A136016 -id:A136016 - cp's OEIS frontend

A202948 a(n+1) = 3A136016a(n).

-3, -72, -7560, -1814400, -778377600, -523069747200, -506854585036800, -669048052248576000, -1154107890128793600000, -2520571632041285222400000, -6797981691615346244812800000

Offset: 1

John M. Campbell, Dec 26 2011

Sums of coefficients from (3n+2)th moments of binomial(m,k)*binomial(2m,k): see Maple code below.

			The evaluation of sum(k^8 binomial(n,k) binomial(2n,k), k=0..n) involves the polynomial 64n^10 + 192n^9 - 1344n^8 - 1056n^7 + 8256n^6 - 3696n^5 - 9940n^4 + 7551n^3 + 348n^2 - 507n + 60, the sum of the coefficients of which is -72=a(2).

Eric W. Weisstein, MathWorld: Binomial Sums

Cf. A136016, A064350.

with(PolynomialTools); polyn := proc (q) options operator, arrow; 3^q*Pi*GAMMA(2*n)*(sum(k^q*binomial(n, k)*binomial(2*n, k), k = 0 .. n))/(27^n*sqrt(3)*GAMMA(n-floor((1/3)*q+1/3)+2/3)*GAMMA(n-floor((1/3)*q)+1/3)) end proc; coefl := proc (q) options operator, arrow; CoefficientList(expand(polyn(q)), n) end proc; coe := proc (j, h) options operator, arrow; coefl(j)[h] end proc; seq(sum(coe(3*r+2, k), k = 1 .. 5*r+3), r = 1 .. 8) ;

print1(a=-3);for(n=2,20,print1(", ",a*=27*n*(n-2)+24)) \\ Charles R Greathouse IV, Dec 27 2011

a(n)=-(1/6)*27^n*GAMMA(n-1/3)*GAMMA(n+1/3)*sqrt(3)/Pi.

A010731 Constant sequence: the all 8's sequence.

Original entry on oeis.org

8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 0

N. J. A. Sloane

Continued fraction expansion of 4+sqrt(17). - Bruno Berselli, Mar 15 2011

Decimal expansion of 8/9. - Elmo R. Oliveira, Feb 20 2024

Tanya Khovanova, Recursive Sequences
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1016
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A136016.

PadRight[{},120,8] (* Harvey P. Dale, Aug 15 2021 *)

a(n)=8 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 8/(1-x). - Bruno Berselli, Mar 15 2011

E.g.f.: 8*exp(x). - Elmo R. Oliveira, Feb 20 2024

A136017 a(n) = 36n^2 - 1.

Original entry on oeis.org

35, 143, 323, 575, 899, 1295, 1763, 2303, 2915, 3599, 4355, 5183, 6083, 7055, 8099, 9215, 10403, 11663, 12995, 14399, 15875, 17423, 19043, 20735, 22499, 24335, 26243, 28223, 30275, 32399, 34595, 36863, 39203, 41615, 44099, 46655, 49283, 51983

Offset: 1

Artur Jasinski, Dec 10 2007

The least common multiple of 6*n+1 and 6*n-1. - Colin Barker, Feb 11 2017

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
X. Gourdon and P. Sebah, Collection of series for Pi
Eric Weisstein's World of Mathematics, Pell Equation
Edward Everett Withford, Pell Equation
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A088878, A023208, A136016, A061037, A016910.

[36*n^2 - 1: n in [1..50]]; // Vincenzo Librandi, Jul 09 2012

Mathematica
```
Table[36n^2 - 1, {n, 1, 100}]
```

a(n)=36*n^2-1 \\ Jaume Oliver Lafont, Oct 20 2009

O.g.f.: x*(-35-38*x+x^2)/(-1+x)^3 = 1-35/(-1+x)-108/(-1+x)^2-72/(-1+x)^3. - R. J. Mathar, Dec 12 2007
a(n) = A061037(12n+10)=(6n-1)*(6n+1). - Paul Curtz, Sep 25 2008
Sum_{k>=1} (-1)^(k+1)/a(k) = (Pi-3)/6. - Jaume Oliver Lafont, Oct 20 2009
E.g.f.: 1 + (36 x^2 + 26 x - 1) exp(x). - Robert Israel, Jun 09 2016
Product_{n >= 1} A016910(n) / a(n) = Pi / 3. - Fred Daniel Kline, Jun 09 2016
Sum_{n>=1} 1/a(n) = 1/2 - sqrt(3)*Pi/12. - Amiram Eldar, Jun 27 2020

A157909 a(n) = 81*n^2 - 9.

Original entry on oeis.org

72, 315, 720, 1287, 2016, 2907, 3960, 5175, 6552, 8091, 9792, 11655, 13680, 15867, 18216, 20727, 23400, 26235, 29232, 32391, 35712, 39195, 42840, 46647, 50616, 54747, 59040, 63495, 68112, 72891, 77832, 82935, 88200, 93627, 99216, 104967, 110880, 116955, 123192

Offset: 1

Vincenzo Librandi, Mar 09 2009

The identity (18*n^2 - 1)^2 - (81*n^2 - 9)*(2*n)^2 = 1 can be written as A157910(n)^2 - a(n)*A005843(n)^2 = 1. - Vincenzo Librandi, Feb 08 2012

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, A136016, A157910.

I:=[72, 315, 720]; [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 08 2012

LinearRecurrence[{3, -3, 1}, {72, 315, 720}, 50] (* Vincenzo Librandi, Feb 08 2012 *)
81*Range[40]^2-9 (* Harvey P. Dale, Oct 14 2023 *)

for(n=1, 40, print1(81*n^2 - 9", ")); \\ Vincenzo Librandi, Feb 08 2012

From Vincenzo Librandi, Feb 08 2012: (Start)
G.f.: -9*x*(8 + 11*x - x^2)/(x - 1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Amiram Eldar, Mar 07 2023: (Start)
Sum_{n>=1} 1/a(n) = 1/18 - Pi/(54*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(27*sqrt(3)) - 1/18. (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: 9*(exp(x)*(9*x^2 + 9*x - 1) + 1).
a(n) = 9*A136016(n). (End)

A144410 a(n) = 4(3n+1)(3n+2).

Original entry on oeis.org

8, 80, 224, 440, 728, 1088, 1520, 2024, 2600, 3248, 3968, 4760, 5624, 6560, 7568, 8648, 9800, 11024, 12320, 13688, 15128, 16640, 18224, 19880, 21608, 23408, 25280, 27224, 29240, 31328, 33488, 35720, 38024, 40400, 42848, 45368, 47960, 50624, 53360, 56168, 59048, 62000, 65024, 68120, 71288, 74528, 77840, 81224, 84680, 88208, 91808, 95480

Offset: 0

Paul Curtz, Sep 30 2008

The sequence lists all numbers k such that k+1 is a square and k+4 is divisible by 12. - Bruno Berselli, Sep 28 2017

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

Cf. A060544, A136016, A244977.

[4*(3*n+1)*(3*n+2): n in [0..40]]; // Vincenzo Librandi, Aug 07 2011

A144410:= n-> 4*(3*n+1)*(3*n+2); seq(A144410(n), n=0..60); # G. C. Greubel, Mar 27 2021

Table[4 (3 n + 1) (3 n + 2), {n, 0, 51}] (* or *)
CoefficientList[Series[8 (1 + 7 x + x^2)/(1 - x)^3, {x, 0, 51}], x] (* Michael De Vlieger, Sep 29 2017 *)

a(n)=4*(3*n+1)*(3*n+2) \\ Charles R Greathouse IV, Jun 17 2017

[4*(3*n+1)*(3*n+2) for n in (0..60)] # G. C. Greubel, Mar 27 2021

G.f.: 8*(1 + 7*x + x^2)/(1 - x)^3. - Michael De Vlieger, Sep 29 2017
a(n) = 8*A060544(n+1).
a(n) = A136016(2*n+1).
a(n) = a(m) + 36*(n - m)*(n + m + 1). For m = n-1, a(n) = a(n-1) + 72*n. - Bruno Berselli, Sep 29 2017
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), n >= 3. - Klaus Purath, Jul 05 2020
E.g.f.: 4*(2 +18*x +9*x^2)*exp(x). - G. C. Greubel, Mar 27 2021
From Amiram Eldar, Dec 10 2022: (Start)
Sum_{n>=0} 1/a(n) = Pi/(12*sqrt(3)) (A244977).
Sum_{n>=0} (-1)^n/a(n) = log(2)/6. (End)

A173102 Number of partitions x + y = z with {x,y,z} in {1,2,3,..,3*n} and z > y >= x.

Original entry on oeis.org

2, 9, 20, 36, 56, 81, 110, 144, 182, 225, 272, 324, 380, 441, 506, 576, 650, 729, 812, 900, 992, 1089, 1190, 1296, 1406, 1521, 1640, 1764, 1892, 2025, 2162, 2304, 2450, 2601, 2756, 2916, 3080, 3249, 3422, 3600, 3782, 3969, 4160, 4356, 4556, 4761, 4970

Offset: 1

Artur Jasinski, Feb 09 2010

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Cf. A016766, A136016.

seq( (-1 +(-1)^n +18*n^2)/8, n=1..50); # G. C. Greubel, Mar 03 2020

aa = {}; Do[i = 0; Do[Do[Do[If[x + y == z, i = i + 1], {x, y, 3 n}], {y, 1, 3 n}], {z, 1, 3 n}]; AppendTo[aa, i], {n, 1, 50}]; aa

vector(50, n, (18*n^2 +(-1)^n -1)/8 ) \\ G. C. Greubel, Mar 03 2020

def A173102(n):
    return (9*n**2 - (n % 2))//4 # Chai Wah Wu, Mar 03 2020

Conjectures from Colin Barker, Sep 04 2013: (Start)
a(n) = (-1 + (-1)^n + 18*n^2)/8.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: x*(2+x)*(1+2*x)/((1-x)^3*(1+x)). (End)
Conjecture: a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n+3)/2). - Wesley Ivan Hurt, Mar 12 2015
From Chai Wah Wu, Mar 03 2020: (Start)
a(n) = (9*n^2-1)/4 if n is odd and a(n) = 9*n^2/4 if n is even.
Proof: z ranges from 2 to 3*n. For each z, since z = y+x >= 2*x, x ranges from 1 to floor(z/2), i.e. there are floor(z/2) partitions. Thus the total number of partitions is a(n) = Sum_{z = 2..3*n} floor(z/2).
For z odd, floor(z/2) = floor((z-1)/2).
As a consequence, if n is odd, 3*n is odd and floor(z/2) occur in pairs, i.e. Sum_{z = 2..3*n} floor(z/2) = 2*(Sum_{w = 1..floor(3*n/2)} w) = 2*(Sum_{w = 1..(3*n-1)/2} w) = 2*((3*n-1)*(3*n+1)/8) = (3*n-1)*(3*n+1)/4 = (9*n^2-1)/4.
If n is even, 3*n is even and floor(z/2) occurs in pairs, except for when z = 3*n where floor(z/2) occurs once. Thus Sum_{z = 2..3*n} floor(z/2) = 2*(Sum_{w = 1..floor(3*n/2)} w) - floor(3*n/2).
This is equal to 2*(Sum_{w = 1..3*n/2} w) - 3*n/2 = (3*n/2)*(3*n/2+1) - 3*n/2 = 9*n^2/4.
This also implies that the above conjectures on the recurrence and g.f. are true.
(End)
E.g.f.: (9*x*(1 + x)*cosh(x) + (-1 + 9*x + 9*x^2)*sinh(x))/4. - Stefano Spezia, Mar 04 2020

cp's OEIS Frontend

A202948 a(n+1) = 3*A136016*a(n).

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A010731 Constant sequence: the all 8's sequence.

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A136017 a(n) = 36n^2 - 1.

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Magma

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PARI

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A157909 a(n) = 81*n^2 - 9.

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Magma

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A144410 a(n) = 4*(3*n+1)*(3*n+2).

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Magma

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A173102 Number of partitions x + y = z with {x,y,z} in {1,2,3,..,3*n} and z > y >= x.

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Maple

Mathematica

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Formula

A202948 a(n+1) = 3A136016a(n).

A144410 a(n) = 4(3n+1)(3n+2).