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A356470 Decimal expansion of (3 - sqrt(5))/(2*sqrt(2)).

2, 7, 0, 0, 9, 0, 7, 5, 6, 7, 3, 7, 7, 2, 6, 4, 5, 3, 6, 0, 1, 5, 4, 3, 1, 5, 7, 0, 4, 9, 0, 9, 3, 9, 2, 5, 4, 9, 7, 3, 6, 5, 1, 2, 1, 7, 0, 1, 4, 0, 6, 8, 4, 8, 1, 6, 8, 1, 3, 3, 5, 9, 0, 2, 9, 4, 9, 0, 0, 7, 4, 9, 1, 8, 6, 7, 7, 0, 7, 2, 3, 8, 0, 1, 7, 2, 8, 6, 2

Offset: 0

Andrew Slattery, Aug 08 2022

Distance between points in the most spread-out hexagonal lattice such that the gap formed by any three mutually tangential unit disks contains a point inside or on its boundary.
Side-length of an equilateral triangle contained in a unit side-length equilateral triangle such that every vertex and midpoint of the smaller triangle is at distance 1/2 from the nearest vertex of the larger triangle.
A quartic number with denominator 2. Minimal polynomial is 4x^4 - 14x^2 + 1. - Charles R Greathouse IV, Sep 29 2022

			0.2700907567377264536...

Andrew Slattery, Interactive
Index entries for algebraic numbers, degree 4

Cf. A002193.

First[RealDigits[N[(3-Sqrt[5])/(2Sqrt[2]),89]]] (* Stefano Spezia, Aug 08 2022 *)

(3-sqrt(5))/sqrt(8) \\ Charles R Greathouse IV, Sep 29 2022

More terms from Stefano Spezia, Aug 08 2022

A333301 a(1) = 1, a(2) = 2. For n>2, if a(n-1) is odd, a(n) = a(n-1) + a(n-2), and otherwise a(n) is the smallest missing number.

Original entry on oeis.org

1, 2, 3, 5, 8, 4, 6, 7, 13, 20, 9, 29, 38, 10, 11, 21, 32, 12, 14, 15, 29, 44, 16, 17, 33, 50, 18, 19, 37, 56, 22, 23, 45, 68, 24, 25, 49, 74, 26, 27, 53, 80, 28, 30, 31, 61, 92, 34, 35, 69, 104, 36, 39, 75, 114, 40, 41, 81, 122, 42, 43, 85, 128, 46, 47, 93, 140, 48, 51, 99, 150, 52, 54, 55, 109, 164

Offset: 1

Andrew Slattery, Jun 08 2020

How many times does each number appear?

			a(10) is even, so a(11) is the least number yet to appear, which is 9.

Cf. A069202, A307023.

A333516 Irregular triangle read by rows in which row n lists the first A000217(n) terms of A002260, n >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6

Offset: 1

Andrew Slattery, Mar 25 2020

a(n) equals the difference between n and the largest number less than n that can be expressed as the sum of the i-th triangular number and the j-th tetrahedral number for integers i < j.

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 2, 1, 2, 3;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7;
  ...

Row sums give A000292.
Right border gives A000027.
Cf. A000217, A002260, A124171.

T:= n-> seq([$1..i][], i=1..n):
seq(T(n), n=1..7);  # Alois P. Heinz, Apr 10 2020

from math import comb, isqrt
from sympy import integer_nthroot
def A333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(n>comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2)+1 # Chai Wah Wu, Nov 10 2024

a(n) = A002260(A124171(n)).

A333236 Largest digit in the decimal expansion of 1/n.

Original entry on oeis.org

1, 5, 3, 5, 2, 6, 8, 5, 1, 1, 9, 8, 9, 8, 6, 6, 9, 5, 9, 5, 9, 5, 9, 6, 4, 8, 7, 8, 9, 3, 9, 5, 3, 9, 8, 7, 7, 9, 6, 5, 9, 9, 9, 7, 2, 9, 9, 8, 9, 2, 9, 9, 9, 8, 8, 8, 9, 9, 9, 6, 9, 9, 8, 6, 8, 5, 9, 9, 9, 8, 9, 8, 9, 5, 3, 9, 9, 8, 8, 5, 9, 9, 9, 9, 9, 9, 9, 6, 9, 1, 9, 9, 8, 9, 9, 6, 9, 9, 1, 1, 9, 9, 9, 9, 9

Offset: 1

Andrew Slattery, Mar 12 2020

			a(50) = 2 because the largest digit in 1/50 = 0.02 is 2.

Index entries for sequences related to decimal expansion of 1/n

Cf. A333237 (a(n) = 9), A333402 (a(n) = 1).

a[n_] := Max@RealDigits[1/n][[1]]; Array[a, 88] (* Giovanni Resta, Mar 12 2020 *)

from sympy import n_order, multiplicity
def A333236(n):
    m2, m5 = multiplicity(2,n), multiplicity(5,n)
    return int(max(str(10**(max(m2,m5)+n_order(10,n//2**m2//5**m5))//n))) # Chai Wah Wu, Feb 07 2022

a(n) = max_{i=0..n} (floor(10^i/n) mod 10).

a(10*n) = a(n) and a(n) = n iff n = 1, 3, 6. - Bernard Schott, Mar 19 2020

More terms from Giovanni Resta, Mar 12 2020

A333237 Numbers k such that 1/k contains at least one '9' in its decimal expansion.

Original entry on oeis.org

11, 13, 17, 19, 21, 23, 29, 31, 34, 38, 41, 42, 43, 46, 47, 49, 51, 52, 53, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 76, 77, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 94, 95, 97, 98, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118

Offset: 1

Andrew Slattery, Mar 12 2020

Almost every prime appears in this sequence.
Among the first 10000 primes, only 2, 3, 5, 7, 37, 79, 239, 4649, and 62003 do not appear in the sequence. - Giovanni Resta, Mar 13 2020
The next primes not in the sequence are 538987, 35121409, and 265371653. - Robert Israel, Mar 18 2020

			5 is not in the sequence because 1/5 = 0.2 does not contain any 9s.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of 1/n

Cf. A333236.
Subsequences (for terms > 1): A000533, A002275, A135577, A252491.
Cf. A216664 (a subsequence).
Cf. A187614.

f:= proc(n) local m,S,r;
   m:= 1; S:= {1};
   do
     r:= floor(m/n);
     if r = 9 then return true fi;
     m:= (m - r*n)*10;
     if member(m,S) then return false fi;
     S:= S union {m};
   od
end proc:
select(f, [$1..1000]); # Robert Israel, Mar 18 2020

Select[Range[120], MemberQ[ Flatten@ RealDigits[1/#][[1]], 9] &] (* Giovanni Resta, Mar 12 2020 *)

from itertools import count, islice
from sympy import n_order, multiplicity
def A333237_gen(startvalue=1): # generator of terms
    for m in count(max(startvalue,1)):
        m2, m5 = multiplicity(2,m), multiplicity(5,m)
        if max(str(10**(max(m2,m5)+n_order(10,m//2**m2//5**m5))//m)) == '9':
            yield m
A333237_list = list(islice(A333237_gen(), 10)) # Chai Wah Wu, Feb 07 2022

A333236(a(n)) = 9.

More terms from Giovanni Resta, Mar 12 2020

A332969 a(n) = [x^n] (Sum_{j>=0} A002193(1-j) * x^j)^2.

Original entry on oeis.org

1, 8, 18, 16, 37, 26, 34, 52, 70, 90, 87, 116, 127, 112, 157, 212, 158, 192, 252, 252, 249, 272, 349, 276, 287, 478, 482, 334, 407, 478, 465, 488, 544, 698, 562, 504, 682, 698, 738, 736, 742, 880, 907, 826, 944, 848, 998, 1110, 976, 1106, 1217, 1112, 1060

Offset: 0

Andrew Slattery, Mar 04 2020

			a(1) = 8 because the coefficient of x^1 in (1 + 4x + ... )^2 is 8.

Cf. A002193.

seq(n)={Vec(Ser(digits(sqrtint(2*100^n)))^2)} \\ Andrew Howroyd, Mar 04 2020

G.f.: (Sum_{j>=0} A002193(1-j) * x^j)^2.
Sum_{k>=0} a(k)/10^k = 2.
a(n) = Sum_{j=0..n} A002193(1-j)*A002193(j-n+1).

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A356470 Decimal expansion of (3 - sqrt(5))/(2*sqrt(2)).

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A333301 a(1) = 1, a(2) = 2. For n>2, if a(n-1) is odd, a(n) = a(n-1) + a(n-2), and otherwise a(n) is the smallest missing number.

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A333516 Irregular triangle read by rows in which row n lists the first A000217(n) terms of A002260, n >= 1.

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A333236 Largest digit in the decimal expansion of 1/n.

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A333237 Numbers k such that 1/k contains at least one '9' in its decimal expansion.

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Maple

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Python

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A332969 a(n) = [x^n] (Sum_{j>=0} A002193(1-j) * x^j)^2.

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