A059268 -id:A059268 - cp's OEIS frontend

A131816 Triangle read by rows: A130321 + A059268 - A000012 as infinite lower triangular matrices, where A130321 = (1; 2,1; 4,2,1; ...), A059268 = (1; 1,2; 1,2,4; ...) and A000012 = (1; 1,1; 1,1,1; ...).

1, 2, 2, 4, 3, 4, 8, 5, 5, 8, 16, 9, 7, 9, 16, 32, 17, 11, 11, 17, 32, 64, 33, 19, 15, 19, 33, 64, 128, 65, 35, 23, 23, 35, 65, 128, 256, 129, 67, 39, 31, 39, 67, 129, 256, 512, 257, 131, 71, 47, 47, 71, 131, 257, 512, 1024, 513, 259, 135, 79, 63, 79, 135, 259, 513, 1024

Offset: 0

Gary W. Adamson, Jul 18 2007

Row sums = A000295: (1, 4, 11, 26, 57, 120, ...).
If we regard the sequence as an infinite square array read by diagonals then it has the formula U(n,k) = (2^n + 2^k)/2 - 1. This appears to coincide with the number of n X k 0..1 arrays colored with only straight tiles, and new values 0..1 introduced in row major order, i.e., no equal adjacent values form a corner. (Fill the array with 0's and 1's. There must never be 3 adjacent identical values making a corner, only same values in a straight line.) Some solutions with n = k = 4 are:
  0 1 0 1     0 1 0 0     0 0 0 1     0 0 1 1     0 1 0 1
  1 0 1 0     1 0 1 1     1 1 1 0     1 1 0 0     1 0 1 0
  1 0 1 0     0 1 0 0     0 0 0 1     0 0 1 1     0 1 0 1
  0 1 0 1     1 0 1 1     1 1 1 0     1 1 0 0     0 1 0 1
(Observation from R. H. Hardin, cf. link.) - M. F. Hasler and N. J. A. Sloane, Feb 26 2013

			First few rows of the triangle:
    1;
    2,  2;
    4,  3,  4;
    8,  5,  5,  8;
   16,  9,  7,  9, 16;
   32, 17, 11, 11, 17, 32;
   64, 33, 19, 15, 19, 33, 64;
  128, 65, 35, 23, 23, 35, 65, 128;
  ...

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
R. H. Hardin, Post to the SeqFan list, Feb 26 2013

Cf. A130321, A059268, A000012.

Row sums give A000295(n+2).

a131816 n k = a131816_tabl !! n !! k
a131816_row n = a131816_tabl !! n
a131816_tabl = map (map (subtract 1)) $
   zipWith (zipWith (+)) a130321_tabl a059268_tabl
-- Reinhard Zumkeller, Feb 27 2013

Table[Table[((2^(m + 1) - 1) + (2^(n - m + 1) - 1))/2, {m, 0, n}], {n, 0, 10}]; Flatten[%] (* Roger L. Bagula, Oct 16 2008 *)

T(n,m) = ((2^(m + 1) - 1) + (2^(n - m + 1) - 1))/2. - Roger L. Bagula, Oct 16 2008

Edited by N. J. A. Sloane, Jul 01 2008 at the suggestion of R. J. Mathar

A130459 A059268 * A097806.

Original entry on oeis.org

1, 3, 2, 3, 6, 4, 3, 6, 12, 8, 3, 6, 12, 24, 16, 3, 6, 12, 24, 48, 32, 3, 6, 12, 24, 48, 96, 64, 3, 6, 12, 24, 48, 96, 192, 128, 3, 6, 12, 24, 48, 96, 192, 384, 256

Offset: 1

Gary W. Adamson, May 26 2007

Row sums = A036563 starting (1, 5, 13, 29, 61, 125, ...).

			First few rows of the triangle:
  1;
  3, 2;
  3, 6,  4;
  3, 6, 12,  8;
  3, 6, 12, 24, 16;
  3, 6, 12, 24, 48, 32;
  ...

Cf. A059268, A097806, A036563.

A059268 * A097806 as infinite lower triangular matrices. A059268 = [1; 1,2; 1,2,4; ...]. A097806 = the pairwise operator.

A130324 A059268^2.

Original entry on oeis.org

1, 3, 4, 7, 12, 16, 15, 28, 48, 64, 31, 60, 112, 192, 256, 63, 124, 240, 448, 768, 1024, 127, 252, 496, 960, 1792, 3072, 4096, 255, 508, 1008, 1984, 3840, 7168, 12288, 16384, 511, 1020, 2032, 4032, 7936, 15360, 28672, 49152, 65536

Offset: 0

Gary W. Adamson, May 24 2007

Row sums = A006095: (1, 7, 35, 155, 651, 2667, ...).

			First few rows of the triangle:
   1;
   3,   4;
   7,  12,  16;
  15,  28,  48,  64;
  31,  60, 112, 192, 256;
  63, 124, 240, 448, 768, 1024;
  ...

Cf. A059268, A006095.

A059268^2 as an infinite lower triangular matrix, where A059268 = (1; 1,2; 1,2,4; ...).

A130329 A059268 * A130321.

Original entry on oeis.org

1, 5, 2, 21, 10, 4, 85, 42, 20, 8, 341, 170, 84, 40, 16, 1365, 682, 340, 168, 80, 32, 5461, 2730, 1364, 680, 336, 160, 64, 21845, 10922, 5460, 2728, 1360, 672, 320, 128, 87381, 43690, 21844, 10920, 5456, 2720, 1344, 640, 256

Offset: 0

Gary W. Adamson, May 24 2007

Row sums = A006095: (1, 7, 35, 155, 651, 2667, ...).
Left border = A002450: (1, 5, 21, 85, 341, 1365, ...).
A130328 = A130321 * A059268.

			First few rows of the triangle:
    1;
    5,   2;
   21,  10,  4;
   85,  42, 20,  8;
  341, 170, 84, 40, 16;
  ...

Cf. A059268, A130321, A006095, A002450, A130328.

A059268 * A130321 as infinite lower triangular matrices.

A131953 A130321 + A059268 - A000012(signed).

Original entry on oeis.org

1, 4, 2, 4, 5, 4, 10, 5, 7, 8, 16, 11, 7, 11, 16, 34, 17, 13, 11, 19, 32, 64, 35, 19, 17, 19, 35, 64, 130, 65, 37, 23, 25, 35, 67, 128, 256, 131, 67, 41, 31, 41, 67, 131, 256, 514, 257, 133, 71, 49, 47, 73, 131, 259, 512

Offset: 0

Gary W. Adamson, Jul 30 2007

Row sums = A101622 starting (1, 6, 13, 30, 61, 126, ...).

			First few rows of the triangle:
   1;
   4,  2;
   4,  5,  4;
  10,  5,  7,  8;
  16, 11,  7, 11, 16;
  34, 17, 13, 11, 19, 32;
  64, 35, 19, 17, 19, 35, 64;
  ...

Cf. A130321, A059268, A101622.

A130321 + A059268 - A000012 (signed + - + -, ... by columns) as infinite lower triangular matrices.

A132110 A007318 + A059268 - A000012 as infinite lower triangular matrices.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 1, 4, 6, 8, 1, 5, 9, 11, 16, 1, 6, 13, 17, 20, 32, 1, 7, 18, 27, 30, 37, 64, 1, 8, 24, 42, 50, 52, 70, 128, 1, 9, 31, 63, 85, 87, 91, 135, 256, 1, 10, 39, 91, 141, 157, 147, 163, 264, 512

Offset: 0

Gary W. Adamson, Aug 09 2007

Row sums = A079583: (1, 3, 8, 19, 42, 89, 184, ...).

			First few rows of the triangle:
  1;
  1, 2;
  1, 3,  4;
  1, 4,  6,  8;
  1, 5,  9, 11, 16;
  1, 6, 13, 17, 20, 32;
  1, 7, 18, 27, 30, 37, 64;
  1, 8, 24, 42, 50, 52, 70, 128;
  ...

Cf. A007318, A059268, A000012, A079583.

A288349 Partial sums of A059268.

Original entry on oeis.org

1, 2, 4, 5, 7, 11, 12, 14, 18, 26, 27, 29, 33, 41, 57, 58, 60, 64, 72, 88, 120, 121, 123, 127, 135, 151, 183, 247, 248, 250, 254, 262, 278, 310, 374, 502, 503, 505, 509, 517, 533, 565, 629, 757, 1013, 1014, 1016, 1020, 1028, 1044, 1076, 1140, 1268, 1524, 2036

Offset: 1

Zhining Yang, Jun 08 2017

nonn

			For n=10, a(10) = 1 + 1 + 2 + 1 + 2 + 4 + 1 + 2 + 4 + 8 = 26.

Cf. A059268.

Accumulate@ Flatten@ Array[2^Range[0, #] &, 10, 0] (* Michael De Vlieger, Jun 11 2017 *)

for(n=1, 100, t=floor(sqrt(2*n)+1/2); print1(2^t+2^(n-t*(t-1)/2)-t-2, ", "));

a(n) = 2^t + 2^(n-t*(t-1)/2) - t - 2, where t = floor(sqrt(2*n) + 1/2), n>=1.

A130453 A097806 * A059268.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 2, 4, 8, 8, 2, 4, 8, 16, 16, 2, 4, 8, 16, 32, 32, 2, 4, 8, 16, 32, 64, 64, 2, 4, 8, 16, 32, 64, 128, 128

Offset: 1

Gary W. Adamson, May 26 2007

Row sums = A033484: (1, 4, 10, 22, 46, 94, ...).

A130459 = A059268 * A097806.

			First few rows of the triangle:
  1;
  2, 2;
  2, 4, 4;
  2, 4, 8,  8;
  2, 4, 8, 16, 16;
  2, 4, 8, 16, 32, 32;
  ...

Cf. A097806, A059268, A033484, A130459.

A097806 * A059268 as infinite lower triangular matrices.

A018900 Sums of two distinct powers of 2.

Original entry on oeis.org

3, 5, 6, 9, 10, 12, 17, 18, 20, 24, 33, 34, 36, 40, 48, 65, 66, 68, 72, 80, 96, 129, 130, 132, 136, 144, 160, 192, 257, 258, 260, 264, 272, 288, 320, 384, 513, 514, 516, 520, 528, 544, 576, 640, 768, 1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2049, 2050, 2052, 2056, 2064, 2080, 2112, 2176, 2304, 2560, 3072

Offset: 1

Jonn Dalton (jdalton(AT)vnet.ibm.com), Dec 11 1996

Appears to give all k such that 8 is the highest power of 2 dividing A005148(k). - Benoit Cloitre, Jun 22 2002
Seen as a triangle read by rows, T(n,k) = 2^(k-1) + 2^n, 1 <= k <= n, the sum of the n-th row equals A087323(n). - Reinhard Zumkeller, Jun 24 2009
Numbers whose base-2 sum of digits is 2. - Tom Edgar, Aug 31 2013
All odd terms are A000051. - Robert G. Wilson v, Jan 03 2014
A239708 holds the subsequence of terms m such that m - 1 is prime. - Hieronymus Fischer, Apr 20 2014

			From _Hieronymus Fischer_, Apr 27 2014: (Start)
a(1) = 3, since 3 = 2^1 + 2^0.
a(5) = 10, since 10 = 2^3 + 2^1.
a(10^2) = 16640
a(10^3) = 35184372089344
a(10^4) = 2788273714550169769618891533295908724670464 = 2.788273714550...*10^42
a(10^5) = 3.6341936214780344527466190...*10^134
a(10^6) = 4.5332938264998904048012398...*10^425
a(10^7) = 1.6074616084721302346802429...*10^1346
a(10^8) = 1.4662184497310967196301632...*10^4257
a(10^9) = 2.3037539289782230932863807...*10^13462
a(10^10) = 9.1836811272250798973464436...*10^42571
(End)

T. D. Noe and Hieronymus Fischer, Table of n, a(n) for n = 1..10000 [terms 1..1000 from T. D. Noe]
Michael Beeler, R. William Gosper, and Richard Schroeppel, HAKMEM, MIT Artificial Intelligence Laboratory report AIM-239, February 1972. Item 175 page 81 by Gosper for iterating. Also HTML transcription.
Tilman Piesk, Square array in reverse binary

Cf. A000120, A001969, A048639, A048645, A057168.
Cf. A000217, A179951, A187813, A239708.
Cf. A000079, A014311, A014312, A014313, A023688, A023689, A023690, A023691 (Hamming weight = 1, 3, 4, ..., 9).
Sum of base-b digits equal b: A226636 (b = 3), A226969 (b = 4), A227062 (b = 5), A227080 (b = 6), A227092 (b = 7), A227095 (b = 8), A227238 (b = 9), A052224 (b = 10). - M. F. Hasler, Dec 23 2016

unsigned hakmem175(unsigned x) {
    unsigned s, o, r;
    s = x & -x; r = x + s;
    o = x ^ r;  o = (o >> 2) / s;
    return r | o;
}
unsigned A018900(int n) {
    if (n == 1) return 3;
    return hakmem175(A018900(n - 1));
} // Peter Luschny, Jan 01 2014

a018900 n = a018900_list !! (n-1)
a018900_list = elemIndices 2 a073267_list  -- Reinhard Zumkeller, Mar 07 2012

a:= n-> (i-> 2^i+2^(n-1-i*(i-1)/2))(floor((sqrt(8*n-1)+1)/2)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 01 2022

Select[ Range[ 1056 ], (Count[ IntegerDigits[ #, 2 ], 1 ]==2)& ]
Union[Total/@Subsets[2^Range[0,10],{2}]] (* Harvey P. Dale, Mar 04 2012 *)

for(m=1,9,for(n=0,m-1,print1(2^m+2^n", "))) \\ Charles R Greathouse IV, Sep 09 2011

is(n)=hammingweight(n)==2 \\ Charles R Greathouse IV, Mar 03 2014

for(n=0,10^5,if(hammingweight(n)==2,print1(n,", "))); \\ Joerg Arndt, Mar 04 2014

a(n)= my(t=sqrtint(n*8)\/2); 2^t + 2^(n-1-t*(t-1)/2); \\ Ruud H.G. van Tol, Nov 30 2024

print([n for n in range(1, 3001) if bin(n)[2:].count("1")==2]) # Indranil Ghosh, Jun 03 2017

A018900_list = [2**a+2**b for a in range(1,10) for b in range(a)] # Chai Wah Wu, Jan 24 2021

from math import isqrt, comb
def A018900(n): return (1<<(m:=isqrt(n<<3)+1>>1))+(1<<(n-1-comb(m,2))) # Chai Wah Wu, Oct 30 2024

distinctPowersOf: b
  "Version 1: Answers the n-th number of the form b^i + b^j, i>j>=0, where n is the receiver.
  b > 1 (b = 2, for this sequence).
  Usage: n distinctPowersOf: 2
  Answer: a(n)"
  | n i j |
  n := self.
  i := (8*n - 1) sqrtTruncated + 1 // 2.
  j := n - (i*(i - 1)/2) - 1.
  ^(b raisedToInteger: i) + (b raisedToInteger: j)
[by Hieronymus Fischer, Apr 20 2014]
------------

distinctPowersOf: b
  "Version 2: Answers an array which holds the first n numbers of the form b^i + b^j, i>j>=0, where n is the receiver. b > 1 (b = 2, for this sequence).
  Usage: n distinctPowersOf: 2
  Answer: #(3 5 6 9 10 12 ...) [first n terms]"
  | k p q terms |
  terms := OrderedCollection new.
  k := 0.
  p := b.
  q := 1.
  [k < self] whileTrue:
         [[q < p and: [k < self]] whileTrue:
                   [k := k + 1.
                   terms add: p + q.
                   q := b * q].
         p := b * p.
         q := 1].
  ^terms as Array
[by Hieronymus Fischer, Apr 20 2014]
------------

floorDistinctPowersOf: b
  "Answers an array which holds all the numbers b^i + b^j < n, i>j>=0, where n is the receiver.
  b > 1 (b = 2, for this sequence).
  Usage: n floorDistinctPowersOf: 2
  Answer: #(3 5 6 9 10 12 ...) [all terms < n]"
  | a n p q terms |
  terms := OrderedCollection new.
  n := self.
  p := b.
  q := 1.
  a := p + q.
  [a < n] whileTrue:
         [[q < p and: [a < n]] whileTrue:
                   [terms add: a.
                   q := b * q.
                   a := p + q].
         p := b * p.
         q := 1.
         a := p + q].
  ^terms as Array
[by Hieronymus Fischer, Apr 20 2014]
------------

invertedDistinctPowersOf: b
  "Given a number m which is a distinct power of b, this method answers the index n such that there are uniquely defined i>j>=0 for which b^i + b^j = m, where m is the receiver;  b > 1 (b = 2, for this sequence).
  Usage: m invertedDistinctPowersOf: 2
  Answer: n such that a(n) = m, or, if no such n exists, min (k | a(k) >= m)"
  | n i j k m |
  m := self.
  i := m integerFloorLog: b.
  j := m - (b raisedToInteger: i) integerFloorLog: b.
  n := i * (i - 1) / 2 + 1 + j.
  ^n
[by Hieronymus Fischer, Apr 20 2014]

a(n) = 2^trinv(n-1) + 2^((n-1)-((trinv(n-1)*(trinv(n-1)-1))/2)), i.e., 2^A002024(n)+2^A002262(n-1). - Antti Karttunen
a(n) = A059268(n-1) + A140513(n-1). A000120(a(n)) = 2. Complement of A161989. A151774(a(n)) = 1. - Reinhard Zumkeller, Jun 24 2009
A073267(a(n)) = 2. - Reinhard Zumkeller, Mar 07 2012
Start with A000051. If n is in sequence, then so is 2n. - Ralf Stephan, Aug 16 2013
a(n) = A057168(a(n-1)) for n>1 and a(1) = 3. - Marc LeBrun, Jan 01 2014
From Hieronymus Fischer, Apr 20 2014: (Start)
Formulas for a general parameter b according to a(n) = b^i + b^j, i>j>=0; b = 2 for this sequence.
a(n) = b^i + b^j, where i = floor((sqrt(8n - 1) + 1)/2), j = n - 1 - i*(i - 1)/2 [for a Smalltalk implementation see Prog section, method distinctPowersOf: b (2 versions)].
a(A000217(n)) = (b + 1)*b^(n-1) = b^n + b^(n-1).
a(A000217(n)+1) = 1 + b^(n+1).
a(n + 1 + floor((sqrt(8n - 1) + 1)/2)) = b*a(n).
a(n + 1 + floor(log_b(a(n)))) = b*a(n).
a(n + 1) = b^2/(b+1) * a(n) + 1, if n is a triangular number (s. A000217).
a(n + 1) = b*a(n) + (1-b)* b^floor((sqrt(8n - 1) + 1)/2), if n is not a triangular number.
The next term can also be calculated without using the index n. Let m be a term and i = floor(log_b(m)), then:
a(n + 1) = b*m + (1-b)* b^i, if floor(log_b(m/(b+1))) + 1 < i,
a(n + 1) = b^2/(b+1) * m + 1, if floor(log_b(m/(b+1))) + 1 = i.
Partial sum:
Sum_{k=1..n} a(k) = ((((b-1)*(j+1)+i-1)*b^(i-j) + b)*b^j - i)/(b-1), where i = floor((sqrt(8*n - 1) + 1)/2), j = n - 1 - i*(i - 1)/2.
Inverse:
For each sequence term m, the index n such that a(n) = m is determined by n := i*(i-1)/2 + j + 1, where i := floor(log_b(m)), j := floor(log_b(m - b^floor(log_b(m)))) [for a Smalltalk implementation see Prog section, method invertedDistinctPowersOf: b].
Inequalities:
a(n) <= (b+1)/b * b^floor(sqrt(2n)+1/2), equality holds for triangular numbers.
a(n) > b^floor(sqrt(2n)+1/2).
a(n) < b^sqrt(2n)*sqrt(b).
a(n) > b^sqrt(2n)/sqrt(b).
Asymptotic behavior:
lim sup a(n)/b^sqrt(2n) = sqrt(b).
lim inf a(n)/b^sqrt(2n) = 1/sqrt(b).
lim sup a(n)/b^(floor(sqrt(2n))) = b.
lim inf a(n)/b^(floor(sqrt(2n))) = 1.
lim sup a(n)/b^(floor(sqrt(2n)+1/2)) = (b+1)/b.
lim inf a(n)/b^(floor(sqrt(2n)+1/2)) = 1.
(End)
Sum_{n>=1} 1/a(n) = A179951. - Amiram Eldar, Oct 06 2020

Edited by M. F. Hasler, Dec 23 2016

A173786 Triangle read by rows: T(n,k) = 2^n + 2^k, 0 <= k <= n.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 36, 40, 48, 64, 65, 66, 68, 72, 80, 96, 128, 129, 130, 132, 136, 144, 160, 192, 256, 257, 258, 260, 264, 272, 288, 320, 384, 512, 513, 514, 516, 520, 528, 544, 576, 640, 768, 1024, 1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048

Offset: 0

Reinhard Zumkeller, Feb 28 2010

Essentially the same as A048645. - T. D. Noe, Mar 28 2011

			Triangle begins as:
     2;
     3,    4;
     5,    6,    8;
     9,   10,   12,   16;
    17,   18,   20,   24,   32;
    33,   34,   36,   40,   48,   64;
    65,   66,   68,   72,   80,   96,  128;
   129,  130,  132,  136,  144,  160,  192,  256;
   257,  258,  260,  264,  272,  288,  320,  384,  512;
   513,  514,  516,  520,  528,  544,  576,  640,  768, 1024;
  1025, 1026, 1028, 1032, 1040, 1056, 1088, 1152, 1280, 1536, 2048;

T. D. Noe, Rows n = 0..100 of triangle, flattened
Wawrzyniec Bieniawski, Piotr Masierak, Andrzej Tomski, and Szymon Łukaszyk, Assembly Theory - Formalizing Assembly Spaces and Discovering Patterns and Bounds, Preprints.org (2025).

Cf. A048645, A118413, A118416.

Cf. also A087112, A370121.

[2^n + 2^k: k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 07 2021

Flatten[Table[2^n + 2^m, {n,0,10}, {m, 0, n}]] (* T. D. Noe, Jun 18 2013 *)

A173786(n) = { my(c = (sqrtint(8*n + 1) - 1) \ 2); 1 << c + 1 << (n - binomial(c + 1, 2)); }; \\ Antti Karttunen, Feb 29 2024, after David A. Corneth's PARI-program in A048645

from math import isqrt, comb
def A173786(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    return (1<Chai Wah Wu, Jun 20 2025

flatten([[2^n + 2^k for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 07 2021

1 <= A000120(T(n,k)) <= 2.
For n>0, 0<=kA048645(n+1,k+2) and T(n,n) = A048645(n+2,1).
Row sums give A006589(n).
Central terms give A161168(n).
T(2*n+1,n) = A007582(n+1).
T(2*n+1,n+1) = A028403(n+1).
T(n,k) = A140513(n,k) - A173787(n,k), 0<=k<=n.
T(n,k) = A059268(n+1,k+1) + A173787(n,k), 0T(n,k) * A173787(n,k) = A173787(2*n,2*k), 0<=k<=n.
T(n,0) = A000051(n).
T(n,1) = A052548(n) for n>0.
T(n,2) = A140504(n) for n>1.
T(n,3) = A175161(n-3) for n>2.
T(n,4) = A175162(n-4) for n>3.
T(n,5) = A175163(n-5) for n>4.
T(n,n-4) = A110287(n-4) for n>3.
T(n,n-3) = A005010(n-3) for n>2.
T(n,n-2) = A020714(n-2) for n>1.
T(n,n-1) = A007283(n-1) for n>0.
T(n,n) = 2*A000079(n).

Typo in first comment line fixed by Reinhard Zumkeller, Mar 07 2010

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A131816 Triangle read by rows: A130321 + A059268 - A000012 as infinite lower triangular matrices, where A130321 = (1; 2,1; 4,2,1; ...), A059268 = (1; 1,2; 1,2,4; ...) and A000012 = (1; 1,1; 1,1,1; ...).

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A130459 A059268 * A097806.

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Formula

A130324 A059268^2.

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Formula

A130329 A059268 * A130321.

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Formula

A131953 A130321 + A059268 - A000012(signed).

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A132110 A007318 + A059268 - A000012 as infinite lower triangular matrices.

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A288349 Partial sums of A059268.

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Mathematica

PARI

Formula

A130453 A097806 * A059268.

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A018900 Sums of two distinct powers of 2.

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