A108946 a(2n) = A001570(n), a(2n+1) = -A007654(n+1).

1, -3, 13, -48, 181, -675, 2521, -9408, 35113, -131043, 489061, -1825200, 6811741, -25421763, 94875313, -354079488, 1321442641, -4931691075, 18405321661, -68689595568, 256353060613, -956722646883, 3570537526921, -13325427460800, 49731172316281

Offset: 0

Author

Creighton Dement, Jul 21 2005

Keywords

• sign
• easy

Comments

In reference to program code, 2baseiseq[X](n) = ((-1)^n)*A001353(n) (a(n)^2 + 1 is a perfect square.) 1tesseq[X](n) = (-1^(n+1))*A097948(n).

Floretion Algebra Multiplication Program, FAMP Code: 1ibaseiseq[X] with X = .5'i + .5i' + 'ii' - .5'jj' + 1.5'kk' - 1 (* Corrected by Creighton Dement, Dec 11 2009 *)

Links

• Robert Munafo, Sequences Related to Floretions
• Index entries for linear recurrences with constant coefficients, signature (-4,0,4,1).

Crossrefs

Cf. A007654, A001570, A076139. See also A117808, A122571 (same except for signs).

Cf. A001353, A097948.

Programs

Magma

/* By definition: */
m:=15; R:=PowerSeriesRing(Integers(), m);
A001570:=Coefficients(R!((1-x)/(1-14*x+x^2)));
A007654:=Coefficients(R!(-3*x^2*(1+x)/(-1+x)/(1-14*x+x^2)));
&cat[[A001570[i],-A007654[i]]: i in [1..m-2]]; // Bruno Berselli, Feb 05 2013

Maple

seriestolist(series((x^2+x+1)/((1-x)*(x+1)*(x^2+4*x+1)), x=0,25));

Mathematica

LinearRecurrence[{-4,0,4,1},{1,-3,13,-48},30] (* Harvey P. Dale, Jun 15 2018 *)

Formula

G.f.: (x^2+x+1)/((1-x)*(x+1)*(x^2+4*x+1)).

Floor(((2 + sqrt(3))^n + (2 - sqrt(3))^n)/4) produces this sequence with a different offset and without signs. - James R. Buddenhagen, May 20 2010

Define c(n) = a(n) - 4*a(n+1) - a(n+2) and d(n) = -a(n) - 4*a(n+1) - a(n+2); Conjectures: I: c(2n) = 24*A076139(n); (Triangular numbers that are one-third of another triangular number) II: c(2n+1) = -A011943(n+1); (Numbers n such that any group of n consecutive integers has integral standard deviation) III: d(2n) = -2; IV: d(2n+1) = -1