A361471 -id:A361471 - cp's OEIS frontend

A361458 Size of the symmetric difference of {1,2,3}, {2,4,6}, ..., {n,2n,3n}.

3, 4, 3, 4, 7, 8, 11, 12, 11, 12, 15, 16, 19, 20, 19, 20, 23, 24, 27, 28, 27, 28, 31, 32, 35, 36, 35, 36, 39, 40, 43, 44, 43, 44, 47, 48, 51, 52, 51, 52, 55, 56, 59, 60, 59, 60, 63, 64, 67, 68, 67, 68, 71, 72, 75, 76, 75, 76, 79, 80, 83, 84, 83, 84, 87, 88, 91

Offset: 1

Guenter Pilz, May 17 2023

a(n) is also the terms of (x+x^2+x^3) + (x^2+x^4+x^6) + ... + (x^n+x^2n+x^3n) in GF(2)[x].

P. Y. Huang, W. F. Ke, and G. F. Pilz, The cardinality of some symmetric differences, Proc. Amer. Math. Soc., 138 (2010), 787-797.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,1,-1).

Cf. A361471.

delta[l_, m_] := Complement[Join[l, m], Intersection[l, m]];
Nabl[s_, n_] := (d = {}; Do[d = delta[d, s*j], {j, Range[n]}]; d);
Table[Length[Nabl[Range[1, 3], n]], {n, 100}]

a(n) = {my(m=0);for(k = 0,n-1,m = bitxor(m, 2^k+2^(2*k+1)+2^(3*k+2))); hammingweight(m)} \\ Thomas Scheuerle, May 17 2023

G.f.: x*(x^5+3*x^4+x^3-x^2+x+3)/(x^7-x^6-x+1). - Alois P. Heinz, May 17 2023

6*a(n) = 1 -(-1)^n +8*n +8*A103368(n-1). - R. J. Mathar, Jan 11 2024

A361472 Size of the symmetric differences of {1,2,3,4,5}, {2,4,6,8,10}, ..., {n,2n,3n,4n,5n}.

Original entry on oeis.org

5, 6, 7, 8, 5, 10, 15, 16, 17, 18, 23, 24, 29, 30, 31, 32, 37, 42, 47, 48, 49, 50, 55, 56, 53, 54, 55, 56, 61, 58, 63, 64, 65, 66, 63, 64, 69, 70, 71, 72, 77, 82, 87, 88, 89, 90, 95, 96, 101, 102, 103, 104, 109, 114, 111, 112, 113, 114, 119, 120, 125, 126, 127

Offset: 1

Guenter Pilz, May 17 2023

Ray Chandler, Table of n, a(n) for n = 1..1000
P. Y. Huang, W. F. Ke, and G. F. Pilz, The cardinality of some symmetric differences, Proc. Amer. Math. Soc., 138 (2010), 787-797.
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A361458, A361471.

delta[l_, m_] := Complement[Join[l, m], Intersection[l, m]];
Nabl[s_, n_] := (d = {}; Do[d = delta[d, s*j], {j, Range[n]}]; d);
Table[Length[Nabl[Range[1, 5], n]], {n, 100}]

G.f.: x*(x^59 +5*x^58 +x^57 +x^56 +x^55 -3*x^54 +5*x^53 +5*x^52 +x^51 +x^50 +x^49 +5*x^48 +x^47 +5*x^46 +x^45 +x^44 +x^43 +5*x^42 +5*x^41 +5*x^40 +x^39 +x^38 +x^37 +5*x^36 +x^35 -3*x^34 +x^33 +x^32 +x^31 +5*x^30 -3*x^29 +5*x^28 +x^27 +x^26 +x^25 -3*x^24 +x^23 +5*x^22 +x^21 +x^20 +x^19 +5*x^18 +5*x^17 +5*x^16 +x^15 +x^14 +x^13 +5*x^12 +x^11 +5*x^10 +x^9 +x^8 +x^7 +5*x^6 +5*x^5 -3*x^4 +x^3 +x^2 +x +5) / (x^61 -x^60 -x +1). - Alois P. Heinz, May 17 2023

cp's OEIS Frontend

A361458 Size of the symmetric difference of {1,2,3}, {2,4,6}, ..., {n,2n,3n}.

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