SiYang Hu - cp's OEIS frontend

User: SiYang Hu

SiYang Hu has authored 5 sequences.

A384694 Sum of the number of cells alive after 2 generations of Conway's game of life for initial 1 X n cells taken in all 2^n combinations of alive or dead.

0, 0, 3, 12, 35, 92, 228, 544, 1264, 2880, 6464, 14336, 31488, 68608, 148480, 319488, 684032, 1458176, 3096576, 6553600, 13828096, 29097984, 61079552, 127926272, 267386880, 557842432, 1161822208, 2415919104, 5016387584, 10401873920, 21541945344, 44560285696, 92073361408, 190052302848, 391915765760

Offset: 0

SiYang Hu, Jun 07 2025

			For n = 5, there are 5 ways for the cells to evolve into a blinker: ..OOO, O.OOO, .OOO., OOO.., OOO.O; 4 ways for the cells to evolve into a beehive predecessor and then a beehive: OOOO., .OOOO; 1 way for it to evolve into 8 cells: OOOOO, so a(5) = 3 * 5 + 6 * 2 + 8 * 1 = 35.

LifeWiki, One cell thick pattern
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A167667 (after one generation).

G.f.: x^2*(3 - x^2)/(1 - 4*x + 4*x^2).
a(n) = 2^(n - 5) * (11*n - 20).
E.g.f.: (9 - 4*x - 2*x^2 + exp(2*x)*(22*x - 9))/16. - Stefano Spezia, Jun 07 2025

A384308 a(1) = 3; for n > 1, a(n) is the smallest number that has not appeared before and has the same set of prime divisors as a(n-1) + 1.

Original entry on oeis.org

3, 2, 9, 10, 11, 6, 7, 4, 5, 12, 13, 14, 15, 8, 27, 28, 29, 30, 31, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 81, 82, 83, 42, 43, 44, 45, 46, 47, 36, 37, 38, 39, 40, 41, 84, 85, 86, 87, 88, 89, 60, 61, 62, 63, 32, 33, 34, 35, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 90, 91, 92, 93, 94, 95, 72, 73, 74, 75

Offset: 1

SiYang Hu, May 25 2025

nonn

Theorem 1: {a(n)} is a permutation of the positive integers greater than 1.
Proof: Suppose the smallest positive integer greater than 1 that does not appear in this sequence is m, whose set of prime divisors is {p1, p2, ..., pk}. Since there are infinitely many numbers with this set of prime divisors, we know that either all of them appear or none of them appear. Since m is the smallest among them, we know that m = p1*p2*...*pk. Therefore, p1*p2*...*pk - 1 does not appear, a contradiction, implying that all integers greater than one appear in {a(n)}. Since there are no repetitions by the definition, we have proven that {a(n)} is a permutation of the positive integers greater than 1.
From Yifan Xie, May 26 2025: (Start)
Theorem 2: The parity of a(n) is the same as the parity of n.
Proof: Since a(1) is odd, we only need to prove that an odd term is immediately followed by an even term, and an even term is immediately followed by an odd term. If a(n-1) is odd, the set of prime divisors of a(n-1) + 1 contains 2, so a(n) is even; If a(n-1) and a(n) are both even, the set of prime divisors of a(n-1) + 1 does not contain 2, so a(n)/2 is a smaller candidate for a(n), a contradiction. (End)

			For a(5) = 11, 11 + 1 = 12, its set of prime divisors is {2, 3}. The smallest number with the same set of prime divisors that has not appeared before is 6, so a(6) = 6.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Similar to A064413 and A257218.

N:= 1000:# for terms before the first term > N
for i from 2 to N do
  S:= numtheory:-factorset(i);
  if assigned(V[S]) then V[S]:= V[S] union {i}
  else V[S]:= {i}
  fi
od:
R:= 3: r:= 3: V[{3}]:= V[{3}] minus {3}:
while r < N do
  S:= numtheory:-factorset(r+1);
  if V[S] = {} then break fi;
  r:= min(V[S]);
  V[S]:= V[S] minus {r};
  R:= R, r;
od:
R; # Robert Israel, May 25 2025

s={3};Do[i=2;While[MemberQ[s,i]||First/@FactorInteger[i]!=First/@FactorInteger[s[[-1]]+1],i++];AppendTo[s,i],{n,2,81}];s (* James C. McMahon, Jun 04 2025 *)

import heapq
from math import prod
from sympy import factorint
from itertools import islice
def bgen(pset): # generator of terms with set of prime divisors = pset
    h = [prod(pset)]
    while True:
        v = heapq.heappop(h)
        yield v
        for p in pset:
            heapq.heappush(h, v*p)
def agen(): # generator of terms
    an, aset = 3, set()
    while True:
        yield an
        aset.add(an)
        an = next(m for m in bgen(set(factorint(an+1))) if m not in aset)
print(list(islice(agen(), 81))) # Michael S. Branicky, May 25 2025

A383766 a(n) is the number of numbers k (0 <= k < n) such that there exist solutions of x^3 + x == y^2 + 1 == k (mod n).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 2, 6, 3, 4, 2, 5, 4, 7, 2, 6, 3, 8, 2, 10, 6, 11, 3, 12, 4, 11, 4, 6, 5, 6, 4, 13, 7, 12, 2, 11, 6, 16, 3, 8, 8, 13, 4, 21, 10, 10, 6, 17, 11, 6, 3, 14, 12, 18, 4, 20, 11, 12, 8, 12, 6, 27, 5, 16, 6, 26, 4, 27, 13, 20, 7, 9, 12, 26, 4, 31, 11, 25

Offset: 1

SiYang Hu, May 09 2025

nonn

			a(7) = 3: k can be 2, 3, 5, for example, when k = 3, x = 2, and y = 4, the equation is satisfied.

SiYang Hu, Table of n, a(n) for n = 1..10000

A383766 = Table[Count[Range[0, n - 1], k_ /; Length[Solve[{x^3 + x == k, y^2 + 1 == k}, {x, y}]] > 0], {n, 1, 50}];

If 8 does not divide n, a(2n) = a(n).

If 8 divides n, a(2n) = 2*a(n).

A383968 Number of distinct subsets S of [1..n] such that for all 1 <= k <= n, there exists two elements x,y in S (not necessarily distinct) such that x+y = 2k.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 17, 30, 58, 107, 205, 392, 768, 1466, 2883, 5597, 11038, 21572, 42675, 83711, 166371, 327893, 651199, 1288480, 2564032, 5082878, 10127472, 20115845, 40104636, 79781149, 159174500, 316962113, 632716744, 1261189166, 2518287361, 5023170116, 10034132101, 20025033970

Offset: 1

SiYang Hu, May 16 2025

nonn

Every subset S of [1..n] must have 1 and n to get 2 and 2*n. For odd n we therefore have 1+n which we need as well. If S is no such subset then no subset S' of S must be tested. - David A. Corneth, May 22 2025

			For n = 5, there are 5 sets S that satisfy the said conditions: {1, 2, 3, 4, 5}, {1, 2, 3, 5}, {1, 2, 4, 5}, {1, 3, 4, 5} and {1, 3, 5}.

David A. Corneth, PARI program
Sean A. Irvine, Java program (github)

def a(n):
    if n == 1: return 1
    c,t = 0, set(k << 1 for k in range(1, n+1))
    for i in range(1 << (n-2), 1 << n):
        s = [j+1 for j in range(n) if (i >> j) & 1]
        if s[0] == 1 and s[-1] == n:
            ss = set(x + y for x in s for y in s if x & 1 == y & 1)
            if t.issubset(ss): c += 1
    return c # Darío Clavijo, May 23 2025

a(23)-a(38) from Sean A. Irvine, May 21 2025

A381975 Number of ways for n competitors to rank in a competition in which each match has 4 possible outcomes in which each competitor gains 0, 1, 2 or 3 points.

Original entry on oeis.org

1, 1, 2, 9, 58, 459, 4370, 48999, 632884, 9254473, 151155362, 2727862751

Offset: 0

SiYang Hu, May 06 2025

Also, the number of maps f:{1, 2, ..., n} -> {1, 2, ..., n} such that f(f(f(x))) >= x for all 1 <= x <= n.

When this definition is changed to f(f(x)), then the result would be the Fubini numbers (A000670).

Cf. A000142, A000312, A000670, A001470.

validMappings :: Int -> Int
validMappings n = validMappingsDFS n [0] 1 where
  checkCondition f = all (\x -> f !! (f !! (f !! (x-1)) - 1) - 1 >= x) [1..n]
  validMappingsDFS n f x
    | x > n = if checkCondition f then 1 else 0
    | otherwise = sum [validMappingsDFS n (take (x-1) f ++ [i] ++ drop x f) (x+1) | i <- [1..n]]

CheckCondition[f_, n_] := AllTrue[Range[n], (f[[f[[f[[#]]]]]] >= # &)]
validMappingsDFS[n_, f_, x_] := Module[{i, f2, count},
  If[x > n,
    If[CheckCondition[f, n], 1, 0],
    count = 0;
    For[i = 1, i <= n, i++,
      f2 = f; f2[[x]] = i;  (* Assign mapping for f[x] *)
      count += validMappingsDFS[n, f2, x + 1];  (* Explore further *)
    ];
    count
  ]
]
A381975[n_] := validMappingsDFS[n, ConstantArray[0, n], 1]
Table[A381975[n], {n, 0, 6}]

def validMappings(n):
    def checkCondition(f, n):
        return all(f[f[f[x]]] >= x for x in range(1, n+1))
    def validMappingsDFS(n, f, x):
        if x > n:
            return 1 if checkCondition(f, n) else 0
        return sum(validMappingsDFS(n, f[:x] + [i] + f[x+1:], x+1) for i in range(1, n+1))
    return validMappingsDFS(n, [0] * (n+1), 1)

validMappings <- function(n) {
  checkCondition <- function(f, n) all(sapply(1:n, function(x) f[f[f[x]]] >= x))
  validMappingsDFS <- function(n, f, x) {
    if (x > n) return(ifelse(checkCondition(f, n), 1, 0))
    sum(sapply(1:n, function(i) { f[x] <- i; validMappingsDFS(n, f, x + 1) }))
  }
  validMappingsDFS(n, rep(0, n), 1)
}

a(11) from Sean A. Irvine, May 13 2025

cp's OEIS Frontend

User: SiYang Hu

A384694 Sum of the number of cells alive after 2 generations of Conway's game of life for initial 1 X n cells taken in all 2^n combinations of alive or dead.

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A384308 a(1) = 3; for n > 1, a(n) is the smallest number that has not appeared before and has the same set of prime divisors as a(n-1) + 1.

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A383766 a(n) is the number of numbers k (0 <= k < n) such that there exist solutions of x^3 + x == y^2 + 1 == k (mod n).

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A383968 Number of distinct subsets S of [1..n] such that for all 1 <= k <= n, there exists two elements x,y in S (not necessarily distinct) such that x+y = 2k.

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A381975 Number of ways for n competitors to rank in a competition in which each match has 4 possible outcomes in which each competitor gains 0, 1, 2 or 3 points.

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Haskell

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