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A127546 a(n) = F(n)^2 + F(n+1)^2 + F(n+2)^2, where F(n) denotes the n-th Fibonacci number. 3
2, 6, 14, 38, 98, 258, 674, 1766, 4622, 12102, 31682, 82946, 217154, 568518, 1488398, 3896678, 10201634, 26708226, 69923042, 183060902, 479259662, 1254718086, 3284894594, 8599965698, 22515002498, 58945041798, 154320122894, 404015326886, 1057725857762 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

The following conjecture, if not already well-known, is probably easy to prove: a(n) = 3a(n-1)-a(n-2)-2(-1)^n, for n=4,5,6,... . (This has been verified up to n=1000.)

a(n)=2*A061646(n+1) = 4*F(n+1)^2-2*(-1)^(n+1). - Emeric Deutsch, Apr 04 2007, Gary Detlefs, Nov 27 2010

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Shalosh B. Ekhad and Doron Zeilberger, Automatic Counting of Tilings of Skinny Plane Regions, arXiv preprint arXiv:1206.4864, 2012.

FORMULA

a(n) = 2*(F(n)^2+F(n+1)^2+F(n)*F(n+1)). - Emeric Deutsch, Apr 04 2007

G.f.: 2(1+x-x^2)/((1+x)(1-3x+x^2)). - R. J. Mathar, Nov 25 2008

EXAMPLE

a(2)=14 because F(2)^2+F(3)^2+F(4)^2=1+4+9=14.

MAPLE

with(combinat): a:=n->fibonacci(n)^2+fibonacci(n+1)^2+fibonacci(n+2)^2: seq(a(n), n=0..32); # Emeric Deutsch, Apr 04 2007

A000045 := proc(n) combinat[fibonacci](n) ; end: A127546 := proc(n) add( A000045(i+1)^2, i=n..n+2) ; end: for n from 1 to 33 do printf("%d, ", A127546(n)) ; od ; # R. J. Mathar, Apr 03 2007

with(combinat): seq(4*fibonacci(n+1)^2-2*(-1)^n, n=0..29)

MATHEMATICA

Total/@(Partition[Fibonacci[Range[0, 30]], 3, 1]^2) (* Harvey P. Dale, Oct 20 2011 *)

PROG

(PARI) for(n=0, 10, print1(4*fibonacci(n+1)^2-2*(-1)^n, ", "))

CROSSREFS

Cf. A061646.

Sequence in context: A275208 A000634 A006654 * A192484 A217861 A188492

Adjacent sequences:  A127543 A127544 A127545 * A127547 A127548 A127549

KEYWORD

nonn

AUTHOR

Simone Severini, Apr 01 2007

EXTENSIONS

Edited and extended by R. J. Mathar, Emeric Deutsch and John W. Layman, Apr 09 2007

STATUS

approved

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Last modified October 20 22:44 EDT 2019. Contains 328291 sequences. (Running on oeis4.)