A152994 -id:A152994 - cp's OEIS frontend

A316466 a(n) = 2n(7*n - 3).

0, 8, 44, 108, 200, 320, 468, 644, 848, 1080, 1340, 1628, 1944, 2288, 2660, 3060, 3488, 3944, 4428, 4940, 5480, 6048, 6644, 7268, 7920, 8600, 9308, 10044, 10808, 11600, 12420, 13268, 14144, 15048, 15980, 16940, 17928, 18944, 19988, 21060, 22160, 23288, 24444, 25628, 26840

Offset: 0

Bruno Berselli, Jul 04 2018

This is the case k = 9 of Sum_{i = 2..k} P(i,n) = (k - 1)*n*((k - 2)*n - (k - 6))/4, where P(k,n) = n*((k - 2)*n - (k - 4))/2 (see Crossrefs for similar sequences and "Square array in A139600" in Links section).

14*x + 9 is a square for x = a(n) or x = a(-n).

Colin Barker, Table of n, a(n) for n = 0..1000
Bruno Berselli, Square array in A139600.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A139600, A218471.

Similar sequences (see the first comment): A000096 (k = 3), A045943 (k = 4), A049451 (k = 5), A033429 (k = 6), A167469 (k = 7), A152744 (k = 8), this sequence (k = 9), A152994 (k = 10).

GAP
```
List([0..50], n -> 2*n*(7*n-3));
```
Julia
```
[2*n*(7*n-3) for n in 0:50] |> println
```
Magma
```
[2*n*(7*n-3): n in [0..50]];
```

Table[2 n (7 n - 3), {n, 0, 50}]
LinearRecurrence[{3,-3,1},{0,8,44},50] (* Harvey P. Dale, Jan 24 2021 *)

Maxima
```
makelist(2*n*(7*n-3), n, 0, 50);
```
PARI
```
vector(50, n, n--; 2*n*(7*n-3))
```

concat(0, Vec(4*x*(2 + 5*x)/(1 - x)^3 + O(x^40))) \\ Colin Barker, Jul 05 2018

Python
```
[2*n*(7*n-3) for n in range(50)]
```
Sage
```
[2*n*(7*n-3) for n in (0..50)]
```

O.g.f.: 4*x*(2 + 5*x)/(1 - x)^3.
E.g.f.: 2*x*(4 + 7*x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
a(n) = 4*A218471(n).

A139609 a(n) = 36*n + 9.

Original entry on oeis.org

9, 45, 81, 117, 153, 189, 225, 261, 297, 333, 369, 405, 441, 477, 513, 549, 585, 621, 657, 693, 729, 765, 801, 837, 873, 909, 945, 981, 1017, 1053, 1089, 1125, 1161, 1197, 1233, 1269, 1305, 1341, 1377, 1413, 1449, 1485, 1521, 1557, 1593, 1629, 1665, 1701

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 9th column of positive numbers in the square array of nonnegative and polygonal numbers A139600.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index to sequences related to polygonal numbers
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016813, A044102, A057145, A139600, A152994.

[9*(4*n + 1): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

Range[9, 7000, 36] (* Vladimir Joseph Stephan Orlovsky, Jul 23 2011 *)
36*Range[0,50]+9 (* or *) LinearRecurrence[{2,-1},{9,45},50] (* Harvey P. Dale, Jan 04 2018 *)

a(n)=36*n+9 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = A057145(n+2,9).
G.f.: 9*(1+3*x)/(x-1)^2. - R. J. Mathar, Jul 28 2016
From Elmo R. Oliveira, Apr 16 2024: (Start)
E.g.f.: 9*exp(x)*(1 + 4*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 9*A016813(n) = A044102(n) + 9 = A152994(n+1) - A152994(n). (End)

A144560 Ten times hexagonal numbers: 10n(2*n-1).

Original entry on oeis.org

0, 10, 60, 150, 280, 450, 660, 910, 1200, 1530, 1900, 2310, 2760, 3250, 3780, 4350, 4960, 5610, 6300, 7030, 7800, 8610, 9460, 10350, 11280, 12250, 13260, 14310, 15400, 16530, 17700, 18910, 20160, 21450, 22780, 24150, 25560, 27010

Offset: 0

Omar E. Pol, Jan 01 2009

Sequence found by reading the line from 0, in the direction 0, 10,..., in the square spiral whose vertices are the generalized dodecagonal numbers A195162. - Omar E. Pol, Sep 18 2011

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

Cf. A000384, A002939, A152745, A152994.

[10*n*(2*n-1): n in [0..40]]; // G. C. Greubel, May 30 2024

10*Binomial[2*Range[0,40], 2] (* G. C. Greubel, May 30 2024 *)

a(n)=10*n*(2*n-1) \\ Charles R Greathouse IV, Jun 17 2017

[10*n*(2*n-1) for n in range(41)] # G. C. Greubel, May 30 2024

a(n) = 20*n^2 - 10*n = 10*A000384(n) = 5*A002939(n) = 2*A152745(n).
a(n) = a(n-1) +40*n -30  (with a(0)=0). - Vincenzo Librandi, Dec 14 2010
From G. C. Greubel, May 30 2024: (Start)
G.f.: 10*x*(1 + 3*x)/(1-x)^3.
E.g.f.: 10*x*(1 + 2*x)*exp(x). (End)

cp's OEIS Frontend

A316466 a(n) = 2*n*(7*n - 3).

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GAP

Julia

Magma

Mathematica

Maxima

PARI

PARI

Python

Sage

Formula

A139609 a(n) = 36*n + 9.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A144560 Ten times hexagonal numbers: 10*n*(2*n-1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A316466 a(n) = 2n(7*n - 3).

A144560 Ten times hexagonal numbers: 10n(2*n-1).