A095121 -id:A095121 - cp's OEIS frontend

A378299 Read the binary representation of n from the most to least significant bit then perform a cumulative XOR and store by reading from least to most significant bit.

0, 1, 1, 2, 1, 6, 2, 5, 1, 14, 6, 9, 2, 13, 5, 10, 1, 30, 14, 17, 6, 25, 9, 22, 2, 29, 13, 18, 5, 26, 10, 21, 1, 62, 30, 33, 14, 49, 17, 46, 6, 57, 25, 38, 9, 54, 22, 41, 2, 61, 29, 34, 13, 50, 18, 45, 5, 58, 26, 37, 10, 53, 21, 42, 1, 126, 62, 65, 30, 97, 33, 94

Offset: 0

Darío Clavijo, Nov 22 2024

a(n) is Gray-coded into the reversed binary representation of n.

Fixed points are 0 and f(n) = 8*f(n-1) + 5 with f(1)=1 or f(n) = (1/14)*(3*(2^(3*n))-10) for n >= 1 (cf. A380001).

			For n = 75 a(75) = 78 because:
75 in base 2 is 1001011 and in base 2:
  m      | x = x XOR (m AND 1) | o
---------+---------------------+----------
1001011  | 1 = 0 XOR 1         |       1
100101   | 0 = 1 XOR 1         |      10
10010    | 0 = 0 XOR 0         |     100
1001     | 1 = 0 XOR 1         |    1001
100      | 1 = 1 XOR 0         |   10011
10       | 1 = 1 XOR 0         |  100111
1        | 0 = 1 XOR 1         | 1001110
And 1001110 in base 10: 78

Paolo Xausa, Table of n, a(n) for n = 0..16383
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2^13-1, with a color function showing a(n) according to floor(log_2(a(n))).
Index entries for sequences related to binary expansion of n

Cf. A000079, A000225, A000051, A000975, A007283, A006068, A030101, A095121.

Cf. A380001 (fixed points).

A378299[n_] := FromDigits[FoldList[BitXor, 0, Reverse[IntegerDigits[n, 2]]], 2];
Array[A378299, 100, 0] (* Paolo Xausa, Dec 13 2024 *)

def a(n):
  m,x,o = n,0,0
  while m > 0:
    x ^= (m & 1)
    o <<= 1
    o |= x
    m >>=1
  return o
print([a(n) for n in range(0,71)])

a(n) = A006068(A030101(n)).
a(A000079(n)) = 1.
a(A007283(n)) = 2.
a(A000225(n)) = A000975(n).
a(A000051(n)) = A095121(n).

A127509 Number of n-tuples where each entry is chosen from the subsets of {1,2,3} such that the intersection of all n entries contains exactly one element.

Original entry on oeis.org

3, 27, 147, 675, 2883, 11907, 48387, 195075, 783363, 3139587, 12570627, 50307075, 201277443, 805208067, 3221028867, 12884508675, 51538821123, 206156857347, 824630575107, 3298528591875, 13194126950403, 52776532967427

Offset: 1

Peter C. Heinig (algorithms(AT)gmx.de), Apr 17 2007

nonn

There is the following general formula: The number T(n,k,r) of n-tuples where each entry is chosen from the subsets of {1,2,..,k} such that the intersection of all n entries contains exactly r elements is: T(n,k,r) = C(k,r) * (2^n - 1)^(k-r). This may be shown by exhibiting a bijection to a set whose cardinality is obviously C(k,r) * (2^n - 1)^(k-r), namely the set of all k-tuples where each entry is chosen from subsets of {1,..,n} in the following way: Exactly r entries must be {1,..,n} itself (there are C(k,r) ways to choose them) and the remaining (k-r) entries must be chosen from the 2^n-1 proper subsets of {1,..,n}, i.e. for each of the (k-r) entries, {1,..,n} is forbidden (there are, independent of the choice of the full entries, (2^n - 1)^(k-r) possibilities to do that, hence the formula). The bijection into this set is given by (X_1,..,X_n) |-> (Y_1,..,Y_k) where for each j in {1,..,k} and each i in {1,..,n}, i is in Y_j if and only if j is in X_i.

			a(1)=3 because the three sequences of length one are: ({1}), ({2}), ({3}).
a(2)=27 because the twenty-seven sequences of length two are:
  ({1},{1}), ({2},{2}), ({3},{3}), ({1},{1,2}),
  ({1},{1,3}), ({2},{1,2}), ({2},{2,3}), ({3},{1,3}),
  ({3},{2,3}), ({1,2},{1}), ({1,3},{1}), ({1,2},{2}),
  ({2,3},{2}), ({1,3},{3}), ({2,3},{3}), ({1},{1,2,3}),
  ({2},{1,2,3}), ({3},{1,2,3}), ({1,2,3},{1}), ({1,2,3},{2}),
  ({1,2,3},{3}), ({1,2},{1,3}), ({1,3},{1,2}), ({1,2},{2,3}),
  ({2,3},{1,2}), ({1,3},{2,3}), ({2,3},{1,3}).

Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A128831, A128832, A128833, A095121.

Maple
```
for k from 1 to 41 do 3*(2^k-1)^2; od;
```

LinearRecurrence[{7,-14,8},{3,27,147},22] (* James C. McMahon, Jan 02 2025 *)

a(n) = 3*(2^n-1)^2.

G.f.: 3*x*(1+2*x)/(1-7*x+14*x^2-8*x^3). [Colin Barker, Feb 08 2012]

A131131 4A007318 - 3A097806.

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 4, 12, 9, 1, 4, 16, 24, 13, 1, 4, 20, 40, 40, 17, 1, 4, 24, 60, 80, 60, 21, 1, 4, 28, 84, 140, 140, 84, 25, 1, 4, 32, 112, 224, 280, 224, 112, 29, 1

Offset: 0

Gary W. Adamson, Jun 16 2007

Row sums = A131130, (1, 2, 10, 26, 52, 98, 190, ...), the binomial transform of (1, 1, 7, 1, 7, 1, ...). Generally, triangles generated from N*A007318 - (N-1)*A097806 have row sums that are binomial transforms of (1, 1, (N-1), 1, (N-1), 1, ...). A095121 = (1, 2, 6, 14, 30, 62, ...), the binomial transform of (1, 1, 3, 1, 3, 1, ...) and = row sums of A131108.

Triangle T(n,k), 0 <= k <= n,read by rows given by [1,3,-4,1,0,0,0,0,0,0,0,...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 18 2007

			First few rows of the triangle:
  1;
  1,  1;
  4,  5,  1;
  4, 12,  9,  1;
  4, 16, 24, 13,  1
  4, 20, 40, 40, 17,  1;
  ...

Cf. A131130, A131129, A131128, A131127, A046055, A131108, A095121, A097806.

4*A007318 - 3*A097806, where A007318 = Pascal's triangle and A097806 = the pairwise operator.

G.f.: (1-x*y+3*x^2+3*x^2*y)/((-1+x+x*y)*(x*y-1)). - R. J. Mathar, Aug 12 2015

A166065 Triangle, read by rows, given by [0,1,1,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 0, 4, 2, 2, 0, 8, 4, 2, 2, 0, 16, 8, 4, 2, 2, 0, 32, 16, 8, 4, 2, 2, 0, 64, 32, 16, 8, 4, 2, 2, 0, 128, 64, 32, 16, 8, 4, 2, 2, 0, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 2048, 1024

Offset: 0

Philippe Deléham, Oct 05 2009

			Triangle begins :
1,
0,2,
0,2,2,
0,4,2,2,
0,8,4,2,2,
0,16,8,4,2,2,
0,32,16,8,4,2,2,
0,64,32,16,8,4,2,2,
0,128,64,32,16,8,4,2,2,
0,256,128,64,32,16,8,4,2,2,
0,512,256,128,64,32,16,8,4,2,2,

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A084247(n), A000007(n), A000079(n), A001787(n+1), A166060(n), A165665(n), A083585(n) for x= -1, 0, 1, 2, 3, 4, 5 respectively. Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A040000(n), A000079(n), A095121(n), A047851(n), A047853(n), A047855(n) for x = 0, 1, 2, 3, 4, 5 respectively.

G.f.: (1-2*x+x*y)/((-1+2*x)*(x*y-1)). - R. J. Mathar, Aug 11 2015

A255049 a(n) = 2*A160552(n).

Original entry on oeis.org

0, 2, 2, 6, 2, 6, 10, 14, 2, 6, 10, 14, 10, 22, 34, 30, 2, 6, 10, 14, 10, 22, 34, 30, 10, 22, 34, 38, 42, 78, 98, 62, 2, 6, 10, 14, 10, 22, 34, 30, 10, 22, 34, 38, 42, 78, 98, 62, 10, 22, 34, 38, 42, 78, 98, 70, 42, 78, 106, 118, 162, 254, 258, 126, 2, 6, 10, 14, 10

Offset: 0

Omar E. Pol, Feb 13 2015

			Written as an irregular triangle in which row lengths are the terms of A011782 the sequence begins:
0;
2;
2,6;
2,6,10,14;
2,6,10,14,10,22,34,30;
2,6,10,14,10,22,34,30,10,22,34,38,42,78,98,62;
2,6,10,14,10,22,34,30,10,22,34,38,42,78,98,62,10,22,34,38,42,78,98,70,42,78,106,118,162,254,258,126;
It appears that row sums give 0 together with A004171, (see also A081294).
It appears that right border gives the nonnegative terms of A000918, (see also A095121).

Cf. A000918, A004171, A081294, A095121, A139250, A160164, A160552, A169707, A169708, A187220.

It appears that a(n) = A169708(n)/2, n >= 1.

Edited by Omar E. Pol, Feb 18 2015

A351839 Triangle read by rows: T(n, k) = A027375(n)Sum_{m=1..floor(n/k)} binomial(n, km).

Original entry on oeis.org

2, 6, 2, 14, 6, 6, 30, 14, 24, 12, 62, 30, 60, 60, 30, 126, 62, 126, 180, 180, 54, 254, 126, 252, 420, 630, 378, 126, 510, 254, 504, 852, 1680, 1512, 1008, 240, 1022, 510, 1014, 1620, 3780, 4536, 4536, 2160, 504, 2046, 1022, 2040, 3060, 7590, 11340, 15120, 10800, 5040, 990

Offset: 1

Stefano Spezia, Feb 21 2022

T(n, k) is the number of k-th roots of unity as eigenvalues of the quantum operator O for a free Motzkin spin chain of length n. For k = 1, it gives the correct result if one excludes the eigenvalue 2.

For the definitions of both a free Motzkin spin chain and the quantum operator O, see Hao et al.

			Triangle begins:
    2;
    6,   2;
   14,   6,   6;
   30,  14,  24,  12;
   62,  30,  60,  60,  30;
  126,  62, 126, 180, 180,  54;
  254, 126, 252, 420, 630, 378, 126;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Kun Hao, Olof Salberger, and Vladimir Korepin, Can a spin chain relate combinatorics to number theory?, arXiv:2202.07647 [quant-ph], 2022. See pp. 9-10.

Cf. A000918 (k = 2), A007318, A024023 (row sums), A027375 (leading diagonal), A095121 (k = 1).

g[n_]:= DivisorSum[n,(2^#)*MoebiusMu[n/#]&]; binomSum[n_,k_]:=Sum[Binomial[n, i],{i,k,n,k}]; T[n_,k_]:=g[k]*binomSum[n,k]; (* See p. 9 in Hao et al. *)
Flatten[Table[T[n,k],{n,10},{k,n}]]

T(n,k) = sumdiv(k,d,moebius(d)*2^(k/d))*sum(m=1,n\k,binomial(n,k*m)) \\ Andrew Howroyd, Feb 21 2022

A154312 Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,0,...] DELTA [2,-1/2,-1/2,2,0,0,0,0,0,0,0 ...] where DELTA is the operator defined in A084938 .

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 0, 3, 5, 0, 0, 0, 7, 9, 0, 0, 0, 0, 15, 17, 0, 0, 0, 0, 0, 31, 33, 0, 0, 0, 0, 0, 0, 63, 65, 0, 0, 0, 0, 0, 0, 0, 127, 129, 0, 0, 0, 0, 0, 0, 0, 0, 255, 257, 0, 0, 0, 0, 0, 0, 0, 0, 0, 511, 513, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1023, 1025, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2047

Offset: 0

Philippe Deléham, Jan 07 2009

Column sums give A003945.

			Triangle begins:
1;
0, 2;
0, 1, 3;
0, 0, 3, 5;
0, 0, 0, 7, 9;
0, 0, 0, 0, 15, 17; ...

Cf. A000225, A094373

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)= A040000(n), A094373(n), A000079(n), A083329(n), A095121(n), A154117(n), A131128(n), A154118(n), A131130(n), A154251(n), A154252(n) for x = -1,0,1,2,3,4,5,6,7,8,9 respectively.
G.f.: (1-x*y+x^2*y-x^2*y^2)/(1-3*x*y+2*x^2*y^2). - Philippe Deléham, Nov 02 2013
T(n,k) = 3*T(n-1,k-1) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(2,0) = 0, T(2,1) = 1, T(2,2) = 3, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 02 2013

cp's OEIS Frontend

A378299 Read the binary representation of n from the most to least significant bit then perform a cumulative XOR and store by reading from least to most significant bit.

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A127509 Number of n-tuples where each entry is chosen from the subsets of {1,2,3} such that the intersection of all n entries contains exactly one element.

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A131131 4*A007318 - 3*A097806.

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A166065 Triangle, read by rows, given by [0,1,1,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A255049 a(n) = 2*A160552(n).

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A351839 Triangle read by rows: T(n, k) = A027375(n)*Sum_{m=1..floor(n/k)} binomial(n, k*m).

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A154312 Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,0,...] DELTA [2,-1/2,-1/2,2,0,0,0,0,0,0,0 ...] where DELTA is the operator defined in A084938 .

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A131131 4A007318 - 3A097806.

A351839 Triangle read by rows: T(n, k) = A027375(n)Sum_{m=1..floor(n/k)} binomial(n, km).