A167469 -id:A167469 - 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).

A167467 a(n) = 25n^3 - n(5*n+1)/2 + 1.

Original entry on oeis.org

23, 190, 652, 1559, 3061, 5308, 8450, 12637, 18019, 24746, 32968, 42835, 54497, 68104, 83806, 101753, 122095, 144982, 170564, 198991, 230413, 264980, 302842, 344149, 389051, 437698, 490240, 546827, 607609, 672736, 742358, 816625, 895687, 979694, 1068796

Offset: 1

A.K. Devaraj, Nov 05 2009

Also the real part of f(x+n*f(x,y,z), y+n*f(x,y,z), z+n*f(x,y,z))/f(x,y,z) for f(x,y,z) = x^3+y^2+z at x=(-1+i*sqrt(3))/2, y=i and z=5.
If f(x,y,z) is a trivariate polynomial, f(x+n*f(x,y,z),y+n*f(x,y,z),z+n*f(x,y,z)) is congruent to 0 (mod f(x,y,z)).
The ratio f(x+n*f,y+n*f,z+n*f)/f of these two functions is decomposed into the real part (this sequence here), and the imaginary part. The imaginary part is 2*n*i + sqrt(3)*A167469(n)*i, where i=sqrt(-1) is the imaginary unit.

			f(x +f(x,y,z), y + f(x,y,z), z + f(x,y,z)) = (23 + 2i + 6*sqrt(3)*i)* f(x,y,z) at 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. A165806, A165808, A165809, A166715, A166957, A167190.

List([1..50], n-> 25*n^3 - n*(5*n+1)/2 + 1); # G. C. Greubel, Sep 01 2019

[25*n^3 - n*(5*n+1)/2 + 1: n in [1..50]]; // G. C. Greubel, Sep 01 2019

f := proc(x,y,z) x^3+y^2+z ; end proc:
A167467 := proc(n) local rho,a ,x,y,z; a := f(x+n*f(x,y,z),y+n*f(x,y,z),z+n*f(x,y,z))/f(x,y,z) ; rho := (-1+I*sqrt(3))/2 ; a := subs({x = rho, y=I,z=5},a) ; a := expand(a) ; Re(a) ; end:
seq(A167467(n),n=1..50) ; # R. J. Mathar, Nov 12 2009

LinearRecurrence[{4,-6,4,-1},{23,190,652,1559},50] (* Harvey P. Dale, Sep 28 2012 *)

a(n)=1+25*n^3-n*(5*n+1)/2 \\ Charles R Greathouse IV, Jul 07 2013

[25*n^3 - n*(5*n+1)/2 + 1 for n in (1..50)] # G. C. Greubel, Sep 01 2019

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(23 + 98*x + 30*x^2 - x^3)/(1-x)^4.
E.g.f.: (2 + 44*x + 145*x^2 + 50*x^3)*exp(x)/2 -1. - G. C. Greubel, Apr 09 2016

a(2) and a(3) corrected, definition simplified and sequence extended by R. J. Mathar, Nov 12 2009

cp's OEIS Frontend

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

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Formula

A167467 a(n) = 25n^3 - n(5*n+1)/2 + 1.

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Author

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Crossrefs

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GAP

Magma

Maple

Mathematica

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Sage

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Julia

Magma

Mathematica

Maxima

PARI

PARI

Python

Sage

Formula

A167467 a(n) = 25*n^3 - n*(5*n+1)/2 + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

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

A167467 a(n) = 25n^3 - n(5*n+1)/2 + 1.