A141759 -id:A141759 - cp's OEIS frontend

A063448 Decimal expansion of Pi * sqrt(2).

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

Offset: 1

Jason Earls, Jul 24 2001

Hypotenuse of the right triangle with legs Pi and Pi. - Zak Seidov, May 04 2005
Circumference of the circumcircle of the unit square. - Jonathan Sondow, Nov 23 2017
Half-perimeter of the closed curve with implicit Cartesian equation x^2 + y^2 = abs(x) + abs(y). - Stefano Spezia, Oct 20 2020

			4.4428829381583662470158809900606936986146216893756902230853...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Peter Bala, Series for sqrt(2)*Pi
Eric Weisstein's World of Mathematics, Gauss Multiplication Formula.
Wikipedia, Bailey-Borwein-Plouffe formula.
Index entries for transcendental numbers

Cf. A063447 (continued fraction), A093954, A153799, A193887, A244976, A247719.

RealDigits[N[Pi*Sqrt[2], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)

PARI
```
\p 400; Pi * sqrt(2)
```

default(realprecision, 20080); x=Pi*sqrt(2); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b063448.txt", n, " ", d)) \\ Harry J. Smith, Aug 21 2009

# Use some guard digits when computing.
# BBP formula (1/8) P(1, 64, 12, (32, 0, 8, 0, 8, 0, -4, 0, -1, 0, -1, 0))
from decimal import Decimal as dec, getcontext
def BBPpisqrt2(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(64)
    for k in range(int(n * 0.5536546824812272) + 1):
        twk = dec(12 * k)
        s += f * ( dec(32) / (twk + 1) + dec(8)  / (twk + 3)
                 + dec(8)  / (twk + 5) - dec(4)  / (twk + 7)
                 - dec(1)  / (twk + 9) - dec(1)  / (twk + 11))
        f /= g
    return s / dec(8)
print(BBPpisqrt2(200))  # Peter Luschny, Nov 03 2023

Equals Gamma(1/4)*Gamma(3/4). - Jean-François Alcover, Nov 24 2014
From Amiram Eldar, Aug 06 2020: (Start)
Equals Integral_{x=0..oo} log(1 + 1/x^4) dx.
Equals Integral_{x=0..oo} log(1 + 2/x^2) dx.
Equals Integral_{x=-oo..oo} exp(x/4)/(exp(x) + 1) dx.
Equals Integral_{x=0..2*Pi} 1/(cos(x)^2 + 1) dx = Integral_{x=0..2*Pi} 1/(sin(x)^2 + 1) dx. (End)
From Andrea Pinos, Jul 03 2023: (Start)
Equals (Product_{k=1..4} Gamma(k/8)*Gamma(1 - k/8))^(1/4).
General result: 2*Pi/(4*y)^(1/(2*y)) = (Product_{k=1..y} Gamma(k/(2*y))*Gamma(1 - k/(2*y)) )^(1/y). (End)
From Peter Bala, Oct 22 2023: (Start)
sqrt(2)*Pi = 4 + 8*Sum_{n >= 0} (-1)^n/(16*n^2 + 32*n + 15). See A141759.
In the following the Eisenstein summation convention is assumed: that is,
Sum_{n = -oo..oo} means Limit_{N -> oo} Sum_{n = -N..N}:
sqrt(2)*Pi = 4*Sum_{n = -oo..oo} (-1)^n/(4*n + 1).
More generally, it appears that for k >= 0, k not of the form 4*m + 1,
sqrt(2)*Pi = -sign(cos(Pi*(k - 3)/4)) * 4*(2^floor(k/2))*k! * Sum_{n = -oo..oo} (-1)^n/((4*n + 1)*(4*n + 3)*...*(4*n + 2*k + 1)) (verified up to k = 50).
sqrt(2)*Pi = (2^4)*Sum_{n >= 0} (-1)^n * (2*n + 1)/((4*n + 1)*(4*n + 3)) = 512/105 - (2^6)*4!*Sum_{n >= 0} (-1)^n * (2*n + 3)/((4*n + 1)*(4*n + 3)*...*(4*n + 11)).
sqrt(2)*Pi = 4 + (2^3)*Sum_{n >= 0} (-1)^n * (4*n + 1)/((4*n + 1)*(4*n + 3)*(4*n + 5)) = 1408/315 - (2^5)*5!*Sum_{n >= 0} (-1)^n * (4*n + 1)/((4*n + 1)*(4*n + 3)*...*(4*n + 13)).
sqrt(2)*Pi = 16/3 - (2^4)*3!*Sum_{n >= 0} (-1)^n/((4*n + 1)*(4*n + 3)*(4*n + 5)*(4*n + 7)) = 14848/3465 + (2^6)*7!*Sum_{n >= 0} (-1)^n/((4*n + 1)*(4*n + 3)*...*(4*n + 15)). (End)
From Peter Bala, Nov 19 2023: (Start)
sqrt(2)*Pi = 512*Sum_{k >= 1} (-1)^(k+1) * k^2/((16*k^2 - 1)*(16*k^2 - 9)).
This is the case n = 1 of the more general result: for n >= 1,
sqrt(2)*Pi = (-1)^(n+1) * 2^(n+7) * (2*n)!/(2*n - 1) * Sum_{k >= 1} (-1)^(k+1) * k^2/( Product_{i = 0..n} (16*k^2 - (2*i+1)^2) ). Cf. A334422. (End)
Equals Integral_{x=-oo..oo} (x^2 + 1)/(x^4 + 1) dx. - Kritsada Moomuang, Jun 04 2025

Edited by N. J. A. Sloane, May 05 2007

Corrected by Neven Juric, Apr 24 2008

A198148 a(n) = n(n+2)(9 - 7*(-1)^n)/16.

Original entry on oeis.org

0, 3, 1, 15, 3, 35, 6, 63, 10, 99, 15, 143, 21, 195, 28, 255, 36, 323, 45, 399, 55, 483, 66, 575, 78, 675, 91, 783, 105, 899, 120, 1023, 136, 1155, 153, 1295, 171, 1443, 190, 1599, 210, 1763, 231, 1935, 253, 2115, 276, 2303, 300, 2499, 325

Offset: 0

Paul Curtz, Oct 21 2011

See, in A181318(n), A060819(n)*A060819(n+p): A060819(n)^2, A064038(n), a(n), A160050(n), A061037(n), A178242(n). The second differences a(n+2)-2*a(n+1)+a(n) = -5, 16, -26, 44, -61, 86, -110, 142, -173, 212, -250, 296, -341, 394, -446, 506, taken modulo 9 are periodic with the palindromic period 4, 7, 1, 8, 2, 5, 7, 7, 7, 5, 2, 8, 1, 7, 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. quadrisections A014105, A001539, A000384, A141759.

Cf. A060819, A000217, A000466, A142705, A000034, A005563, A010689, A028552, A050534.

List([0..60], n -> n*(n+2)*(9-7*(-1)^n)/16); # G. C. Greubel, Feb 21 2019

[n*(n+2)*(9-7*(-1)^n)/16: n in [0..60]]; // Vincenzo Librandi, Nov 25 2011

A198148:=n->n*(n+2)*(9-7*(-1)^n)/16; seq(A198148(k), k=0..100); # Wesley Ivan Hurt, Oct 16 2013

LinearRecurrence[{0,3,0,-3,0,1},{0,3,1,15,3,35},60] (* Vincenzo Librandi, Nov 25 2011 *)

a(n)=n*(n+2)*(9-7*(-1)^n)/16 \\ Charles R Greathouse IV, Oct 16 2015

[n*(n+2)*(9-7*(-1)^n)/16 for n in (0..60)] # G. C. Greubel, Feb 21 2019

a(n) = A060819(n)*A060819(n+2).
a(2n) = n*(n+1)/2 = A000217(n).
a(2n+1) = (2*n+1)*(2*n+3) = A000466(n+1).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6), n>5.
a(n+1) - a(n) = (7*(-1)^n *(2*n^2+6*n+3) +18*n +27)/16.
a(n) = A142705(n) / A000034(n+1).
a(n) = A005563(n) / A010689(n+1). - Franklin T. Adams-Watters, Oct 21 2011
G.f. x*(3 +x +6*x^2 -x^4)/(1-x^2)^3. - R. J. Mathar, Oct 25 2011
a(n)*a(n+1) = a(A028552(n)) = A050534(n+2). - Bruno Berselli, Oct 26 2011
a(n) = numerator( binomial((n+2)/2,2) ). - Wesley Ivan Hurt, Oct 16 2013
E.g.f.: x*((24+x)*cosh(x) + (3+8*x)*sinh(x))/8. - G. C. Greubel, Sep 20 2018
Sum_{n>=1} 1/a(n) = 5/2. - Amiram Eldar, Aug 12 2022

A282284 Least common multiple of 3n+1 and 3n-1.

Original entry on oeis.org

1, 4, 35, 40, 143, 112, 323, 220, 575, 364, 899, 544, 1295, 760, 1763, 1012, 2303, 1300, 2915, 1624, 3599, 1984, 4355, 2380, 5183, 2812, 6083, 3280, 7055, 3784, 8099, 4324, 9215, 4900, 10403, 5512, 11663, 6160, 12995, 6844, 14399, 7564, 15875, 8320, 17423

Offset: 0

Colin Barker, Feb 11 2017

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. A000466, A136017, A141759, A282285, A282286.

Table[LCM@@{3n+1,3n-1},{n,0,50}] (* or *) LinearRecurrence[{0,3,0,-3,0,1},{1,4,35,40,143,112,323},60] (* Harvey P. Dale, Sep 05 2020 *)

PARI
```
vector(60, n, n--; lcm(3*n+1, 3*n-1))
```

Vec((1+4*x+32*x^2+28*x^3+41*x^4+4*x^5-2*x^6) / ((1-x)^3*(1+x)^3) + O(x^60))

a(n) = 9*n^2-1 for n>0 and even.
a(n) = (9*n^2-1)/2 for n odd.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
G.f.: (1+4*x+32*x^2+28*x^3+41*x^4+4*x^5-2*x^6) / ((1-x)^3*(1+x)^3).

A282285 Least common multiple of 5n+1 and 5n-1.

Original entry on oeis.org

1, 12, 99, 112, 399, 312, 899, 612, 1599, 1012, 2499, 1512, 3599, 2112, 4899, 2812, 6399, 3612, 8099, 4512, 9999, 5512, 12099, 6612, 14399, 7812, 16899, 9112, 19599, 10512, 22499, 12012, 25599, 13612, 28899, 15312, 32399, 17112, 36099, 19012, 39999, 21012

Offset: 0

Colin Barker, Feb 11 2017

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. A000466, A136017, A141759, A282284, A282286.

PARI
```
vector(60, n, n--; lcm(5*n+1, 5*n-1))
```

Vec((1+12*x+96*x^2+76*x^3+105*x^4+12*x^5-2*x^6) / ((1-x)^3*(1+x)^3) + O(x^60))

a(n) = 25*n^2-1 for n>0 and even.
a(n) = (25*n^2-1)/2 for n odd.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
G.f.: (1+12*x+96*x^2+76*x^3+105*x^4+12*x^5-2*x^6) / ((1-x)^3*(1+x)^3).

A282286 Least common multiple of 7n+1 and 7n-1.

Original entry on oeis.org

1, 24, 195, 220, 783, 612, 1763, 1200, 3135, 1984, 4899, 2964, 7055, 4140, 9603, 5512, 12543, 7080, 15875, 8844, 19599, 10804, 23715, 12960, 28223, 15312, 33123, 17860, 38415, 20604, 44099, 23544, 50175, 26680, 56643, 30012, 63503, 33540, 70755, 37264, 78399

Offset: 0

Colin Barker, Feb 11 2017

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. A000466, A136017, A141759, A282284, A282285.

PARI
```
vector(60, n, n--; lcm(7*n+1, 7*n-1))
```

Vec((1+24*x+192*x^2+148*x^3+201*x^4+24*x^5-2*x^6) / ((1-x)^3*(1+x)^3) + O(x^60))

a(n) = 49*n^2-1 for n>0 and even.
a(n) = (49*n^2-1)/2 for n odd.
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n>6.
G.f.: (1+24*x+192*x^2+148*x^3+201*x^4+24*x^5-2*x^6) / ((1-x)^3*(1+x)^3).

A158487 a(n) = 64*n^2 - 8.

Original entry on oeis.org

56, 248, 568, 1016, 1592, 2296, 3128, 4088, 5176, 6392, 7736, 9208, 10808, 12536, 14392, 16376, 18488, 20728, 23096, 25592, 28216, 30968, 33848, 36856, 39992, 43256, 46648, 50168, 53816, 57592, 61496, 65528, 69688, 73976, 78392, 82936, 87608, 92408, 97336, 102392

Offset: 1

Vincenzo Librandi, Mar 20 2009

The identity (16*n^2 - 1)^2 - (64*n^2 - 8)*(2*n)^2 = 1 can be written as A141759(n)^2 - a(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 09 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A141759, A157914.

I:=[56, 248, 568]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 09 2012

LinearRecurrence[{3, -3, 1}, {56, 248, 568}, 50] (* Vincenzo Librandi, Feb 09 2012 *)

for(n=1, 40, print1(64*n^2 - 8", ")); \\ Vincenzo Librandi, Feb 09 2012

From Vincenzo Librandi, Feb 09 2012: (Start)
G.f.: -8*x*(7 + 10*x - x^2)/(x - 1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(2)))*Pi/(2*sqrt(2)))/16.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(2)))*Pi/(2*sqrt(2)) - 1)/16. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 8*(exp(x)*(8*x^2 + 8*x - 1) + 1).
a(n) = 8*A157914(n). (End)

A158562 a(n) = 256*n^2 - 16.

Original entry on oeis.org

240, 1008, 2288, 4080, 6384, 9200, 12528, 16368, 20720, 25584, 30960, 36848, 43248, 50160, 57584, 65520, 73968, 82928, 92400, 102384, 112880, 123888, 135408, 147440, 159984, 173040, 186608, 200688, 215280, 230384, 246000, 262128, 278768, 295920, 313584, 331760

Offset: 1

Vincenzo Librandi, Mar 21 2009

The identity (32*n^2 - 1)^2 - (256*n^2 - 16)*(2*n)^2 = 1 can be written as A158563(n)^2 - a(n)*A005843(n)^2 = 1. [rewritten by R. J. Mathar, Oct 16 2009]

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A141759, A158563.

I:=[240,1008,2288]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

16(16Range[40]^2-1) (* or *) LinearRecurrence[{3,-3,1},{240,1008,2288},40] (* Harvey P. Dale, Sep 13 2011 *)

for(n=1, 50, print1(256*n^2-16", ")); \\ Vincenzo Librandi, Feb 15 2012

G.f.: 16*x*(-15 - 18*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 09 2023: (Start)
Sum_{n>=1} 1/a(n) = (4 - Pi)/128.
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)*Pi - 4)/128. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 16*(exp(x)*(16*x^2 + 16*x - 1) + 1).
a(n) = 16*A141759(n-1). (End)

A158730 a(n) = 68*n^2 - 1.

Original entry on oeis.org

67, 271, 611, 1087, 1699, 2447, 3331, 4351, 5507, 6799, 8227, 9791, 11491, 13327, 15299, 17407, 19651, 22031, 24547, 27199, 29987, 32911, 35971, 39167, 42499, 45967, 49571, 53311, 57187, 61199, 65347, 69631, 74051, 78607, 83299, 88127, 93091, 98191, 103427, 108799

Offset: 1

Vincenzo Librandi, Mar 25 2009

The identity (68*n^2 - 1)^2 - (1156*n^2 - 34)*(2*n)^2 = 1 can be written as a(n)^2 - A158729(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A141759, A158729.

I:=[67, 271, 611]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 20 2012

LinearRecurrence[{3, -3, 1}, {67, 271, 611}, 50] (* Vincenzo Librandi, Feb 20 2012 *)

for(n=1, 40, print1(68*n^2 - 1", ")); \\ Vincenzo Librandi, Feb 20 2012

G.f.: x*(-67 - 70*x + x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 22 2023: (Start)
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(2*sqrt(17)))*Pi/(2*sqrt(17)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(2*sqrt(17)))*Pi/(2*sqrt(17)) - 1)/2. (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: exp(x)*(68*x^2 + 68*x - 1) + 1.
a(n) = A141759(2*n). (End)

Comment rewritten and formula replaced by R. J. Mathar, Oct 22 2009

A335574 Numbers of the form 16n^2 + 32n + 15 for which the central region of its symmetric representation of sigma consists of two subparts of sizes 4n+7 and 4n+1, n>=0.

Original entry on oeis.org

15, 63, 143, 255, 399, 575, 783, 1023, 1295, 1599, 1935, 2303, 2703, 3599, 4623, 5183, 6399, 7055, 7743, 8463, 9215, 9999, 10815, 11663, 12543, 16383, 17423, 18495, 19599, 20735, 21903, 23103, 24335, 25599, 26895, 28223, 29583, 32399, 36863, 38415, 39999

Offset: 0

Hartmut F. W. Hoft, Jan 26 2021

nonn

The sequence is a subsequence of A141759. An alternate description is that for any divisor d <= row(a(k)) - see A235791 - of a(k) = (4n+3)(4n+5) the inequalities d != 4n+3, d != 4n+5 and 2d < 4n+3 hold in addition to 2*(4n+3) > row(a(k)). These conditions state that the symmetric representation of sigma consists of an odd number of regions and that the central region has maximum width 2. With the triangular function T in A235791 we get T[a(k), 4n+3] = T[(4n+3)(4n+5), 4n+3 ] = 2n + 4 and T[a(k), 4n+5] = T[(4n+3)(4n+5), 4n+5 ] = 2n + 1 determining the lengths of the two subparts - see A279387 - as 2*(2n+4) - 1 = 4n + 7 and 2*(2n+1) - 1 = 4n + 1 which results in the pattern [ 3 width 1, (4n + 1) width 2, 3 width 1 ] of unit cells and a total area of 8*(n+1) for the central region. The first region has area 8*(n+1)^2.

			a(3) = 255 = 3*5*17 = 15*17 = A141759(3) is in the sequence since 2*3 < 15 and 2*5 < 15 with row(255) = 22, and the central region of its symmetric representation of sigma has maximum width 2 and area 32 with subparts 4*3+7 = 19 and 4*3+1= 13.
3173 = 3*5*11*19 = 55*57 = A141759(13) is the first number in A141759 not in this sequence since the central region of the symmetric representation of sigma for 3173 has width 3 and also 2*(3*11) = 66 > 55.
a(37) = 32399 = 179*181 = A141759(44)  is in the sequence since the divisor conditions are vacuously true and the central region of its symmetric representation of sigma has maximum width 2 and area 8*45 = 360 with subparts 4*44 + 7 = 183 and 4*44 + 1 = 177.
35343 = 3*3*3*7*11*17 = (11*17)*(7*27) = 187*189 = A141759(46) is not in the sequence since 2*99, 2*119 and 2*153 exceed 187. While the area of the first region of its symmetric representation of sigma is 8*47^2 = 17672, the area of the central region is 21992 and of maximum width 5.

Cf. A141759, A235791, A237270, A237593, A279387.

(* function segments[ ] is defined in A237270 *)
centerQ[n_] := Module[{s=Select[segments[n], First[#]!=0&], len}, len=Length[s]; OddQ[len]&&Max[s[[(len+1)/2]]]==2]
a335574[n_] := Select[Map[(4#+3)(4#+5)&, Range[0, n]], centerQ]
a335574[50] (* sequence data *)
(* alternative function based on divisors - much faster computation *)
divisorQ[n_] := Module[{a=4n+3, b=4n+5, d, r}, r=Floor[(Sqrt[8 a b + 1] - 1)/2]; d=Select[Divisors[a b],#<=r&&#!=a&&#!=b&]; r<2a&&AllTrue[d, 2#

a(k) = (4n+3)(4n+5) for n = sqrt(a(k)+1)/4 - 1, i.e., a(k) = A141759(n), for k>=0.

cp's OEIS Frontend

A063448 Decimal expansion of Pi * sqrt(2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Python

Formula

Extensions

A198148 a(n) = n*(n+2)*(9 - 7*(-1)^n)/16.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A282284 Least common multiple of 3*n+1 and 3*n-1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A282285 Least common multiple of 5*n+1 and 5*n-1.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A282286 Least common multiple of 7*n+1 and 7*n-1.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A158487 a(n) = 64*n^2 - 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A158562 a(n) = 256*n^2 - 16.

Views

Author

Keywords

Comments

Links

A198148 a(n) = n(n+2)(9 - 7*(-1)^n)/16.

A282284 Least common multiple of 3n+1 and 3n-1.

A282285 Least common multiple of 5n+1 and 5n-1.

A282286 Least common multiple of 7n+1 and 7n-1.