A006095 -id:A006095 - cp's OEIS frontend

A177730 Expansion of (6x + 1) / ((x - 1)(2x - 1)(4x - 1)(8*x - 1)).

1, 21, 245, 2325, 20181, 168021, 1370965, 11075925, 89042261, 714081621, 5719635285, 45785027925, 366392038741, 2931583636821, 23454458533205, 187642826282325, 1501171242849621, 12009484474209621, 96076333921424725, 768612503886583125, 6148907361161794901

Offset: 0

Roger L. Bagula, May 12 2010

Muniru A Asiru, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (15,-70,120,-64).

Cf. A006095, A006100.

a := List([0..200],n->((2^(n+1)-1)^2*(2^(n+2)-1))/3); # Muniru A Asiru, Jan 27 2018

a := seq(((2^(n+1)-1)^2 * (2^(n+2)-1))/3, n = 0..200); # Muniru A Asiru, Jan 27 2018

CoefficientList[Series[(6x+1)/((x-1)(2x-1)(4x-1)(8x-1)),{x,0,30}],x] (* or *) LinearRecurrence[{15,-70,120,-64},{1,21,245,2325},30] (* Harvey P. Dale, Jul 16 2018 *)

Vec((6*x + 1) / ((x - 1)*(2*x - 1)*(4*x - 1)*(8*x - 1)) + O(x^30)) \\ Colin Barker, Jan 27 2018

From Colin Barker, Jan 27 2018: (Start)
a(n) = ((2^(n+1)-1)^2 * (2^(n+2)-1)) / 3.
a(n) = 15*a(n-1) - 70*a(n-2) + 120*a(n-3) - 64*a(n-4) for n>3.
(End)

Heavily edited, with the blessing of Michel Marcus and Joerg Arndt, by Colin Barker, Jan 27 2018

A296306 a(n) = A001157(n)/A050999(n).

Original entry on oeis.org

1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 341, 1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 1365, 1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 341, 1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 5461, 1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 341, 1, 5, 1, 21, 1, 5, 1, 85, 1, 5, 1, 21, 1, 5, 1, 1365

Offset: 1

Ivan N. Ianakiev, Dec 10 2017

a(n) is the sum of the second powers of the divisors of n divided by the sum of the second powers of the odd divisors of n.
Conjecture 1: For any nonnegative integer k and positive integer n, the sum of the k-th powers of the divisors of n is divisible by the sum of the k-th powers of the odd divisors of n.
Conjecture 2: Distinct terms form A002450 without A002450(0). In other words, a(2^(n-1)) = A002450(n), for n > 0.
Conjecture 3: For n > 0, the list of the first 2^n - 1 terms is palindromic.
Conjecture 4: For n > 0, the sum of the first 2^n - 1 terms equals A006095(n+1).
To prove Conjecture 1 all one needs to do is to prove that the sum of the k-th powers of the divisors of n divided by the sum of the k-th powers of the odd divisors of n equals: a) A001511(n), for k = 0, and b) ((2^k)^A001511(n) - 1)/(2^k - 1), for k > 0. - Ivan N. Ianakiev, Jan 29 2020
Conjecture 1 indeed follows from multiplicativity of sigma_k, in particular sigma_k(2^j (2m+1)) = sigma_k(2^j) sigma_k(2m+1). Conjecture 3 follows from Radcliffe's formula, since A007814 has this property. - M. F. Hasler, Jan 31 2020
a(n) is the sum of squares of powers of 2 that divide n. - Amiram Eldar, Nov 12 2020

			A001157(4) = 21 and A050999(4) = 1, therefore a(4) = A001157(4)/A050999(4) = 21.

Antti Karttunen, Table of n, a(n) for n = 1..16383

Cf. A001157, A038712, A050999, A318935.

[DivisorSigma(2,n)/&+[d^2:d in Divisors(n)|IsOdd(d)]:n in [1..100]]; // Marius A. Burtea, Jan 29 2020

f[n_]:=DivisorSigma[2,n]/Total[Select[Divisors[n],OddQ]^2]; f/@Range[100]
Table[(4^(IntegerExponent[n, 2] + 1) - 1)/3, {n, 1, 100}] (* Amiram Eldar, Nov 12 2020 *)

a(n) = sigma(n, 2)/sumdiv(n, d, d^2*(d % 2)); \\ Michel Marcus, Dec 11 2017

def A296306(n): return ((1<<((n&-n).bit_length()<<1))-1)//3 # Chai Wah Wu, Jul 16 2022

a(n) = (4^(A007814(n) + 1) - 1)/3. - David Radcliffe, Dec 11 2017
Multiplicative with a(2^e) = (4^(e+1)-1)/3, and a(p^e) = 1 for odd primes p. - Amiram Eldar, Nov 12 2020
G.f.: Sum_{k>=0} 4^k * x^(2^k) / (1 - x^(2^k)). - Ilya Gutkovskiy, Dec 14 2020
Dirichlet g.f.: zeta(s)/(1-4/2^s). - Amiram Eldar, Dec 31 2022

A346465 Numbers k such that (4^k - 2)(4^k - 1)/Clausen(2k, 1) is not squarefree, where Clausen(n, m) = A160014(n, m).

Original entry on oeis.org

9, 11, 18, 27, 32, 36, 45, 50, 53, 54, 63, 68, 72, 74, 78, 81, 90, 95, 99, 100, 108, 116, 117, 126, 127, 135, 137, 144, 147, 150, 153, 155, 158, 162, 171, 179, 180, 182, 189, 198, 200, 204, 207, 216, 221, 225, 233, 234, 242, 243, 250, 252, 261, 263, 270, 279

Offset: 1

Peter Luschny, Jul 20 2021

nonn

Also numbers k such that 6*GaussBinomial(2*k, 2, 2)/denominator(Bernoulli(2*k, 1)) is not squarefree.

Cf. A006095, A002445, A007947, A160014, A346463, A346464.

with(NumberTheory): isa := n -> not IsSquareFree(((4^n - 2)*(4^n - 1))/
mul(i, i = select(isprime, map(i -> i+1, Divisors(2*n))))):
select(isa, [$(1..100)]);

q[n_] := Product[k, {k, Select[Table[d + 1, {d, Divisors[2 n]}], PrimeQ]}];
isA[n_] := ! SquareFreeQ[((4^n - 2) (4^n -1)) / q[n]];
Select[Range[50],  isA]

The positive multiples of 9 form a subsequence.

k is a term if and only if A346463(k) > A007947(A346463(k)).

More terms from Jinyuan Wang, Jul 23 2021

A372230 Triangular array read by rows. T(n,k) is the number of size k circuits in the linear matroid M[A] where A is the n X 2^n-1 matrix whose columns are the nonzero vectors in GF(2)^n, n>=2, 3<=k<=n+1.

Original entry on oeis.org

1, 7, 7, 35, 105, 168, 155, 1085, 5208, 13888, 651, 9765, 109368, 874944, 3999744, 2667, 82677, 1984248, 37039296, 507967488, 4063739904, 10795, 680085, 33732216, 1349288640, 43177236480, 1036253675520, 14737830051840

Offset: 2

Geoffrey Critzer, Apr 28 2024

For n>=2 and 3<=k<=n, to construct a size k circuit of M[A]: Choose a basis b_1,b_2,...,b_{k-1} of a k-1 dimensional subspace of GF(2)^n. Append the vector b_1 + b_2 + ... + b_{k-1}.

			Triangle begins ...
    1;
    7,    7;
   35,   105,     168;
  155,  1085,    5208,    13888;
  651,  9765,  109368,   874944,   3999744;
 2667, 82677, 1984248, 37039296, 507967488, 4063739904;
...

J. Oxley, Matroid Theory, Oxford Graduate Texts in Mathematics, 1992, page 8.

Cf. A022166, A053601, A006095, A372350 (row sums).

nn = 8; Map[Select[#, # > 0 &] &, Table[Table[PadRight[Table[Product[(2^n - 2^i)/(2^k - 2^i), {i, 0, k - 1}], {k, 2, n}], nn], {n, 2, nn}][[All, j]]*    Table[Product[2^n - 2^i, {i, 0, n - 1}]/(n + 1)!, {n, 2, nn}][[j]], {j, 1, nn - 1}] // Transpose] // Grid

T(n,k) = A022166(n,k-1)*A053601(k-1)/k.
T(n,3) = A006095.
T(n,n+1) = A053601(n)/(n+1).

A374981 a(n) = (n^2 - 1)/6 - Sum_{t=0..n-1} [ A000217(t)/n ], where [ x ] means the fractional part of x here.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 7, 11, 13, 16, 19, 24, 27, 33, 35, 42, 47, 53, 58, 67, 71, 78, 85, 95, 102, 112, 118, 128, 138, 147, 155, 170, 178, 191, 200, 213, 224, 238, 248, 263, 277, 290, 302, 322, 331, 347, 361, 380, 395, 413, 427, 445, 463, 482, 496, 519, 534, 554, 573, 594, 612, 637, 651, 678, 698

Offset: 1

Thomas Scheuerle, Jul 26 2024

a(c^n) for some constant c can be expressed as a linear recurrence with constant coefficients.

Cf. A000217, A006095, A374968.

a(n) = (n^2-1)/6-sum(k=1,n,(k*(k+1)/2)%n)/n+((n+1)%2)/2

a(n) = (n^2 - 1 - A374968(n))/6.
a(2^n) = A006095(n).
a(3^n) has the ordinary generating function: x*(1 - 2*x + 5*x^2 + 12*x^3)/(1 - 13*x + 36*x^2 + 12*x^3 - 117*x^4 + 81*x^5).

A006108 Gaussian binomial coefficient [ 2n,n ] for q=4.

Original entry on oeis.org

1, 5, 357, 376805, 6221613541, 1634141006295525, 6857430062381149327845, 460250514083576206796548772325, 494205307747746503853075131001823990245

Offset: 0

N. J. A. Sloane

nonn

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

A270218 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 129", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 4, 28, 140, 620, 2604, 10668, 43180, 173740, 697004, 2792108, 11176620, 44722860, 178924204, 715762348, 2863180460

Offset: 0

Robert Price, Mar 13 2016

Initialized with a single black (ON) cell at stage zero.

It appears Rules 385, 425, 465 and 553 also generate this sequence. - Lars Blomberg, Apr 30 2016 (It would be nice to have a proof! - N. J. A. Sloane, May 09 2016)

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A270217.

CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
code=129; stages=128;
rule=IntegerDigits[code,2,10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
PrependTo[ca,a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)

Conjectures from Colin Barker, Mar 13 2016: (Start)
a(n) = 4*(1-3*2^n+2^(1+2*n))/3.
a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.
G.f.: (1-3*x+14*x^2-8*x^3) / ((1-x)*(1-2*x)*(1-4*x)).
(End)
a(n) = 4*A006095(n+1) (conjectured). - Michal Stajszczak, May 20 2020

a(8)-a(15) from Lars Blomberg, Apr 30 2016

A374839 Triangle read by rows: T(n,k) is the number of strong subtrees of height k in the complete binary tree of height n.

Original entry on oeis.org

1, 3, 1, 7, 7, 1, 15, 35, 23, 1, 31, 155, 403, 279, 1, 63, 651, 6603, 71827, 65815, 1, 127, 2667, 106299, 18394315, 4313323667, 4295033111, 1, 255, 10795, 1703451, 4709050939, 282677998234827, 18447026751295461523, 18446744078004584727, 1

Offset: 1

John V Siratt, Jul 21 2024

Informally, a strong subtree is one that preserves meets, the relative level of vertices, and the number of immediate successors to non-terminal vertices.

The collection of strong subtrees with height k of some tree T is often denoted by S_k(T). T(n,k) is the cardinality of S_k(2^(

There is a general expression for T(n,k) given in terms of an auxiliary function that can be eliminated for the diagonals, with the following examples:

T(m+1,m) = 3 + Sum_{j=1..m-1} 2^(2^i).

T(m+2,m) = 7 + 3*(Sum_{j=1..m-1} 2^(2^i)) + Sum_{1<=i,j<=m-1} 2^(2^i + 2^j).

T(m+3,m) = 15 + 7*(Sum_{j=1..m-1} 2^(2^i)) + 3*(Sum_{1<=i,j<=m-1} 2^(2^i + 2^j)) + Sum_{1<=i,j,k<=m-1} 2^(2^i + 2^j + 2^k).

			Triangle begins:
    1;
    3,    1;
    7,    7,      1;
   15,   35,     23,        1;
   31,  155,    403,      279,          1;
   63,  651,   6603,    71827,      65815,          1;
  127, 2667, 106299, 18394315, 4313323667, 4295033111, 1;
  ...
Formatted as a transposed array:
T(n,k) | n=1  2  3   4    5      6           7                     8
--------------------------------------------------------------------
k=1    |   1  3  7  15   31     63         127                   255
2      |   0  1  7  35  155    651        2667                 10795
3      |   0  0  1  23  403   6603      106299               1703451
4      |   0  0  0   1  279  71827    18394315            4709050939
5      |   0  0  0   0    1  65815  4313323667       282677998234827
6      |   0  0  0   0    0      1  4295033111  18447026751295461523
7      |   0  0  0   0    0      0           1  18446744078004584727
8      |   0  0  0   0    0      0           0                     1

John V. Siratt, Some applications of formal mathematics, Doctoral dissertation, University of Notre Dame (2024).

Column 1 is A000225.

Column 2 appears to be A006095.

T(n,k)={my(s=0); forvec(x=vector(k,i,[1,n]),s+=prod(i=1, k,  (2^(x[i] - if(i>1, x[i-1]) - 1))^(2^(i - 1))), 2); s}
{ for(n=1, 8, print(vector(n,k,T(n,k)))) } \\ Andrew Howroyd, Jul 23 2024

T(n,k) = Sum_{1 <= x_1 < ... < x_k <= n} Product_{i=1..k} (2^(x_i - x_{i-1} - 1))^(2^(i - 1)), where x_0 = 0.

A377278 Denominators in a harmonic triangle; q-analog of A126615, here q = 2.

Original entry on oeis.org

1, 3, 3, 3, 21, 7, 3, 21, 105, 15, 3, 21, 105, 465, 31, 3, 21, 105, 465, 1953, 63, 3, 21, 105, 465, 1953, 8001, 127, 3, 21, 105, 465, 1953, 8001, 32385, 255, 3, 21, 105, 465, 1953, 8001, 32385, 130305, 511, 3, 21, 105, 465, 1953, 8001, 32385, 130305, 522753, 1023

Offset: 1

Werner Schulte, Oct 22 2024

The harmonic triangle uses the terms of this sequence as denominators, numerators = 1. The inverse of the harmonic triangle has entries -2^(n-k-1) for 1<=k

Conjecture: Row sums of the harmonic triangle are A204243(n) / A005329(n).

			Triangle T(n, k) for 1 <= k <= n starts:
n\ k :  1   2    3    4     5     6      7       8       9    10
================================================================
   1 :  1
   2 :  3   3
   3 :  3  21    7
   4 :  3  21  105   15
   5 :  3  21  105  465    31
   6 :  3  21  105  465  1953    63
   7 :  3  21  105  465  1953  8001    127
   8 :  3  21  105  465  1953  8001  32385     255
   9 :  3  21  105  465  1953  8001  32385  130305     511
  10 :  3  21  105  465  1953  8001  32385  130305  522753  1023
  etc.
The harmonic triangle starts:
  [1]  1/1
  [2]  1/3   1/3
  [3]  1/3  1/21    1/7
  [4]  1/3  1/21  1/105   1/15
  [5]  1/3  1/21  1/105  1/465    1/31
  [6]  1/3  1/21  1/105  1/465  1/1953  1/63
  etc.
The inverse of the harmonic triangle starts:
  [1]    1
  [2]   -1   3
  [3]   -2  -1   7
  [4]   -4  -2  -1  15
  [5]   -8  -4  -2  -1  31
  [6]  -16  -8  -4  -2  -1  63
  etc.

Cf. A000225, A005329, A006095, A126615, A204243.

PARI
```
T(n,k)=if(k
				
```

T(n, k) = (2^k - 1) * (2^(k+1) - 1) for 1 <= k < n; T(n, n) = 2^n - 1.
Sum_{k=1..n} 2^(k-1) / T(n, k) = 1.
Product_{k=1..n} T(n, k)^((-1)^k) = 1.
Row sums are n + 4 * (2^n - 1) * (2^(n-1) - 1) / 3 = n + 4 * A006095(n).
G.f.: x*y*(1 + 2*x - 4*x*y + 4*x^2*y)/((1 - x)*(1 - x*y)(1 - 2*x*y)*(1 - 4*x*y)). - Stefano Spezia, Oct 23 2024

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cp's OEIS Frontend

A177730 Expansion of (6*x + 1) / ((x - 1)*(2*x - 1)*(4*x - 1)*(8*x - 1)).

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A296306 a(n) = A001157(n)/A050999(n).

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A346465 Numbers k such that (4^k - 2)*(4^k - 1)/Clausen(2*k, 1) is not squarefree, where Clausen(n, m) = A160014(n, m).

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A372230 Triangular array read by rows. T(n,k) is the number of size k circuits in the linear matroid M[A] where A is the n X 2^n-1 matrix whose columns are the nonzero vectors in GF(2)^n, n>=2, 3<=k<=n+1.

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A374981 a(n) = (n^2 - 1)/6 - Sum_{t=0..n-1} [ A000217(t)/n ], where [ x ] means the fractional part of x here.

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A006108 Gaussian binomial coefficient [ 2n,n ] for q=4.

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A270218 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 129", based on the 5-celled von Neumann neighborhood.

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A374839 Triangle read by rows: T(n,k) is the number of strong subtrees of height k in the complete binary tree of height n.

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A177730 Expansion of (6x + 1) / ((x - 1)(2x - 1)(4x - 1)(8*x - 1)).

A346465 Numbers k such that (4^k - 2)(4^k - 1)/Clausen(2k, 1) is not squarefree, where Clausen(n, m) = A160014(n, m).