya Gutkovskiy, Feb 27 2017
a(2k) = 2*a(k) + a(k-1); a(2k+1) = 2*a(k) + a(k+1). - Michael J. Collins, Dec 25 2018
a(n) = n^log_2(3) + Psi_D(log_2(n)) + O(n^log_2(phi)), where phi is the golden ratio (A001622) and Psi_D is a 1-periodic continuous function which is Hölder continuous with any exponent smaller than log_2(3/phi) (Heuberger et al., 2022). - Amiram Eldar, May 18 2023

A166556 Triangle read by rows, A000012 * A047999.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 2, 1, 5, 2, 2, 1, 1, 6, 3, 2, 1, 2, 1, 7, 3, 3, 1, 3, 1, 1, 8, 4, 4, 2, 4, 2, 2, 1, 9, 4, 4, 2, 4, 2, 2, 1, 1, 10, 5, 4, 2, 4, 2, 2, 1, 2, 1, 11, 5, 5, 2, 4, 2, 2, 1, 3, 1, 1, 12, 6, 6, 3, 4, 2, 2, 1, 4, 2, 2, 1

Offset: 0

Gary W. Adamson, Oct 17 2009

			First few rows of the triangle =
   1;
   2, 1;
   3, 1, 1;
   4, 2, 2, 1;
   5, 2, 2, 1, 1;
   6, 3, 2, 1, 2, 1;
   7, 3, 3, 1, 3, 1, 1;
   8, 4, 4, 2, 4, 2, 2, 1;
   9, 4, 4, 2, 4, 2, 2, 1, 1;
  10, 5, 4, 2, 4, 2, 2, 1, 2, 1;
  11, 5, 5, 2, 4, 2, 2, 1, 3, 1, 1;
  12, 6, 6, 3, 4, 2, 2, 1, 4, 2, 2, 1;
  13, 6, 6, 3, 5, 2, 2, 1, 5, 2, 2, 1, 1;
  ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A000027, A004524, A004526, A047999, A080100.

Sums include: A006046 (row), A007729 (diagonal).

A166556:= func< n,k | (&+[(Binomial(j,k) mod 2): j in [k..n]]) >;
[A166556(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Dec 02 2024

A166556 := proc(n,k)
    local j;
    add(A047999(j,k),j=k..n) ;
end proc: # R. J. Mathar, Jul 21 2016

A166556[n_, k_]:= Sum[Mod[Binomial[j,k], 2], {j,k,n}];
Table[A166556[n,k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2024 *)

def A166556(n,k): return sum(binomial(j,k)%2 for j in range(k,n+1))
print(flatten([[A166556(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 02 2024

Triangle read by rows, A000012 * A047999; where A000012 = an infinite lower triangular matrix with all 1's: [1; 1,1; 1,1,1;..]; and A047999 = Sierpinski's gasket.
The operation takes partial sums of Sierpinski's gasket terms, by columns.
From G. C. Greubel, Dec 02 2024: (Start)
T(n, k) = Sum_{j=k..n} (binomial(j,k) mod 2).
T(n, 0) = A000027(n+1).
T(n, 1) = A004526(n+1).
T(n, 2) = A004524(n+1).
T(2*n, n) = A080100(n).
Sum_{k=0..n} T(n, k) = A006046(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = A006046(floor(n/2)+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A007729(n). (End)

cp's OEIS Frontend

A177443 Triangle, row sums = A007729; derived from the generator for A002487, Stern's diatomic series.

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A072170 Square array T(n,k) (n >= 0, k >= 2) read by antidiagonals, where T(n,k) is the number of ways of writing n as Sum_{i>=0} c_i 2^i, c_i in {0,1,...,k-1}.

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A174868 Partial sums of Stern's diatomic series A002487.

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A166556 Triangle read by rows, A000012 * A047999.

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Magma

Maple

Mathematica

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