Guenter Pilz - cp's OEIS frontend

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

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

A361471 Size of the symmetric difference of {1,2,3,4}, {2,4,6,8}, ..., {n,2n,3n,4n}.

Original entry on oeis.org

4, 4, 4, 4, 8, 12, 16, 16, 16, 16, 20, 20, 24, 24, 24, 24, 28, 32, 36, 36, 36, 36, 40, 40, 44, 44, 44, 44, 48, 52, 56, 56, 56, 56, 60, 60, 64, 64, 64, 64, 68, 72, 76, 76, 76, 76, 80, 80, 84, 84, 84, 84, 88, 92, 96, 96, 96, 96, 100, 100, 104, 104, 104, 104, 108

Offset: 1

Guenter Pilz, May 17 2023

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 (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A361458.

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, 4], n]], {n, 100}]

G.f.: 4*x*(x^10+x^6+x^5+x^4+1)/(x^13-x^12-x+1). - Alois P. Heinz, May 17 2023

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

Original entry on oeis.org

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

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

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A361471 Size of the symmetric difference of {1,2,3,4}, {2,4,6,8}, ..., {n,2n,3n,4n}.

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

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