A033568 -id:A033568 - cp's OEIS frontend

A125200 a(n) = n(4n^2 + n - 1)/2.

2, 17, 57, 134, 260, 447, 707, 1052, 1494, 2045, 2717, 3522, 4472, 5579, 6855, 8312, 9962, 11817, 13889, 16190, 18732, 21527, 24587, 27924, 31550, 35477, 39717, 44282, 49184, 54435, 60047, 66032, 72402, 79169, 86345, 93942, 101972, 110447, 119379

Offset: 1

Reinhard Zumkeller, Nov 24 2006

a(n) = Sum_{k=1..n} (4*n*k - n - k), sums of rows of the triangle in A125199.

A003415(A003415(a(n))) = 2*A016969(n-1).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Cf. A003415, A016969, A033568, A125199.

[n*(4*n^2 +n-1)div 2:n in [1..40]]; // Vincenzo Librandi, Dec 27 2010

LinearRecurrence[{4,-6,4,-1},{2,17,57,134},40] (* Harvey P. Dale, Feb 05 2013 *)

a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - R. J. Mathar, Feb 12 2010
G.f.: x*(2+9*x+x^2)/(x-1)^4. - R. J. Mathar, Feb 12 2010
a(n) = Sum_{i=1..n} A033568(i). - Bruno Berselli, Jul 22 2013

Definition corrected by Vincenzo Librandi, Dec 27 2010

A033589 a(n) = (2n-1)(3n-1)(4*n-1).

Original entry on oeis.org

-1, 6, 105, 440, 1155, 2394, 4301, 7020, 10695, 15470, 21489, 28896, 37835, 48450, 60885, 75284, 91791, 110550, 131705, 155400, 181779, 210986, 243165, 278460, 317015, 358974, 404481, 453680, 506715

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A060747, A033568, A033590.

[(2*n-1)*(3*n-1)*(4*n-1): n in [0..30]]; // G. C. Greubel, Mar 05 2020

seq( mul(j*n-1, j=2..4), n=0..30); # G. C. Greubel, Mar 05 2020

Table[Times@@(n*Range[2,4]-1),{n,0,30}] (* or *) LinearRecurrence[{4,-6,4,-1},{-1,6,105,440},30] (* Harvey P. Dale, Sep 22 2014 *)

vector(31, n, my(m=n-1); prod(j=2,4, j*m-1) ) \\ G. C. Greubel, Mar 05 2020

[product(j*n-1 for j in (2..4)) for n in (0..30)] # G. C. Greubel, Mar 05 2020

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Harvey P. Dale, Sep 22 2014
G.f.: (-1 +10*x +75*x^2 +60*x^3)/(1-x)^4. - R. J. Mathar, Feb 06 2017
From G. C. Greubel, Mar 05 2020: (Start)
a(n) = n^3 * Pochhammer(2 - 1/n, 3) = Product_{j=2..4} (j*n-1).
E.g.f.: (-1 + 7*x + 46*x^2 + 24*x^3)*exp(x). (End)
Sum_{n>=1} 1/a(n) = (sqrt(3)/2-1)*Pi + 8*log(2) - 9*log(3)/2. - Amiram Eldar, Feb 22 2022

A033590 a(n) = (2n-1)(3n-1)(4n-1)(5*n-1).

Original entry on oeis.org

1, 24, 945, 6160, 21945, 57456, 124729, 238680, 417105, 680680, 1052961, 1560384, 2232265, 3100800, 4201065, 5571016, 7251489, 9286200, 11721745, 14607600, 17996121, 21942544, 26504985, 31744440

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A060747, A033568, A033589.

[(2*n-1)*(3*n-1)*(4*n-1)*(5*n-1): n in [0..40]]; // G. C. Greubel, Mar 05 2020

seq( mul(j*n-1, j=2..5), n=0..40); # G. C. Greubel, Mar 05 2020

Table[(2*n-1)*(3*n-1)*(4*n-1)*(5*n-1), {n,0,40}] (* G. C. Greubel, Mar 05 2020 *)

vector(41, n, my(m=n-1); prod(j=2,5, j*m-1) ) \\ G. C. Greubel, Mar 05 2020

[product(j*n-1 for j in (2..5)) for n in (0..40)] # G. C. Greubel, Mar 05 2020

G.f.: (1 + 19*x + 835*x^2 + 1665*x^3 + 360*x^4)/(1-x)^5. - R. J. Mathar, Feb 06 2017
From G. C. Greubel, Mar 05 2020: (Start)
a(n) = n^4 * Pochhammer(2 - 1/n, 4) = Product_{j=2..5} (j*n-1).
E.g.f.: (1 + 23*x + 449*x^2 + 566*x^3 + 120*x^4)*exp(x). (End)

A104586 Pentagonal wave sequence triangle.

Original entry on oeis.org

1, 7, 2, 12, 5, 1, 26, 15, 7, 2, 35, 22, 12, 5, 1, 57, 40, 26, 15, 7, 2, 70, 51, 35, 22, 12, 5, 1, 100, 77, 57, 40, 26, 15, 7, 2

Offset: 1

Gary W. Adamson, Mar 17 2005

Row sums = A086500: 1, 9, 18, 50, 75, 147, 196...

			The first few rows are:
1;
7, 2;
12, 5, 1;
26, 15, 7, 2;
35, 22, 12, 5, 1;
57, 40, 26, 15, 7, 2;
70, 51, 35, 22, 12, 5, 1;
...

Cf. A086500, A104584, A104585, A000326, A005449, A049452, A033570, A049453, A033568.

Odd columns are terms of A104584, pentagonal wave sequence of the first kind, (starting with 1): 1, 7, 12, 26, 35, 57, 70... Even columns are terms of A104585, pentagonal wave sequence of the second kind (starting with 2): 2, 5, 15, 22, 40, 51... Odd rows are pentagonal numbers (A000326) starting with "1" at the right. Even rows are second pentagonal numbers (A005449) starting with 2 at the right. The triangle is extracted from a matrix product A * B, A = [1; 1, 2; 1, 2, 1; 1, 2, 1, 2;...], B = [1; 3, 1; 5, 3, 1; 7, 5, 3, 1;...] (both infinite lower triangular matrices, with the rest zeros).

cp's OEIS Frontend

A125200 a(n) = n*(4*n^2 + n - 1)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Formula

Extensions

A033589 a(n) = (2*n-1)*(3*n-1)*(4*n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A033590 a(n) = (2*n-1)*(3*n-1)*(4*n-1)*(5*n-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Formula

A104586 Pentagonal wave sequence triangle.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A125200 a(n) = n(4n^2 + n - 1)/2.

A033589 a(n) = (2n-1)(3n-1)(4*n-1).

A033590 a(n) = (2n-1)(3n-1)(4n-1)(5*n-1).