Ralf Stephan - cp's OEIS frontend

A385110 Terms of A198587 congruent {1, 3, 7} (mod 8).

17, 35, 75, 151, 1137, 2275, 2417, 4835, 4849, 9699, 19417, 38833, 38835, 72817, 77667, 145635, 154737, 309475, 310385, 620771, 621377, 1242737, 1242755, 2485361, 2485475, 4660337, 4970723, 4971025, 9320675, 9903217, 9942051, 19806435, 19864689, 19884107, 39729379

Offset: 1

Ralf Stephan, Jun 18 2025

nonn

Sorted set of all A385109(A198587(i)), i>0.

Cf. A198587, A385109.

N=3;for(n=1, 1000000, s=n; t=0; while(s!=1, if(s%2==0, s=s/2, s=(3*s+1)/2; t++); if(s==1&&t==N&&(n%8==1||n%8==3||n%8==7), print1(n,", ") ); ))

a(22)-a(31) from Michel Marcus, Jun 19 2025

a(32)-a(35) added using A198587 by Jinyuan Wang, Jun 26 2025

A385109 If n is 5 (mod 8) then apply n = (n-1)/4 until the result is not equivalent 5 (mod 8); otherwise a(n) = n.

Original entry on oeis.org

0, 1, 2, 3, 4, 1, 6, 7, 8, 9, 10, 11, 12, 3, 14, 15, 16, 17, 18, 19, 20, 1, 22, 23, 24, 25, 26, 27, 28, 7, 30, 31, 32, 33, 34, 35, 36, 9, 38, 39, 40, 41, 42, 43, 44, 11, 46, 47, 48, 49, 50, 51, 52, 3, 54, 55, 56, 57, 58, 59, 60, 15, 62, 63, 64, 65, 66, 67, 68, 17, 70, 71, 72, 73, 74, 75, 76, 19

Offset: 0

Ralf Stephan, Jun 18 2025

nonn

a(8*n + 5) = A347840(n).

			a(37): n is 5 (mod 8), so n = (37-1)/4 = 9, which is 1 (mod 8), so a(37) = 9.
a(53): n is 5 (mod 8), so n = (53-1)/4 = 13, which is 5 (mod 8), so n = (13-1)/4 = 3, and, as 3 is 3 (mod 8), a(53) = 3.

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A347840.

A385109[n_] := NestWhile[(# - 1)/4 &, n , Mod[#, 8] == 5 &];
Array[A385109, 100, 0] (* Paolo Xausa, Jun 25 2025 *)

a(n) = if(n%8!=5, n, m=n; while(m%8==5, m=(m-1)/4); m)

Recurrence: a(1) = 1, a(2n) = 2n, a(4n+3) = 4n+3, a(8n+1) = 8n+1, a(8n+5) = a(2n+1).

A375071 Smallest k such that Product_{i=1..k} (n+i) divides Product_{i=1..k} (n+k+i), or 0 if there is no such k.

Original entry on oeis.org

1, 5, 4, 207, 206, 2475, 984, 8171, 8170, 45144, 45143, 3648830, 3648829, 7979077, 7979076, 58068862, 58068861, 255278295, 255278294

Offset: 0

Ralf Stephan, Jul 29 2024

Erdős and Straus asked if a(n) exists for all n.

			3*4*5*6 ∣ 7*8*9*10, so a(2) = 4.

Thomas Bloom, Problem 389, Erdős Problems.

a(n) = my(k=1); while(prod(i=1, k, n+k+i)%prod(i=1, k, n+i), k++); k;

from itertools import count
def A375071(n):
    a, b = n+1, n+2
    for k in count(1):
        if not b%a:
            return k
        a *= n+k+1
        b = b*(n+2*k+1)*(n+2*k+2)//(n+k+1) # Chai Wah Wu, Aug 01 2024

a(10)-a(18) from Bhavik Mehta, Aug 16 2024

A375081 Smallest k>n such that the denominator of Sum {i=n..k} (1/i) is larger than the denominator of Sum {i=n..k+1} (1/i).

Original entry on oeis.org

5, 5, 5, 17, 17, 14, 14, 14, 14, 14, 32, 34, 34, 34, 27, 27, 27, 27, 23, 23, 27, 51, 51, 51, 51, 44, 44, 44, 44, 44, 39, 39, 39, 39, 39, 44, 74, 74, 74, 74, 74, 74, 74, 74, 65, 65, 65, 65, 65, 65, 65, 65, 65, 65, 59, 71, 71, 71, 71, 71, 71, 71, 71, 76, 76, 76

Offset: 1

Ralf Stephan, Jul 29 2024

			1/3+1/4+1/5=47/60 and 1/3+1/4+1/5+1/6=19/20, and 60>20, so a(3)=5.

Bhavik Mehta, Table of n, a(n) for n = 1..10000
Thomas Bloom, Problem 290, Erdős Problems.
Wouter van Doorn, On the non-monotonicity of the denominator of generalized harmonic sums, arXiv:2411.03073 [math.NT], 2024.

a(n) = for(k=0, oo, my(s=sum(n=n, n+k, 1/n)); if(denominator(s)>denominator(s+1/(n+k+1)), return(n+k); break))

from fractions import Fraction
from itertools import count
def A375081(n):
    a = Fraction((n<<1)+1,n*(n+1))
    for k in count(n+1):
        if a.denominator > (a:=a+Fraction(1,k+1)).denominator:
            return k # Chai Wah Wu, Jul 30 2024

a(n) < 4.374*n for all n > 1. - Wouter van Doorn, Nov 06 2024

a(n) > n + 0.54*log(n) for all large enough n, and there are infinitely many n with a(n) < n + 0.61*log(n). - Wouter van Doorn, Feb 06 2025

a(56) onwards from Bhavik Mehta, Jul 31 2024

A375077 Smallest k such that Product_{i=0..n} (k-i) divides C(2k,k).

Original entry on oeis.org

2, 2480, 8178, 45153, 3648841, 7979090, 101130029

Offset: 1

Ralf Stephan, Jul 29 2024

Thomas Bloom, Problem 396, Erdős Problems.

Cf. A000984, A120626, A129489.

for(n=1,20,for(k=n+1,100000,if(binomial(2*k,k)%prod(i=0,n,k-i)==0,print(n," ",k);break)))

from math import prod, comb
def A375077(n):
    a, c, k = prod(n+1-i for i in range(n+1)), comb(n+1<<1,n+1), n+1
    while c%a:
        k += 1
        a = a*k//(k-n-1)
        c = c*((k<<1)-1<<1)//k
    return k # Chai Wah Wu, Jul 30 2024

a(5) from Chai Wah Wu, Aug 01 2024

a(6)-a(7) from Max Alekseyev, Feb 25 2025

A249328 Position where n occurs first in the continued fraction expansion of e^3.

Original entry on oeis.org

2, 3, 5, 4, 7, 18, 32, 91, 142, 454, 1, 245, 15, 16, 126, 10, 304, 74, 202, 0, 296, 409, 227, 89, 865, 425, 183, 384, 616, 634, 1200, 1467, 1226, 159, 233, 2023, 584, 1384, 745, 764, 770, 329, 1582, 1721, 585, 85, 392, 136, 3984, 740, 302, 890

Offset: 1

Ralf Stephan, Jan 12 2015

nonn

Cf. A058282.

l = continued_fraction(exp(3))[:10000]
[next((i for i, x in enumerate(l) if x == n), None) for n in range(1,100)]

A242983 n/2 * (n^3 - 2n^2 - 2n + 5).

Original entry on oeis.org

0, 1, 1, 12, 58, 175, 411, 826, 1492, 2493, 3925, 5896, 8526, 11947, 16303, 21750, 28456, 36601, 46377, 57988, 71650, 87591, 106051, 127282, 151548, 179125, 210301, 245376, 284662, 328483, 377175, 431086, 490576, 556017

Offset: 0

Ralf Stephan, Jun 09 2014

For n>1, number of ways to place two dominoes horizontally on an n X n chessboard.

Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A242856, A242861.

Table[n/2 (n^3-2n^2-2n+5),{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{0,1,1,12,58},40] (* Harvey P. Dale, Jul 19 2018 *)

a(n) = A019582(n) + A077414(n-2), n>1.

G.f.: x*(-2*x^3 + 17*x^2 - 4*x + 1) / (1-x)^5.

A243256 Smallest distance of the n-th Fibonacci number to the set of all square integers.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 3, 4, 2, 6, 8, 0, 8, 16, 15, 26, 3, 17, 44, 41, 79, 22, 96, 143, 51, 289, 169, 285, 140, 296, 669, 267, 1449, 343, 1979, 144, 592, 665, 4223, 699, 5283, 2872, 19604, 6477, 21826, 17999, 16008, 46080, 31240, 102696, 8638, 45526, 95764

Offset: 0

Ralf Stephan, Jun 02 2014

nonn

a(n) = 0 if and only if n = 0, 1, 2, 12.

The sorted unique members: 0, 1, 2, 3, 4, 6, 8, 15, 16, 17, 22, 26 ...

{a(n) = my(f, i); if( n<0, 0, i = sqrtint( f = fibonacci(n))); min(f - i^2, (i+1)^2 - f)}; /* Michael Somos, Jun 02 2014 */

def a(n):
    f = fibonacci(n)
    return min((floor(sqrt(f))+1)^2 - f, f - floor(sqrt(f))^2)

a(n) = min(A190993(n), A216223(n)).

A242861 Triangle T(n,k) by rows: number of ways k dominoes can be placed on an n X n chessboard, k>=0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 12, 44, 56, 18, 1, 24, 224, 1044, 2593, 3388, 2150, 552, 36, 1, 40, 686, 6632, 39979, 157000, 407620, 695848, 762180, 510752, 192672, 35104, 2180, 1, 60, 1622, 26172, 281514, 2135356, 11785382, 48145820, 146702793, 333518324, 562203148

Offset: 0

Ralf Stephan, May 24 2014

Also, coefficients of the matching-generating polynomial of the n X n grid graph.

In the n-th row there are floor(n^2/2)+1 values.

			Triangle starts:
  1
  1
  1  4   2
  1 12  44   56    18
  1 24 224 1044  2593   3388   2150    552     36
  1 40 686 6632 39979 157000 407620 695848 762180 510752 192672 35104 2180
  ...

Alois P. Heinz, Rows n = 0..14, flattened
Ralf Stephan, Two dominoes on the 3x3 chessboard, illustration of T(3,2)=44.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Matching-Generating Polynomial
Wikipedia, Matching polynomial
Index entries for sequences related to dominoes
Index entries for sequences related to matchings

Cf. A046741, A096713.

b:= proc(n, l) option remember; local k;
      if n=0 then 1
    elif min(l[])>0 then b(n-1, map(h->h-1, l))
    else for k while l[k]>0 do od; expand(`if`(n>1,
         x*b(n, subsop(k=2, l)), 0) +`if`(k (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [0$n])):
seq(T(n), n=0..8); # Alois P. Heinz, Jun 01 2014

b[n_, l_List] := b[n, l] =  Module[{k}, Which[n == 0, 1, Min[l]>0, b[n-1, l-1], True, For[k=1, l[[k]]>0, k++]; Expand[If[n>1, x*b[n, ReplacePart[l, k -> 2]], 0] + If[k 1, k + 1 -> 1}]], 0] + b[n, ReplacePart[l, k -> 1]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, Array[0&, n]]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jun 16 2015, after Alois P. Heinz *)

def T(n,k):
   G = Graph(graphs.Grid2dGraph(n,n))
   G.relabel()
   mu = G.matching_polynomial()
   return abs(mu[n^2-2*k])

T(n,1) = A046092(n-1), T(n,2) = A242856(n).
T(n,floor(n^2/2)) = A137308(n), T(2n,2n^2) = A004003(n).
sum(k>=0, T(n,k)) = A210662(n,n) = A028420(n).
T(n,3) = A243206(n), T(n,4) = A243215(n), T(n,5) = A243217(n), T(n,floor(n^2/4)) = A243221(n). - Alois P. Heinz, Jun 01 2014

A242856 Number of 2-matchings of the n X n grid graph.

Original entry on oeis.org

2, 44, 224, 686, 1622, 3272, 5924, 9914, 15626, 23492, 33992, 47654, 65054, 86816, 113612, 146162, 185234, 231644, 286256, 349982, 423782, 508664, 605684, 715946, 840602, 980852, 1137944, 1313174, 1507886, 1723472, 1961372, 2223074, 2510114, 2824076, 3166592

Offset: 2

Ralf Stephan, May 24 2014

Number of ways two dominoes can be placed on an n X n chessboard.

Alois P. Heinz, Table of n, a(n) for n = 2..1000
Ralf Stephan, In how many ways can we place two dominoes on the n x n chessboard? Proof of formula.
Ralf Stephan, Two dominoes on the 3x3 chessboard, illustration of a(3)=44.
Index entries for sequences related to dominoes.
Index entries for sequences related to matchings.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Second column of A242861. Cf. A016742, A046092, A054000, A210662.

LinearRecurrence[{5, -10, 10, -5, 1}, {2, 44, 224, 686, 1622}, 50] (* Paolo Xausa, May 20 2024 *)

Vec(-2*x^2*(x^4-7*x^3+12*x^2+17*x+1)/(x-1)^5 + O(x^100)) \\ Colin Barker, Jun 26 2014

def a(n):
    G = Graph(graphs.Grid2dGraph(n,n))
    G.relabel()
    return G.matching_polynomial()[n^2-4]

a(n) = 2*n^4 - 4*n^3 - 5*n^2 + 13*n - 4.
G.f.: -2*x^2*(x^4-7*x^3+12*x^2+17*x+1) / (x-1)^5. - Colin Barker, Jun 26 2014
a(n + 1) = (1/2)*A046092(n)*(A046092(n) - 1) - A016742(n) - A054000(n). - Nicolas Bělohoubek, May 15 2024
E.g.f.: 4 - 2*x + exp(x)*(2*x^4 + 8*x^3 - 3*x^2 + 6*x - 4). - Stefano Spezia, Jun 04 2024

a(7)-a(36) from Alois P. Heinz, Jun 01 2014

cp's OEIS Frontend

User: Ralf Stephan

A385110 Terms of A198587 congruent {1, 3, 7} (mod 8).

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A385109 If n is 5 (mod 8) then apply n = (n-1)/4 until the result is not equivalent 5 (mod 8); otherwise a(n) = n.

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A375071 Smallest k such that Product_{i=1..k} (n+i) divides Product_{i=1..k} (n+k+i), or 0 if there is no such k.

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A375081 Smallest k>n such that the denominator of Sum {i=n..k} (1/i) is larger than the denominator of Sum {i=n..k+1} (1/i).

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A375077 Smallest k such that Product_{i=0..n} (k-i) divides C(2k,k).

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A249328 Position where n occurs first in the continued fraction expansion of e^3.

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A242983 n/2 * (n^3 - 2*n^2 - 2*n + 5).

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A243256 Smallest distance of the n-th Fibonacci number to the set of all square integers.

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A242983 n/2 * (n^3 - 2n^2 - 2n + 5).