A145910 -id:A145910 - cp's OEIS frontend

A085001 a(n) = (3n+1)(3*n+4).

4, 28, 70, 130, 208, 304, 418, 550, 700, 868, 1054, 1258, 1480, 1720, 1978, 2254, 2548, 2860, 3190, 3538, 3904, 4288, 4690, 5110, 5548, 6004, 6478, 6970, 7480, 8008, 8554, 9118, 9700, 10300, 10918, 11554, 12208, 12880, 13570, 14278, 15004, 15748, 16510, 17290, 18088

Offset: 0

Gary W. Adamson, Jun 17 2003

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 38.

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

Cf. A145910.

[(3*n+1)*(3*n+4): n in [0..50]]; // Vincenzo Librandi, Jul 08 2012

CoefficientList[Series[2*(2+8x-x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 08 2012 *)
Table[(3n+1)(3n+4),{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{4,28,70},50] (* Harvey P. Dale, Apr 07 2019 *)

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

Sum_{k=0..n} 3/a(k) = 3*(n+1)/(3*n+4). [Corrected by Gary Detlefs, Mar 14 2018]
Sum_{k>=0} 3/a(k) = 1.
From Gary W. Adamson, Jan 03 2007: (Start)
Sum_{k>=0} 1/a(k) = 1/3.
Sum_{k=0..n} 1/a(k) = (n+1)/(3*n+4) [Jolley]. (End) [Corrected by Gary Detlefs, Mar 14 2018]
G.f.: 2*(2+8*x-x^2)/(1-x)^3. - R. J. Mathar, Sep 17 2008
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jul 08 2012
Sum_{n>=0} (-1)^n/a(n) = 2*Pi/(9*sqrt(3)) + 2*log(2)/9 - 1/3. - Amiram Eldar, Oct 08 2023
From Elmo R. Oliveira, Nov 15 2024: (Start)
E.g.f.: exp(x)*(4 + 24*x + 9*x^2).
a(n) = 2*A145910(n). (End)

Edited by Don Reble, Nov 13 2005

A178977 a(n) = (3n+2)(3*n+5)/2.

Original entry on oeis.org

5, 20, 44, 77, 119, 170, 230, 299, 377, 464, 560, 665, 779, 902, 1034, 1175, 1325, 1484, 1652, 1829, 2015, 2210, 2414, 2627, 2849, 3080, 3320, 3569, 3827, 4094, 4370, 4655, 4949, 5252, 5564, 5885, 6215, 6554, 6902, 7259, 7625, 8000, 8384, 8777, 9179, 9590, 10010

Offset: 0

Paul Curtz, Jan 02 2011

Companion to A145910.

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

Cf. A008585, A145910, A168233, A168300, A016789, A178971, A235332.

[n*(n+3)/2: n in [2..135 by 3]]; // Bruno Berselli, Apr 14 2011

A178977:=n->(3*n+2)*(3*n+5)/2: seq(A178977(n), n=0..50); # Wesley Ivan Hurt, Oct 23 2014

f[n_] := (3 n + 2) (3 n + 5)/2; Array[f, 45, 0]
LinearRecurrence[{3,-3,1},{5,20,44},50] (* Harvey P. Dale, Apr 19 2013 *)

a(n)=(3*n+2)*(3*n+5)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 6 + 9*n.
a(n) = A178971(3*n+2).
a(n) = A145910(n) + 3 + 3*n = A145910(n) + A008585(n+1).
a(n) = A168233(n+1)*A168300(n+1).
G.f.: (-5-5*x+x^2)/(x-1)^3. [Adapted to the offset by Bruno Berselli, Apr 14 2011]
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Apr 19 2013
From Amiram Eldar, Mar 10 2022: (Start)
Sum_{n>=0} 1/a(n) = 1/3.
Sum_{n>=0} (-1)^n/a(n) = 4*Pi/(9*sqrt(3)) - 1/3 - 4*log(2)/9. (End)
From Elmo R. Oliveira, Oct 30 2024: (Start)
E.g.f.: exp(x)*exp(x)*(5 + 15*x + 9*x^2/2).
a(n) = A016789(n)*A016789(n+1)/2. (End)

A178978 a(n) = A144448(n+1)/8.

Original entry on oeis.org

0, 2, 5, 1, 14, 20, 1, 35, 44, 2, 65, 77, 10, 104, 119, 5, 152, 170, 7, 209, 230, 28, 275, 299, 4, 350, 377, 5, 434, 464, 55, 527, 560, 22, 629, 665, 26, 740, 779, 91, 860, 902, 35, 989, 1034, 40, 1127, 1175, 136, 1274, 1325, 17

Offset: 0

Paul Curtz, Jan 02 2011

Differs from A178971 for indices n > 23.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A061039, A145910, A145911, A178971, A178977, A178978, A144448.

A061039 := proc(n) numer(1/9-1/n^2) ; end proc:
A144448 := proc(n) A061039(1+2*n) ; end proc:
A178978 := proc(n) A144448(n+1)/8 ; end proc:
seq(A178978(n),n=0..80) ; # R. J. Mathar, Jan 06 2011

Table[Numerator[1/9 -1/(2*n+3)^2]/8, {n, 0, 75}] (* G. C. Greubel, Mar 06 2022 *)

[numerator(1/9 -1/(2*n+3)^2)/8 for n in (0..75)] # G. C. Greubel, Mar 06 2022

Trisections:
  a(3*n)   = A145911(n);
  a(3*n+1) = A145910(n);
  a(3*n+2) = A178977(n).
a(n) = 3*a(n-27) - 3*a(n-54) + a(n-81). - G. C. Greubel, Mar 06 2022

cp's OEIS Frontend

A085001 a(n) = (3n+1)(3*n+4).

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A178977 a(n) = (3n+2)(3*n+5)/2.

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A178978 a(n) = A144448(n+1)/8.

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cp's OEIS Frontend

A085001 a(n) = (3*n+1)*(3*n+4).

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Magma

Mathematica

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A178977 a(n) = (3*n+2)*(3*n+5)/2.

Views

Author

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Magma

Maple

Mathematica

PARI

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A178978 a(n) = A144448(n+1)/8.

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Maple

Mathematica

Sage

Formula

A085001 a(n) = (3n+1)(3*n+4).

A178977 a(n) = (3n+2)(3*n+5)/2.