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

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

Author

A.K. Devaraj, Nov 05 2009

Keywords

• nonn
• easy

Comments

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.

Examples

			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.
		

Links

• 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).

Crossrefs

Cf. A165806, A165808, A165809, A166715, A166957, A167190.

Programs

GAP

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

Magma

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

Maple

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

Mathematica

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

PARI

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

Sage

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

Formula

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

Extensions

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