A101463 Expansion of g.f. (x^3+x^2+2*x+1)/(x^4+5*x^2+1).

1, 2, -4, -9, 19, 43, -91, -206, 436, 987, -2089, -4729, 10009, 22658, -47956, -108561, 229771, 520147, -1100899, -2492174, 5274724, 11940723, -25272721, -57211441, 121088881, 274116482, -580171684, -1313370969, 2779769539, 6292738363, -13318676011

Offset: 0

Author

Creighton Dement, Jan 20 2005

Keywords

• easy
• sign

Comments

A floretion-generated sequence relating to Pythagoras' theorem generalized.

Floretion Algebra Multiplication Program. FAMP code: em[J* ]sigcycseq[ + .75'i + .5'k + .25i' + .5j' + .5k' - .25'ii' + .25'jj' - .25'kk' - .75'jk' + .5'ki' - .25'kj' + .25e]

References

• F. A. Haight, On a generalization of Pythagoras' theorem, pp. 73-77 of J. C. Butcher, editor, A Spectrum of Mathematics. Auckland University Press, 1971.

Links

• James A. Sellers, Domino Tilings and Products of Fibonacci and Pell Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.2.
• Index entries for linear recurrences with constant coefficients, signature (0,-5,0,-1)

Crossrefs

Elements of even index in the sequence gives A004253. Elements of odd index in the sequence gives A002310.

Cf. A004253, A002310.

Programs

Mathematica

CoefficientList[Series[(x^3+x^2+2x+1)/(x^4+5x^2+1),{x,0,30}],x] (* or *) LinearRecurrence[{0,-5,0,-1},{1,2,-4,-9},31] (* Harvey P. Dale, Apr 15 2012 *)

Formula

Let b(1)=1, b(2)=2, b(3)=4 and b(n)=(b(n-1)*b(n-2)+(3+(-1)^n)/2)/b(n-3) then b(n)=abs(a(n)) - Benoit Cloitre, Mar 03 2007

a(n) = -5*a(n-2)-a(n-4), n>3. [Harvey P. Dale, Apr 15 2012]

G.f.: ( 1+2*x+x^2+x^3 ) / ( 1+5*x^2+x^4 ). - R. J. Mathar, Jun 18 2014

a(n) = -3a(n-1)+2a(n-2) if n even. a(n) = (5*a(n-1)+a(n-2))/2 if n odd. - R. J. Mathar, Jun 18 2014