A010701 -id:A010701 - cp's OEIS frontend

A363099 Triangle T(n,k) in which the n-th row encodes the inverse of a 3n+1 X 3n+1 Jacobi matrix, with 1's on the lower, main, and upper diagonals in GF(2), where the encoding consists of the decimal representations for the binary rows (n >= 1, 1 <= k <= 3n+1).

11, 3, 12, 13, 91, 27, 96, 107, 3, 108, 109, 731, 219, 768, 859, 27, 864, 875, 3, 876, 877, 5851, 1755, 6144, 6875, 219, 6912, 7003, 27, 7008, 7019, 3, 7020, 7021, 46811, 14043, 49152, 55003, 1755, 55296, 56027, 219, 56064, 56155, 27, 56160, 56171, 3, 56172, 56173, 374491, 112347, 393216, 440027, 14043, 442368

Offset: 1

Nei Y. Soma, May 20 2023

Each term in the sequence encodes a line of the inverse of a Jacobi matrix that has 1s on its lower, main, and upper diagonals in GF(2). The associated inverse matrix column values come from the binary representation of that base-10 number, being a bit per column. These matrices start with a 4 X 4 matrix and the consecutive terms came by adding ascending and consecutive multiples of 3. If the binary number has fewer bits than the number of columns, it must be zero-padded to the left. To obtain the inverse matrices in real numbers instead of GF(2), alternate between + and - between the 1s in a row. If a row is a multiple of 3, alternate between - and +. The determinants of these 3m+1 X 3m+1 Jacobi matrices are 1 in GF(2), and alternate between -1 and 1 in R if m is odd or even. These properties were proven by Sutner (1989) and Melo (1997), respectively.

			For m = 1, the Jacobi 4 X 4 matrix has as rows
     1, 1, 0, 0
     1, 1, 1, 0
     0, 1, 1, 1
     0, 0, 1, 1
Its inverse has the rows
     1, 0, 1, 1
     0, 0, 1, 1
     1, 1, 0, 0
     1, 1, 0, 1
Representing these rows as binary numbers in base 10 the first three terms of the sequence are: 11, 3, 12, 13.
The next numbers in the sequence occur for m = 2, given a sequence of six numbers. The Jacobi 7 X 7 matrix has as its rows:
     1, 1, 0, 0, 0, 0, 0
     1, 1, 1, 0, 0, 0, 0
     0, 1, 1, 1, 0, 0, 0
     0, 0, 1, 1, 1, 0, 0
     0, 0, 0, 1, 1, 1, 0
     0, 0, 0, 0, 1, 1, 1
     0, 0, 0, 0, 0, 1, 1
Its inverse has as rows:
     1, 0, 1, 1, 0, 1, 1
     0, 0, 1, 1, 0, 1, 1
     1, 1, 0, 0, 0, 0, 0
     1, 1, 0, 1, 0, 1, 1
     0, 0, 0, 0, 0, 1, 1
     1, 1, 0, 1, 1, 0, 0
     1, 1, 0, 1, 1, 0, 1
These 7 latter rows from binary to base 10 give the next 7 terms of the sequence: 91, 27, 96, 107, 3, 108, 109.
Triangle T(n,k) begins:
    11,    3,   12,   13;
    91,   27,   96,  107,   3,  108,  109;
   731,  219,  768,  859,  27,  864,  875,  3,  876,  877;
  5851, 1755, 6144, 6875, 219, 6912, 7003, 27, 7008, 7019, 3, 7020, 7021;
  ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Boston, 2nd Ed., 12th printing, 2002, pp. 24-25.
P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, Boston, 1985, p. 35.
J. P. Melo, Reversibility of John von Neumann cellular automata, M.Sc. Thesis, Division of Computer Science, Instituto Tecnológico de Aeronáutica, 1997 (in Portuguese), p. 18.

Nei Y. Soma, Rows n = 1..30, flattened
K. Sutner, Linear Cellular Automata and the Garden-of-Eden, The Mathematical Intelligencer, 11(2), 1989, pp. 49-53; see p. 52

Column k=1 gives A245599(n+1).
Column k=2 gives A083713.
Column k=3 gives A204623.
T(n,3n-1) gives A010701.
Cf. A038184 One-dimensional cellular automaton (Rule 150) in a tape with 3m cells has as adjacency matrix the Jacobi matrices, 3m X 3m, with 1s on the lower, main and upper diagonals and the operations on it are in GF(2). And A363146 for the inverse of Jacobi matrices 3m X 3m, with 1s on the lower, main, and upper diagonals in GF(2).

T:= n-> (M-> seq(add(abs(M[j, n*3+1-i])*2^i, i=0..n*3), j=1..n*3+1))
               (Matrix(n*3+1, (i, j)-> `if`(abs(i-j)<2, 1, 0))^(-1)):
seq(T(n), n=1..6);  # Alois P. Heinz, May 20 2023

sequence = {};
For[k = 1, k <= 50, k++, {
  n = 3*k + 1;
  J = ConstantArray[0, {n, n}];
  For[i = 1, i < n, i++,
   J[[i, i]] = J[[i + 1, i]] = J[[i, i + 1]] = 1];
  J[[1, 1]] = J[[n, n]] = 1;
  InvJ = Mod[Inverse[J], 2];
  For[i = 1, i <= n, i++, AppendTo[sequence, FromDigits[InvJ[[i]], 2]]]
  }
 ]
sequence

row(n)=my(m=3*n+1, A=lift(matrix(m, m, i, j, Mod(abs(i-j)<=1, 2))^(-1))); vector(m, i, fromdigits(A[i,], 2)) \\ Andrew Howroyd, May 20 2023

The recurrence has as its base:
     r(1, 1) = 11;
     r(2, 1) =  3;
     r(3, 1) = 12;
     r(4, 1) = 13.
For 2 <= k <= m, and i = 1, 2, 3, ..., 3k - 2:
    r(i, k) = 8*r(i, k-1) + r(2, 1) (i != 0 (mod 3)).
And r(3k-1, k) = r(2, 1);
    r(3k,   k) = 8*r(3(k-1), k-1) + r(3,1);
    r(3k+1, k) = 8*r(3(k-1), k-1) + r(4,1).

A383641 a(n) is the difference between the sum of even composites and the sum of the odd composites in the first n positive integers.

Original entry on oeis.org

0, 0, 0, 4, 4, 10, 10, 18, 9, 19, 19, 31, 31, 45, 30, 46, 46, 64, 64, 84, 63, 85, 85, 109, 84, 110, 83, 111, 111, 141, 141, 173, 140, 174, 139, 175, 175, 213, 174, 214, 214, 256, 256, 300, 255, 301, 301, 349, 300, 350, 299, 351, 351, 405, 350, 406, 349, 407, 407

Offset: 1

Felix Huber, May 08 2025

nonn

			Of the first 9 positive integers, 4, 6, and 8 are even composites and 9 is an odd composite, so a(9) = 4 + 6 + 8 - 9 = 9.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Composite Number

Cf. A000720, A002808, A004526, A010701, A028552, A034387, A066247, A071904, A101256, A193356, A262044, A383259.

A383641:=n->`if`(n=1,0,floor((n-2)/2)-n*(n mod 2)+add(ithprime(i),i=2..NumberTheory:-pi(n)));seq(A383641(n),n=1..59);

lim=59;cn=Select[Range[lim],CompositeQ];a[n_]:=Total[Select[cn,EvenQ[#]&&#<=n&]]-Total[Select[cn,OddQ[#]&&#<=n&]];Array[a,lim] (* James C. McMahon, May 14 2025 *)

a(n) = floor((n-2)/2) - n*(n mod 2) + Sum_{i=2..pi(n)} prime(i) for n > 1.
a(n) = A004526(n) - A193356(n) - A010701(n) + A034387(A000720(n)) for n > 1.
a(n) = Sum_{i=1..n} ((-1)^i*i*A066247(i)).

A387235 Decimal expansion of 2*log(2)/3.

Original entry on oeis.org

4, 6, 2, 0, 9, 8, 1, 2, 0, 3, 7, 3, 2, 9, 6, 8, 7, 2, 9, 4, 4, 8, 2, 1, 4, 1, 4, 3, 0, 5, 4, 5, 1, 0, 4, 5, 3, 8, 3, 6, 6, 6, 7, 5, 6, 2, 4, 0, 1, 7, 0, 1, 6, 9, 4, 1, 3, 7, 8, 6, 6, 7, 2, 9, 9, 5, 5, 9, 5, 7, 4, 7, 9, 7, 9, 7, 9, 6, 4, 7, 7, 0, 7, 0, 5, 7, 5, 5, 5, 1, 3, 3, 0, 9, 4, 5, 7, 9, 1, 6

Offset: 0

Stefano Spezia, Aug 23 2025

Area enclosed by the curve of the equation x^6 + y^6 - x^3*y + x*y^3 = 0.

The asymptotic mean of A256232. - Amiram Eldar, Aug 23 2025

			0.46209812037329687294482141430545104538366675624...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 436.
Index entries for transcendental numbers.

Cf. A002162, A001504, A010701, A010722, A016627, A193535, A256232.

Mathematica
```
RealDigits[2Log[2]/3,10,100][[1]]
```

Equals A002162*A010722.
Equals log(4)/3 = A010701*A016627.
Equals Sum_{k>=0} (-1)^k/((3*k + 1)*(3*k + 2)) = Integral_{x=0..1} x^2*log(1 + 1/x^3) = -Integral_{x=0..1} log[1 - x^6]/x^4. [Shamos]
Equals A016627/3 = 2*A193535. - Hugo Pfoertner, Aug 23 2025

A174971 Periodic sequence: Repeat 3, -3.

Original entry on oeis.org

3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3, 3, -3

Offset: 0

Klaus Brockhaus, Apr 04 2010

Interleaving of A010701 and -A010701; signed version of A010701.
Essentially first differences of A010674.
Inverse binomial transform of 3 followed by A000004.
Second inverse binomial transform of A010701.
Third inverse binomial transform of A007283.
Fourth inverse binomial transform of A000244 without initial term 1.
Fifth inverse binomial transform of A164346.
Sixth inverse binomial transform of A005053 without initial term 1.
Seventh inverse binomial transform of A169604.
Eighth inverse binomial transform of A169634.
Ninth inverse binomial transform of A103333 without initial term 1.
Tenth inverse binomial transform of A013708.
Eleventh inverse binomial transform of A093138 without initial term 1.

Index entries for linear recurrences with constant coefficients, signature (-1).

Cf. A010701 (all 3's sequence), A000004 (all zeros sequence), A007283 (3*2^n), A000244 (powers of 3), A164346 (3*4^n), A005053 (expand (1-2x)/(1-5x)), A169604 (3*6^n), A169634 (3*7^n), A103333 (expand (1-5x)/(1-8x)), A013708 (3^(2n+1)), A093138 (expand (1-7x)/(1-10x)).

&cat[ [3, -3]: n in [0..41] ];
[ 3*(-1)^n: n in [0..83] ];

PadRight[{},120,{3,-3}] (* or *) NestList[-1#&,3,120] (* Harvey P. Dale, Dec 30 2023 *)

a(n)=3*(-1)^n \\ Charles R Greathouse IV, Jun 13 2013

a(n) = 3*(-1)^n.
a(n) = -a(n-1) for n > 0; a(0) = 3.
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = -3.
G.f.: 3/(1+x).

A229945 Triangle read by rows in which row n lists the union of the primes <= n and the divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 5, 1, 2, 3, 5, 6, 1, 2, 3, 5, 7, 1, 2, 3, 4, 5, 7, 8, 1, 2, 3, 5, 7, 9, 1, 2, 3, 5, 7, 10, 1, 2, 3, 5, 7, 11, 1, 2, 3, 4, 5, 6, 7, 11, 12, 1, 2, 3, 5, 7, 11, 13, 1, 2, 3, 5, 7, 11, 13, 14, 1, 2, 3, 5, 7, 11, 13, 15, 1, 2, 3, 4, 5, 7, 8, 11, 13, 16

Offset: 1

Omar E. Pol, Nov 04 2013

Also row n lists the divisors of n and the primes < n that do not divide n, in increasing order.
Also row n lists the nonprime divisors of n and the primes <= n, in increasing order.
Note that if n is 1 or prime then row n lists the first A036234(n) terms of A008578.
The motivation for this sequence is A046022 which is also the union of the odd primes and the divisors of 4. Here the n-th row of triangle can be interpreted as the initial terms of the infinite sequence defined as the union of the prime numbers and the divisors of n.

			For n = 10, the divisors of 10 are 1, 2, 5, 10. The primes less than 10 that do not divide 10 are 3 and 7. So row 10 is 1, 2, 3, 5, 7, 10.
On the other hand, the primes <= n are 2, 3, 5, 7. The nonprime divisors of n are 1, 10. So row 10 is 1, 2, 3, 5, 7, 10.
Written as an irregular triangle the sequence begins:
1;
1, 2;
1, 2, 3;
1, 2, 3, 4;
1, 2, 3, 5;
1, 2, 3, 5, 6;
1, 2, 3, 5, 7;
1, 2, 3, 4, 5, 7, 8;
1, 2, 3, 5, 7, 9;
1, 2, 3, 5, 7, 10;
1, 2, 3, 5, 7, 11;
1, 2, 3, 4, 5, 6, 7, 11, 12;
1, 2, 3, 5, 7, 11, 13;
1, 2, 3, 5, 7, 11, 13, 14;
1, 2, 3, 5, 7, 11, 13, 15;
1, 2, 3, 4, 5, 7, 8, 11, 13, 16;
1, 2, 3, 5, 7, 11, 13, 17;
1, 2, 3, 5, 6, 7, 9, 11, 13, 17, 18;
1, 2, 3, 5, 7, 11, 13, 17, 19;
1, 2, 3, 4, 5, 7, 10, 11, 13, 17, 19, 20;
1, 2, 3, 5, 7, 11, 13, 17, 19, 21;
1, 2, 3, 5, 7, 11, 13, 17, 19, 22;
1, 2, 3, 5, 7, 11, 13, 17, 19, 23;
1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 17, 19, 23, 24;

Columns 1-3: A000012, A007395, A010701.
Right border gives A000027.
Cf. A000005, A000040, A000720, A008578, A027750, A036234, A046022.

A239545 Decimal expansion of Sum_{k>=0} (-1)^k/((2k+1)(2k+3)(2k+5)).

Original entry on oeis.org

0, 5, 9, 3, 6, 5, 7, 4, 8, 3, 6, 5, 3, 9, 0, 8, 2, 1, 4, 7, 4, 4, 9, 7, 0, 8, 9, 5, 7, 6, 6, 0, 4, 5, 2, 7, 1, 9, 1, 3, 1, 2, 8, 4, 1, 5, 8, 8, 5, 5, 4, 8, 9, 4, 2, 8, 8, 5, 3, 4, 7, 4, 0, 7, 0, 5, 1, 4, 3, 7, 1, 7, 4, 5, 2, 4, 4, 2, 7, 9, 1, 4, 9, 5, 1, 7, 1, 0

Offset: 0

Bruno Berselli, Mar 21 2014

			0.0593657483653908214744970895766045271913128415885548942885347407051...

L. B. W. Jolley, Summation of Series, Dover, 1961, p. 44 (series n. 240)

Cf. A000796, A001539, A005408 (odd numbers) A021016 (Sum_{k>=0} 1/((2k+1)*(2k+3)*(2k+5))), A061550.

Mathematica
```
RealDigits[Pi/8 - 1/3, 10, 90, -1][[1]]
```

Pi/8-1/3 \\ Charles R Greathouse IV, May 05 2014

Equals Pi/8 - 1/3 = A019675 - A010701.

Equals Sum_{k>=1} 1/((4*k+1)*(4*k+3)) = Sum_{k>=1} 1/A001539(k). - Amiram Eldar, Jul 04 2020

A267942 Interleave (n-1)^2 + 2 and (n+1)^2 + 2.

Original entry on oeis.org

3, 3, 2, 6, 3, 11, 6, 18, 11, 27, 18, 38, 27, 51, 38, 66, 51, 83, 66, 102, 83, 123, 102, 146, 123, 171, 146, 198, 171, 227, 198, 258, 227, 291, 258, 326, 291, 363, 326, 402, 363, 443, 402, 486, 443, 531, 486, 578, 531, 627, 578, 678, 627, 731, 678, 786, 731

Offset: 0

Paul Curtz, Jan 22 2016

Trisections:
3,  6,  6,  27, 27, 66,  66, ... = 3*(1, 2, 2, 9, 9, 22, 22, ... ). See A056105.
3,  3,  18, 18, 51, 51, 102, ... = 3*(1, 1, 6, 6, 17, 17, ... ). See A056109.
2, 11,  11, 38, 38, 83,  83, ...   (== 2 (mod 9)).
The trisections also have the signature (1,2,-2,-1,1). The corresponding main sequence is 0, 0, 0, 0, 1, 1, 3, 3, ... = A161680(n) with each term duplicated.

			a(0) = (2+13)/5, a(1) = (13+2)/5, a(2) = (5+5)/5, a(3) = (29+1)/5, ... (using first formula).

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A000290, A007395, A010701, A022998, A056105, A056109, A059100, A161680, A174474, A193356, A255844, A261327.

&cat [[(n-1)^2+2, (n+1)^2+2]: n in [0..50]]; // Vincenzo Librandi, Jan 23 2016

Flatten[Table[{n^2 - 2 n + 3, n^2 + 2 n + 3}, {n, 0, 30}]] (* Vincenzo Librandi, Jan 23 2016 *)
CoefficientList[Series[(3 - 7 x^2 + 4 x^3 + 2 x^4)/((1 - x)^3 (1 + x)^2), {x, 0, 56}], x] (* Michael De Vlieger, Jan 24 2016 *)

Vec((3-7*x^2+4*x^3+2*x^4)/((1-x)^3*(1+x)^2) + O(x^100)) \\ Colin Barker, Jan 22 2016

a(n) = (A261327(n+2) + A261327(n-3))/5.
a(n+1) = a(n) + (-1)^n * A022998(n), a(0)=3.
a(n+3) = a(n) + 3*A193356(n), a(0)=a(1)=3, a(2)=2.
a(n) = 3 + A174474(n).
a(2n) + a(2n+1) = A255844(n).
From Colin Barker, Jan 22 2016: (Start)
a(n) = (2*n^2 - 6*(-1)^n*n - 2*n + 3*(-1)^n + 21)/8.
a(n) = (n^2 - 4*n + 12)/4 for n even.
a(n) = (n^2 + 2*n + 9)/4 for n odd.
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n > 4.
G.f.: (3 - 7*x^2 + 4*x^3 + 2*x^4) / ((1-x)^3*(1+x)^2).
(End)

More terms from Colin Barker, Jan 22 2016

A320661 a(n) = 2^(n+3) - 6*n - 7.

Original entry on oeis.org

1, 3, 13, 39, 97, 219, 469, 975, 1993, 4035, 8125, 16311, 32689, 65451, 130981, 262047, 524185, 1048467, 2097037, 4194183, 8388481, 16777083, 33554293, 67108719, 134217577, 268435299, 536870749, 1073741655, 2147483473, 4294967115, 8589934405, 17179868991

Offset: 0

Paul Curtz, Nov 14 2018

Companion to A247618 which has the same recurrence.

For this recurrence the main sequence is A000295.

Index entries for linear recurrences with constant coefficients, signature (4,-5,2).

Cf. A000225, A000295, A010701, A016921, A100402, A129756, A247618.

List([0..40], n -> 2^(n+3) -6*n -7); # G. C. Greubel, Nov 15 2018

[2^(n+3) -6*n -7: n in [0..40]]; // G. C. Greubel, Nov 15 2018

a[n_]:=2^(n+3) - 6*n - 7; Array[a,32,0] (* Amiram Eldar, Nov 14 2018 *)

vector(40, n, n--; 2^(n+3) -6*n -7) \\ G. C. Greubel, Nov 15 2018

[2^(n+3) -6*n -7 for n in range(40)] # G. C. Greubel, Nov 15 2018

a(n) = 4*a(n-1) - 5*a(n-2) + 2*a(n-3).
a(n+1) = a(n-1) + 12*A000225(n). a(-1) = 3.
a(2*n) mod 9 = period 3: repeat [1, 4, 7].
a(2*n+1) mod 9 = 3.
a(n) mod 9 = period 6: repeat [1, 3, 4, 3, 7, 3].
a(n) mod 10 = period 20: repeat [1, 3, 3, 9, 7, 9, 9, 5, 3, 5, 5, 1, 9, 1, 1, 7, 5, 7, 7, 3] = Im(n). Im(n-1) = [3, 1, 3, 3,  9, 7, 9, 9,  5, 3, 5, 5,  1, 9, 1, 1,  7, 5, 7, 7].  Disordered [1, 1, 1, 1, 3, 3, 3, 3, 5, 5, 5, 5, 7, 7, 7, 7, 9, 9, 9, 9].
a(n+1) - a(n) = 2^(n+3) - 6.
From G. C. Greubel, Nov 15 2018: (Start)
G.f.: (1-x+6*x^2)/((1-2*x)*(1-x)^2).
E.g.f.: 8*exp(2*x) - (7 + 6*x)*exp(x). (End)

More terms from Amiram Eldar, Nov 14 2018

A376322 (1/4) times obverse convolution (2)**(2^n + 1); see Comments.

Original entry on oeis.org

1, 5, 35, 385, 7315, 256025, 17153675, 2247131425, 582007039075, 299733625123625, 307826433001962875, 631352014087025856625, 2587911905742718986305875, 21207938067561582092776645625, 347534481113131645754330891856875, 11389052480558437163015177657041650625

Offset: 0

Clark Kimberling, Sep 20 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences.

Cf. A000051, A000079, A007395, A010701, A374848, A375054, A139030.

s[n_] := 2; t[n_] := 2^n + 1;
u[n_] := (1/4) Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]
(* or *)
Table[2^(n*(n+1)/2 - 2) * QPochhammer[-3, 1/2, n+1], {n, 0, 15}] (* Vaclav Kotesovec, Sep 20 2024 *)

a(n) = a(n-1)*A062709(n) for n>=1.

a(n) = (1/4)((3)**(2^n)) = (1/4)(A010701(n)**A000079(n)) for n>=0.

A376323 (1/5) times obverse convolution (3)**(2^n + 1); see Comments.

Original entry on oeis.org

1, 6, 48, 576, 11520, 414720, 28200960, 3722526720, 967856947200, 499414184755200, 513397781928345600, 1053492248516965171200, 4319318218919557201920000, 35401132122264690826936320000, 580153753219673753271832412160000, 19012798800515148242224491811307520000

Offset: 0

Clark Kimberling, Sep 20 2024

nonn

See A374848 for the definition of obverse convolution and a guide to related sequences.

Cf. A000051, A010701, A052548, A374848, A375053, A139030.

s[n_] := 3; t[n_] := 2^n + 1;
u[n_] := (1/5) Product[s[k] + t[n - k], {k, 0, n}];
Table[u[n], {n, 0, 20}]
(* or *)
Table[2^(n*(n+1)/2) * QPochhammer[-4, 1/2, n+1]/5, {n, 0, 15}] (* Vaclav Kotesovec, Sep 20 2024 *)

a(n) = 2 a(n-1)*A052548(n-1) for n>=1.

cp's OEIS Frontend

A363099 Triangle T(n,k) in which the n-th row encodes the inverse of a 3n+1 X 3n+1 Jacobi matrix, with 1's on the lower, main, and upper diagonals in GF(2), where the encoding consists of the decimal representations for the binary rows (n >= 1, 1 <= k <= 3n+1).

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A383641 a(n) is the difference between the sum of even composites and the sum of the odd composites in the first n positive integers.

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A387235 Decimal expansion of 2*log(2)/3.

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A174971 Periodic sequence: Repeat 3, -3.

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A229945 Triangle read by rows in which row n lists the union of the primes <= n and the divisors of n.

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A239545 Decimal expansion of Sum_{k>=0} (-1)^k/((2k+1)*(2k+3)*(2k+5)).

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A267942 Interleave (n-1)^2 + 2 and (n+1)^2 + 2.

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A239545 Decimal expansion of Sum_{k>=0} (-1)^k/((2k+1)(2k+3)(2k+5)).