Thomas Otten - cp's OEIS frontend

A374532 Number of complete unit squares that fit inside a circle of radius sqrt(n^2+1) centered at the origin of a square lattice.

0, 4, 12, 24, 40, 68, 96, 132, 180, 224, 284, 340, 408, 492, 564, 656, 740, 848, 960, 1060, 1184, 1304, 1444, 1576, 1704, 1868, 2024, 2196, 2356, 2520, 2716, 2892, 3104, 3292, 3504, 3720, 3916, 4160, 4384, 4628, 4872, 5108, 5372, 5640, 5916, 6188, 6456, 6764, 7036

Offset: 0

Thomas Otten, Jul 10 2024

nonn

Thomas Otten, Illustration of initial terms.

Cf. A119677 (case for radius of n), A237526.

Cf. A046092, A000328 (quadrant width 1 cell).

a(n) = my(s=n^2+1); 4*sum(k=1, sqrtint(s), sqrtint(s-k^2)) \\ Andrew Howroyd, Jul 11 2024

def A374532(n): return sum(isqrt(k*((n<<1)-k)+1) for k in range(n))<<2 # Chai Wah Wu, Jul 18 2024

a(n) = 4*A237526(n^2 + 1).

A373349 Twin primes where both the prime and its twin have prime reversals.

Original entry on oeis.org

3, 5, 7, 11, 13, 71, 73, 149, 151, 179, 181, 311, 313, 1031, 1033, 1151, 1153, 1229, 1231, 3299, 3301, 3371, 3373, 3389, 3391, 3467, 3469, 3851, 3853, 7457, 7459, 7949, 7951, 9011, 9013, 9437, 9439, 10007, 10009, 10067, 10069, 10457, 10459, 10499, 10501, 10889

Offset: 1

Thomas Otten, Jun 01 2024

Robert Israel, Table of n, a(n) for n = 1..10001

Cf. A001097, A007500.

Cf. A004086.

R:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||n):
q:= p-> andmap(isprime, [p, R(p)]) and ormap(
    t-> andmap(isprime, [p+t, R(p+t)]), [-2, +2]):
select(q, [$2..12000])[];  # Alois P. Heinz, Jun 06 2024

isrevp(p) = isprime(p) && isprime(fromdigits(Vecrev(digits(p))));
isok(p) = isprime(p) && ((isprime(p+2) && isrevp(p+2)) || (isprime(p-2) &&  isrevp(p-2))) && isrevp(p); \\ Michel Marcus, Jun 06 2024

A367480 Decimal expansion of the radius of a common circle surrounded by seven tangent unit circles.

Original entry on oeis.org

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

Offset: 1

Thomas Otten, Dec 23 2023

The radius of a common circle surrounded by n tangent unit circles (n > 2) is r = 1/sin(Pi/n) - 1.

n=7 is the smallest number for which the radius cannot be expressed using square roots, since the regular heptagon formed by the centers of the tangent circles is non-constructible (see A246724, A188582, and A121570 for n=3, 4, 5).

			1.3047648709624865052...

Andrew M. Gleason, Angle Trisection, the Heptagon, and the Triskaidecagon, The American Mathematical Monthly 95, no. 3 (March 1988), pp. 185-194.
Index entries for algebraic numbers, degree 6.

Cf. A121598.

Cf. A121570, A188582, A246724.

RealDigits[Csc[Pi/7] - 1, 10, 120][[1]] (* Amiram Eldar, Dec 28 2023 *)

PARI
```
1/sin(Pi/7) - 1
```

Equals 1 / sin(Pi/7) - 1.
Equals A121598 - 1.
Largest of the 6 real-valued roots of 7*x^6+ 42*x^5 +49*x^4 -84*x^3 -119*x^2 +42*x-1=0. - R. J. Mathar, Aug 29 2025

More digits from Jon E. Schoenfield, Dec 24 2023

Comments edited by Michal Paulovic, Dec 26 2023

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User: Thomas Otten

A374532 Number of complete unit squares that fit inside a circle of radius sqrt(n^2+1) centered at the origin of a square lattice.

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A373349 Twin primes where both the prime and its twin have prime reversals.

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A367480 Decimal expansion of the radius of a common circle surrounded by seven tangent unit circles.

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