Ethan Ji - cp's OEIS frontend

A385820 Number of equivalence classes of finitely-supported integer functions on Z^2 modulo moves that add + or -1 to every cell whose coordinates form an arithmetic progression of length n.

1, 2, 27, 1024, 9765625, 272097792, 558545864083284007, 295147905179352825856, 1144561273430837494885949696427, 305175781250000000000000000000000000, 1890591424712781041871514584574319778449301246603238034051, 98746073676238604311280222171685832518740805156864

Offset: 1

Ethan Ji, Jul 09 2025

nonn

Ethan Ji, Table of n, a(n) for n = 1..36

Cf. A076113.

a[n_Integer?Positive] := Module[{pairs = FactorInteger[n]}, Times @@ (#1^(n^2*(#2 #1^(2 #2) - (#1^#2 (#1^#2 - 1))/(#1 - 1))/(2 #1^(2 #2))) & @@@ pairs)]

a(n) = my(f=factor(n)); for (i=1, #f~, my(p=f[i,1], k=f[i,2]); f[i,2] = n^2*(k*p^(2*k) - p^k*(p^k-1)/(p-1))/(2*p^(2*k))); factorback(f); \\ Michel Marcus, Jul 10 2025

a(n) = Product_{p^k | n : prime p, k = p-adic order of n} p^(n^2*(k*p^(2k) - p^k(p^k - 1)/(p - 1)) / (2*p^(2k))).

a(p) = A076113(p), for prime p.

A378196 Number of 2-colorings of length n without an arithmetic progression of length 4.

Original entry on oeis.org

1, 2, 4, 8, 14, 26, 48, 78, 132, 230, 356, 548, 842, 1078, 1344, 1764, 1744, 1850, 1948, 1708, 1442, 1342, 1032, 702, 524, 316, 168, 136, 136, 144, 152, 160, 168, 176, 28, 0

Offset: 0

Ethan Ji, Nov 19 2024

nonn

After 0, the sequence will continue to be 0. A sequence satisfying this property cannot have a subsequence which violates it, thus there must exist a sequence of length n-1 if there exists a sequence of length n.

First 0 index given by A005346.

Cf. A378195, A378197.

HasEquallySpacedKBits[bits_, k_] :=
 If[k == 1, True,
  Module[{n = Length[bits], found = False},
   Do[If[Count[Table[bits[[start + gap*i]], {i, 0, k - 1}],
       bits[[start]]] == k, found = True; Break[]], {gap, 1,
     Floor[n/(k - 1)]}, {start, 1, n - gap*(k - 1)}];
   found]]
BitSequence[k_] :=
 Module[{prevSequences = {{}}, currSequences, n = 0, ExtendSequence},
  ExtendSequence[seq_] :=
   Module[{newSeq0, newSeq1, result = {}}, newSeq0 = Join[seq, {0}];
    newSeq1 = Join[seq, {1}];
    If[! HasEquallySpacedKBits[newSeq0, k], AppendTo[result, newSeq0]];
    If[! HasEquallySpacedKBits[newSeq1, k], AppendTo[result, newSeq1]];
    result];
  Function[targetN,
   Print["k=", k, ", n=", n, ": count=", Length[prevSequences]];
   While[n < targetN, n++;
    currSequences = Flatten[ExtendSequence /@ prevSequences, 1];
    prevSequences = currSequences;
    Print["k=", k, ", n=", n, ": count=", Length[prevSequences]];];]]
BitSequence[4][35]
(* Ethan Ji, Nov 19 2024 *)

A378195 Number of 2-colorings of length n without an arithmetic progression of length 3.

Original entry on oeis.org

1, 2, 4, 6, 10, 14, 20, 16, 6, 0

Offset: 0

Ethan Ji, Nov 19 2024

nonn

After 0, the sequence will continue to be 0. A sequence satisfying this property cannot have a subsequence which violates it, thus there must exist a sequence of length n-1 if there exists a sequence of length n.

			a(3) = 6 since we have [0,0,1],[0,1,0],[0,1,1],[1,0,0],[1,0,1],[1,1,0].

First 0 index given by A005346.

Cf. A378196, A378197.

HasEquallySpacedKBits[bits_, k_] :=
 If[k == 1, True,
  Module[{n = Length[bits], found = False},
   Do[If[Count[Table[bits[[start + gap*i]], {i, 0, k - 1}],
       bits[[start]]] == k, found = True; Break[]], {gap, 1,
     Floor[n/(k - 1)]}, {start, 1, n - gap*(k - 1)}];
   found]]
BitSequence[k_] :=
 Module[{prevSequences = {{}}, currSequences, n = 0, ExtendSequence},
  ExtendSequence[seq_] :=
   Module[{newSeq0, newSeq1, result = {}}, newSeq0 = Join[seq, {0}];
    newSeq1 = Join[seq, {1}];
    If[! HasEquallySpacedKBits[newSeq0, k], AppendTo[result, newSeq0]];
    If[! HasEquallySpacedKBits[newSeq1, k], AppendTo[result, newSeq1]];
    result];
  Function[targetN,
   Print["k=", k, ", n=", n, ": count=", Length[prevSequences]];
   While[n < targetN, n++;
    currSequences = Flatten[ExtendSequence /@ prevSequences, 1];
    prevSequences = currSequences;
    Print["k=", k, ", n=", n, ": count=", Length[prevSequences]];];]]
BitSequence[3][9]
(* Ethan Ji, Nov 19 2024 *)

A378197 Number of 2-colorings of length n without an arithmetic progression of length 5.

Original entry on oeis.org

1, 2, 4, 8, 16, 30, 58, 112, 216, 400, 740, 1398, 2638, 4710, 8444, 15118, 27690, 48406, 84382, 146928, 255844, 402998, 625824, 956370, 1447476, 2066828, 3225856, 5020232, 7823236, 10975318, 15264202, 21500308, 30004914, 39030820, 50728472, 65402746, 88886116

Offset: 0

Ethan Ji, Nov 19 2024

nonn

After a(178) = 0, the sequence will continue to be 0. A sequence satisfying this property cannot have a subsequence which violates it, thus there must exist a sequence of length n-1 if there exists a sequence of length n.

Michael De Vlieger, Table of n, a(n) for n = 0..178

First 0 index given by A005346.

Cf. A378195, A378196.

HasEquallySpacedKBits[bits_, k_] :=
 If[k == 1, True,
  Module[{n = Length[bits], found = False},
   Do[If[Count[Table[bits[[start + gap*i]], {i, 0, k - 1}],
       bits[[start]]] == k, found = True; Break[]], {gap, 1,
     Floor[n/(k - 1)]}, {start, 1, n - gap*(k - 1)}];
   found]]
BitSequence[k_] :=
 Module[{prevSequences = {{}}, currSequences, n = 0, ExtendSequence},
  ExtendSequence[seq_] :=
   Module[{newSeq0, newSeq1, result = {}}, newSeq0 = Join[seq, {0}];
    newSeq1 = Join[seq, {1}];
    If[! HasEquallySpacedKBits[newSeq0, k], AppendTo[result, newSeq0]];
    If[! HasEquallySpacedKBits[newSeq1, k], AppendTo[result, newSeq1]];
    result];
  Function[targetN,
   Print["k=", k, ", n=", n, ": count=", Length[prevSequences]];
   While[n < targetN, n++;
    currSequences = Flatten[ExtendSequence /@ prevSequences, 1];
    prevSequences = currSequences;
    Print["k=", k, ", n=", n, ": count=", Length[prevSequences]]; ]; ]]
BitSequence[5][178]
(* Ethan Ji, Nov 19 2024 *)

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User: Ethan Ji

A385820 Number of equivalence classes of finitely-supported integer functions on Z^2 modulo moves that add + or -1 to every cell whose coordinates form an arithmetic progression of length n.

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A378196 Number of 2-colorings of length n without an arithmetic progression of length 4.

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A378195 Number of 2-colorings of length n without an arithmetic progression of length 3.

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A378197 Number of 2-colorings of length n without an arithmetic progression of length 5.

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