A014641 -id:A014641 - cp's OEIS frontend

A289873 Related to perfect Wichmann rulers: a(n) = ( n^2 - (mod(n, 6) - 3)^2 ) / 3.

1, 3, 5, 7, 9, 15, 21, 27, 33, 39, 45, 55, 65, 75, 85, 95, 105, 119, 133, 147, 161, 175, 189, 207, 225, 243, 261, 279, 297, 319, 341, 363, 385, 407, 429, 455, 481, 507, 533, 559, 585, 615, 645, 675, 705, 735, 765, 799, 833, 867, 901, 935, 969, 1007, 1045, 1083, 1121, 1159, 1197, 1239, 1281, 1323, 1365

Offset: 2

Hugo Pfoertner, Jul 14 2017

Leading term in length A289761 of longest perfect Wichmann ruler with n segments.

Hugo Pfoertner, Table of n, a(n) for n = 2..10001

Cf. A004137, A193802, A193803, A289761.

A014641 is a subsequence.

p := (n, x) -> (2*n - 3*(1 + x))*(1 + x):
a := n -> p(n, 2*floor(n/6)):
seq(a(n), n = 2..64); # Peter Luschny, Jul 14 2017

Table[(n^2 - (Mod[n, 6] - 3)^2)/3, {n, 2, 64}] (* Michael De Vlieger, Jul 14 2017 *)

def A289873(n): return (n+(m:=n%6))*(n-(k:=m-3))//3+k-n # Chai Wah Wu, Jun 20 2024

a(n) = A289761(n) - n.
G.f.: x^2*(1 + x - x^2)*(1 + x^2 - x^3 + 2*x^4 + x^5) / ((1 - x)^3*(1 + x)*(1 - x + x^2)*(1 + x + x^2)) (conjectured). - Colin Barker, Jul 14 2017
Can be seen as a family of parabolas p_{n}(x) = (2*n - 3*(1 + x))*(1 + x) evaluated at x = 2*floor(n/6). - Peter Luschny, Jul 14 2017

A329404 Interleave 2n(3n-1), (2n+1)(6n+1) for n >= 0.

Original entry on oeis.org

0, 1, 4, 21, 20, 65, 48, 133, 88, 225, 140, 341, 204, 481, 280, 645, 368, 833, 468, 1045, 580, 1281, 704, 1541, 840, 1825, 988, 2133, 1148, 2465, 1320, 2821, 1504, 3201, 1700, 3605, 1908, 4033, 2128, 4485, 2360, 4961

Offset: 0

Paul Curtz, Nov 13 2019

a(n) + a(n+3) = 21, 21, 69, 69, 153, 153, ...
Hexagonal spiral for A026741:
.
                   33--17--35--18
                   /
                 16   8--17---9--19
                 /   /             \
               31  15   5---3---7  10
               /   /   /         \   \
             15   7   2   0===1===4==21==>
               \   \   \     /   /   /
               29  13   3---1   9  11
                 \   \         /   /
                 14   6--11---5  23
                   \             /
                   27--13--25--12
.
a(n) is the horizontal sequence from 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A005449, A014641, A026741, A033579, A165355.

LinearRecurrence[{0,3,0,-3,0,1},{0,1,4,21,20,65},100] (* Paolo Xausa, Nov 13 2023 *)

concat(0, Vec(x*(1 + 4*x + 18*x^2 + 8*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^3) + O(x^45))) \\ Colin Barker, Nov 13 2019

a(n) = n * A165355(n-1).
From Colin Barker, Nov 13 2019: (Start)
G.f.: x*(1 + 4*x + 18*x^2 + 8*x^3 + 5*x^4) / ((1 - x)^3*(1 + x)^3).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n > 5.
a(n) = (1/4)*(-1)*((-3 + (-1)^n)*n*(-2+3*n)). (End)
From Amiram Eldar, Dec 27 2024: (Start)
Sum_{n>=1} 1/a(n) = Pi/(8*sqrt(3)) + 9*log(3)/8.
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*Pi/(8*sqrt(3)) - 3*log(3)/8. (End)

A181093 p*(p+2)/3 where p and p+4 are primes.

Original entry on oeis.org

5, 21, 65, 133, 481, 645, 1541, 2133, 3201, 3605, 4033, 5461, 8965, 12545, 16725, 17633, 25761, 31621, 32865, 40833, 48133, 52801, 64533, 69921, 71765, 79381, 83333, 125665, 138245, 151425, 182533, 191521, 197633, 226325, 243105, 246533, 256961, 260485, 274821

Offset: 1

Zak Seidov, Jan 23 2011

nonn

For p>3, p == 1 mod 6 and p(p+2) == 0 mod 3, hence, except for the first term, a(n) = subsequence of A014641 Odd octagonal numbers: (2n+1)(6n+1).

			p=3,p+4=7 are primes and a(1)=3*5/3=3,
p=7,p+4=11 are primes and a(2)=7*9/3=21=A014641(2),
p=13,p+4=17 are primes and a(3)=13*15/3=65=A014641(3).

Cf. A014641.

# (# + 2)/3 & /@ Select[Prime@Range@140, PrimeQ[# + 4] &]

{forprime (p=3,10^3,isprime(p+4)&print1(p*(p+2)/3,","))}

More terms from Michel Marcus, Mar 04 2014

cp's OEIS Frontend

A289873 Related to perfect Wichmann rulers: a(n) = ( n^2 - (mod(n, 6) - 3)^2 ) / 3.

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A329404 Interleave 2n(3n-1), (2n+1)(6n+1) for n >= 0.

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A181093 p*(p+2)/3 where p and p+4 are primes.

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cp's OEIS Frontend

A289873 Related to perfect Wichmann rulers: a(n) = ( n^2 - (mod(n, 6) - 3)^2 ) / 3.

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Maple

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Formula

A329404 Interleave 2*n*(3*n-1), (2*n+1)*(6*n+1) for n >= 0.

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Mathematica

PARI

Formula

A181093 p*(p+2)/3 where p and p+4 are primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A329404 Interleave 2n(3n-1), (2n+1)(6n+1) for n >= 0.