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A278417 a(n) = n*((2+sqrt(3))^n + (2-sqrt(3))^n)/2. 1
0, 2, 14, 78, 388, 1810, 8106, 35294, 150536, 632034, 2620870, 10759342, 43804812, 177105266, 711809378, 2846259390, 11330543632, 44929049794, 177540878718, 699402223118, 2747583822740, 10766828545746, 42095796462874, 164244726238366, 639620518118424, 2486558615814050, 9651161613824822, 37403957244654702 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This was originally based on a graph theory formula in the Wikipedia which turned out to be wrong.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (8,-18,8,-1).

FORMULA

From Colin Barker, Nov 21 2016: (Start)

a(n) = 7*a(n-1) - 10*a(n-2) - 10*a(n-3) + 7*a(n-4) - a(n-5) for n>6.

G.f.: 2*x^3*(39 - 118*x + 55*x^2 - 7*x^3) / (1 - 4*x + x^2)^2.

(End)

MAPLE

f:=n->expand(n*((2+sqrt(3))^n + (2-sqrt(3))^n)/2); # N. J. A. Sloane, May 13 2017

MATHEMATICA

Table[Simplify[(n/2) (((2 + #)^n + (2 - #)^n)) &@ Sqrt@ 3], {n, 3, 27}] (* or *)

Drop[#, 3] &@ CoefficientList[Series[2 x^3*(39 - 118 x + 55 x^2 - 7 x^3)/(1 - 4 x + x^2)^2, {x, 0, 27}], x] (* Michael De Vlieger, Nov 24 2016 *)

LinearRecurrence[{8, -18, 8, -1}, {0, 2, 14, 78}, 30] (* Harvey P. Dale, Jan 01 2021 *)

PROG

(Python)

import math

def p(n):

m=math.sqrt(3)

n=float(n)

x=2+m

y=2-m

return round((n/2)*(x**n+y**n), 0)

for i in range(3, 531):

print str(i)+" "+str(int(p(i))) \\Indranil Ghosh, Nov 21 2016

(PARI) vector(25, n, n+=2; n*((2+sqrt(3))^n + ((2-sqrt(3))^n))/2) \\ Colin Barker, Nov 21 2016

(PARI) Vec(2*x^3*(39 - 118*x + 55*x^2 - 7*x^3) / (1 - 4*x + x^2)^2 + O(x^30)) \\ Colin Barker, Nov 21 2016

CROSSREFS

Cf. A030019, A069996, A139400, A193153.

Sequence in context: A277297 A185055 A034573 * A339240 A133224 A183577

Adjacent sequences: A278414 A278415 A278416 * A278418 A278419 A278420

KEYWORD

nonn,easy

AUTHOR

Indranil Ghosh, Nov 21 2016

EXTENSIONS

Entry revised by N. J. A. Sloane, May 13 2017

STATUS

approved

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Last modified December 8 01:51 EST 2022. Contains 358672 sequences. (Running on oeis4.)