A022272 -id:A022272 - cp's OEIS frontend

A022288 a(n) = n(31n-1)/2.

0, 15, 61, 138, 246, 385, 555, 756, 988, 1251, 1545, 1870, 2226, 2613, 3031, 3480, 3960, 4471, 5013, 5586, 6190, 6825, 7491, 8188, 8916, 9675, 10465, 11286, 12138, 13021, 13935, 14880, 15856, 16863, 17901

Offset: 0

N. J. A. Sloane

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

Cf. A000217, A022289.

Cf. similar sequences of the form n*((2*k+1)*n - 1)/2: A161680 (k=0), A000326 (k=1), A005476 (k=2), A022264 (k=3), A022266 (k=4), A022268 (k=5), A022270 (k=6), A022272 (k=7), A022274 (k=8), A022276 (k=9), A022278 (k=10), A022280 (k=11), A022282 (k=12), A022284 (k=13), A022286 (k=14), this sequence (k=15).

Table[n (31 n - 1)/2, {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 15, 61}, 40] (* Harvey P. Dale, Mar 31 2014 *)

a(n)=n*(31*n-1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 31*n + a(n-1) - 16 for n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
a(0)=0, a(1)=15, a(2)=61; for n>2, a(n) = 3*a(n-1)-3*a(n-2)+a(n-3). - Harvey P. Dale, Mar 31 2014
G.f.: x*(15 + 16*x)/(1 - x)^3. - R. J. Mathar, Sep 02 2016
a(n) = A000217(16*n-1) - A000217(15*n-1). In general, n*((2*k+1)*n - 1)/2 = A000217((k+1)*n-1) - A000217(k*n-1), and the ordinary generating function is x*(k + (k+1)*x)/(1 - x)^3. - Bruno Berselli, Oct 14 2016
E.g.f.: (x/2)*(31*x + 30)*exp(x). - G. C. Greubel, Aug 24 2017

A064761 a(n) = 15*n^2.

Original entry on oeis.org

0, 15, 60, 135, 240, 375, 540, 735, 960, 1215, 1500, 1815, 2160, 2535, 2940, 3375, 3840, 4335, 4860, 5415, 6000, 6615, 7260, 7935, 8640, 9375, 10140, 10935, 11760, 12615, 13500, 14415, 15360, 16335, 17340, 18375, 19440, 20535, 21660, 22815

Offset: 0

Roberto E. Martinez II, Oct 18 2001

Number of edges in a complete 6-partite graph of order 6n, K_n,n,n,n,n,n.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A000290, A008585, A008587, A022272, A033428, A033429, A033581, A033583.

Cf. A008597, A195046.

Table[15*n^2, {n, 0, 45}] (* Amiram Eldar, Feb 03 2021 *)

a(n)=15*n^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 15*A000290(n) = 5*A033428(n) = 3*A033429(n). - Omar E. Pol, Dec 13 2008
a(n) = A008587(n)*A008585(n). - Reinhard Zumkeller, Apr 12 2010
a(n) = a(n-1) + 30*n - 15 for n > 0, a(0)=0. - Vincenzo Librandi, Dec 15 2010
a(n) = A022272(n) + A022272(-n). - Bruno Berselli, Mar 31 2015
a(n) = t(6*n) - 6*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(6*n) - 6*A000217(n). - Bruno Berselli, Aug 31 2017
From Amiram Eldar, Feb 03 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/90.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/180.
Product_{n>=1} (1 + 1/a(n)) = sqrt(15)*sinh(Pi/sqrt(15))/Pi.
Product_{n>=1} (1 - 1/a(n)) = sqrt(15)*sin(Pi/sqrt(15))/Pi. (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 15*x*(1 + x)/(1 - x)^3.
E.g.f.: 15*x*(1 + x)*exp(x).
a(n) = n*A008597(n) = A195046(2*n).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A022273 a(n) = n(15n + 1)/2.

Original entry on oeis.org

0, 8, 31, 69, 122, 190, 273, 371, 484, 612, 755, 913, 1086, 1274, 1477, 1695, 1928, 2176, 2439, 2717, 3010, 3318, 3641, 3979, 4332, 4700, 5083, 5481, 5894, 6322, 6765, 7223, 7696, 8184, 8687, 9205, 9738, 10286, 10849, 11427, 12020, 12628, 13251, 13889

Offset: 0

N. J. A. Sloane

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

Cf. A022272, A064761, A110449.

Cf. similar sequences listed in A022289.

[n*(15*n + 1)/2: n in [0..45]]; // Vincenzo Librandi, Mar 31 2015

Table[n (15 n + 1)/2, {n, 0, 40}] (* Bruno Berselli, Mar 12 2015 *)
CoefficientList[Series[x (8 + 7 x) / (1 - x)^3, {x, 0, 40}], x]; (* Vincenzo Librandi, Mar 31 2015 *)

a(n)=n*(15*n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = A110449(n, 7) for n>6.
a(n) = 15*n + a(n-1) - 7 for n>0, a(0)=0. - Vincenzo Librandi, Aug 04 2010
G.f.: x*(8+7*x)/(1-x)^3. - Vincenzo Librandi, Mar 31 2015
a(n) = 3*a(n-1) - 3*a(n-2) - a(n-3) for n>2. - Vincenzo Librandi, Mar 31 2015
a(n) = A022272(-n). - Bruno Berselli, Mar 31 2015
a(n) + a(-n) = A064761(n). - Bruno Berselli, Mar 31 2015
a(n) = A000217(8*n) - A000217(7*n). - Bruno Berselli, Oct 13 2016
E.g.f.: (x/2)*(15*x + 16)*exp(x). - G. C. Greubel, Aug 23 2017

More terms from Vincenzo Librandi, Mar 31 2015

A360962 Square array T(n,k) = k((3+6n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 1, 0, 4, 5, 0, 7, 17, 12, 0, 10, 29, 39, 22, 0, 13, 41, 66, 70, 35, 0, 16, 53, 93, 118, 110, 51, 0, 19, 65, 120, 166, 185, 159, 70, 0, 22, 77, 147, 214, 260, 267, 217, 92, 0, 25, 89, 174, 262, 335, 375, 364, 284, 117, 0, 28, 101, 201, 310, 410, 483, 511, 476, 360, 145

Offset: 0

Paul Curtz, Feb 27 2023

The main diagonal is A024394.

The antidiagonals sums are A000537.

			The rows are:
  0  1  5  12  22  35  51  70 ... = A000326
  0  4 17  39  70 110 159 217 ... = A022266
  0  7 29  66 118 185 267 364 ... = A022272
  0 10 41  93 166 260 375 511 ... = A022278
  0 13 53 120 214 335 483 658 ... = A022284
  ... .
Columns: A000004, A016777, A017581, A154266=3*A017209, 2*A348845, 5*A161447, 3*A158057(n+1), ... (coefficients from A026741).
Difference between two consecutive rows are: A033428.
This square array read by antidiagonals leads to the triangle
  0
  0  1
  0  4  5
  0  7 17 12
  0 10 29 39  22
  0 13 41 66  70  35
  0 16 53 93 118 110 51
  ... .

Cf. A000326, A000537, A022266, A022272, A022278, A022284, A024394.
Cf. A033428.
Cf. A000004, A016777, A017209, A017581, A026741, A154266, A158057, A161447, A348845.

T:= (n,k)-> k*(k*(3+6*n)-1)/2:
seq(seq(T(d-k,k), k=0..d), d=0..10);  # Alois P. Heinz, Feb 28 2023

T[n_, k_] := ((6*n + 3)*k - 1)*k/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 27 2023 *)

T(n,k) = k*((3+6*n)*k-1)/2; \\ Michel Marcus, Feb 27 2023

Take successively sequences n*(3*n-1)/2, n*(9*n-1)/2, n*(15*n-1)/2, n*(21*n-1)/2, ... listed in the EXAMPLE section.
From Stefano Spezia, Feb 21 2024: (Start)
G.f.: y*(1 + 2*y + x*(2 + y))/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2 + 3*y + 6*x*(1 + y))/2. (End)

cp's OEIS Frontend

A022288 a(n) = n*(31*n-1)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A064761 a(n) = 15*n^2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A022273 a(n) = n*(15*n + 1)/2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A360962 Square array T(n,k) = k*((3+6*n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A022288 a(n) = n(31n-1)/2.

A022273 a(n) = n(15n + 1)/2.

A360962 Square array T(n,k) = k((3+6n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.