Sequence relating to Pell numbers; a new technique

[creigh at o2online.de]

Dear fellow Seqfans,

The following is perhaps an interesting sequence relating to the Pell numbers:

(a(n)) = (2, 5, 13, 32, 78, 189, 457, 1104, 2666, 6437, 15541, 37520, ) 

a(n) = A000129(n+2) + A000129(n+1) - A024537(n)
or, using the existing formula,
A024537(n+1) =  1/2 * (P(n+1)+P(n)+1), with P(n) = Pell numbers (A000129)
we have 
a(n) = 1/2 * ( 2*P(n+2) + P(n+1) - P(n) - 1 )  
 
Notes:

- This was found by applying ves, tes, to the floretion:
z = .5( - 'i' - i' - 'ii' + 'jj' + 'ij' + 'ji' +'jk' + kj') + 1 

- Also, the relation A048654(n) = 2P(n+2) - 3P(n+1) was found. 
(see below for the description of a new technique!)
 

TEST: APPLY VARIOUS TRANSFORMATIONS TO a(n) -from above- AND LOOK IT 
UP IN THE ENCYCLOPEDIA AGAIN 

SUCCESS 
(limited to 40 matches): 

Transformation T030 gave a match with: 
%I A048655 
%S A048655 1,5,11,27,65,157,379,915,2209,5333,12875,31083,75041,181165,
[snip]
%N A048655 Generalized Pellian with second term equal to 5. 
%D A048655 M. Bicknell, A Primer on the Pell Sequence and related sequences, 
Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349. 
%D A048655 A. F. Horadam, Basic Properties of a Certain Generalized Sequence 
of Numbers, Fibonacci Quarterly, Vol. 3, No. 3, 1965, pp. 161-176. 
%D A048655 A. F. Horadam, Special Properties of the Sequence W(a, b; p, 
q), Fibonacci Quarterly, Vol. 5, No. 5, 1967, pp. 424-434. 
%D A048655 A. F. Horadam, Pell Identities, Fibonacci Quarterly, Vol. 9, 
No. 3, 1971, pp. 245-252. 
%F A048655 a(n)=2*a(n-1)+a(n-2); a(0)=1, a(1)=5. 
%F A048655 a(n)=[ (4+sqrt(2))(1+sqrt(2))^n - (4-sqrt(2))(1-sqrt(2))^n ]
/2*sqrt(2). 
[snip]
%K A048655 easy,nice,nonn 
%O A048655 0,2 
%A A048655 Barry E. Williams 

Transformation T050 gave a match with: 
%I A076736 
%S A076736 1,1,1,2,1,4,2,8,4,16,8,32,16,64,32,128,64,256,128,512,256,1024,512, 
[snip]
%N A076736 Let u(1)=u(2)=u(3)=2, u(n)=(1+u(n-1)u(n-2))/u(n-3) then a(n) 
is the denominator of u(n). 
%C A076736 The sequence 1,4,2,8,4,... has g.f. (1+4x)/(1-2x^2) and a(n)
=(2^(n/2)(1+2sqrt(2)+(1-2sqrt(2))(-1)^n)/2. - Paul Barry (pbarry(AT)wit.
ie), Apr 26 2004 
%C A076736 The sequence 2,1,4,2,8,... has a(n)=2^(n/2)(1+2sqrt(2)-(1-2sqrt(2))
(-1)^n)/(2sqrt(2)) and is essentially the pair-reversal of A016116. - Paul 
Barry (pbarry(AT)wit.ie), Apr 26 2004 
%F A076736 For n>4, a(n)=2^A028242(n-4) 
[snip]
%K A076736 frac,nonn 
%O A076736 1,4 
%A A076736 Benoit Cloitre (abcloitre(AT)wanadoo.fr), Nov 24 2002 
%E A076736 More terms from Paul Barry (pbarry(AT)wit.ie), Apr 26 2004 

Transformation T018 gave a match with: 
%I A078343 
%S A078343 1,2,3,8,19,46,111,268,647,1562,3771,9104,21979,53062,128103,
[snip]
%N A078343 a(0) = -1, a(1) = 2; a(n) = 2*a(n-1) + a(n-2). 
%D A078343 H. S. M. Coxeter, 1998, Numerical distances among the circles 
in a loxodromic sequence, Nieuw Arch. Wisk, 16, pp. 1-9. 
%F A078343 For the unsigned version: a(1)=1; a(2)=2; a(n) = sum(k=2,n-1,
a(k) + a(k-1) ). 
%F A078343 a(n) is asymptotic to (1/4)*(8-5*sqrt(2))*(1+sqrt(2))^n. 
%F A078343 a(n) = A048746(n-3) + 2, for n>2. - Ralf Stephan (ralf(AT)ark.
in-berlin.de), Oct 17 2003 
[snip]
%O A078343 0,2 
%A A078343 Benoit Cloitre (abcloitre(AT)wanadoo.fr), Nov 22 2002 
%E A078343 Entry revised by njas, Apr 29 2004 

Transformation T019 gave a match with: 
%I A048655 
%S A048655 1,5,11,27,65,157,379,915,2209,5333,12875,31083,75041,181165,
[snip]
%N A048655 Generalized Pellian with second term equal to 5. 
%D A048655 M. Bicknell, A Primer on the Pell Sequence and related sequences, 
Fibonacci Quarterly, Vol. 13, No. 4, 1975, pp. 345-349. 
%D A048655 A. F. Horadam, Basic Properties of a Certain Generalized Sequence 
of Numbers, Fibonacci Quarterly, Vol. 3, No. 3, 1965, pp. 161-176. 
%D A048655 A. F. Horadam, Special Properties of the Sequence W(a, b; p, 
q), Fibonacci Quarterly, Vol. 5, No. 5, 1967, pp. 424-434. 
%D A048655 A. F. Horadam, Pell Identities, Fibonacci Quarterly, Vol. 9, 
No. 3, 1971, pp. 245-252. 
%F A048655 a(n)=2*a(n-1)+a(n-2); a(0)=1, a(1)=5. 
%F A048655 a(n)=[ (4+sqrt(2))(1+sqrt(2))^n - (4-sqrt(2))(1-sqrt(2))^n ]
/2*sqrt(2). 
[snip]
%K A048655 easy,nice,nonn 
%O A048655 0,2 
%A A048655 Barry E. Williams 


List of transformations used: 
T018 sequence u[j+1]-u[j] 
T019 sequence u[j+2]-2*u[j+1]+u[j] 
T030 sequence u[j+2]-u[j] 
T050 jth coefficient of Sn(z)*(1-z)^j 

Abbreviations used in the above list of transformations: 
u[j] = j-th term of the sequence 
v[j] = u[j]/(j-1)! 


****************

New technique used successfully:
 
Somewhat naively, as the definitions of les, jes, etc. :

tes( (x_1)'i + (x_2)'j + ... + (x_16)1 ) = x_16 
ves( (x_1)'i + (x_2)'j + ... + (x_16)1 ) = x_1 + x_2 + ... + x_16 
jes( (x_1)'i + (x_2)'j + (x_3)'k + (x_4)i' + (x_5)j' + (x_6)k' 
+(x_7)'ii' + ... + (x_16)1 ) = x_1 + x_2 + x_3 + x_4 + x_5 + x_6 

were found to be useful, I was beginning to forget that 
the definitions in use were originally based on certain
symmetries and that these symmetries were only some among many. 
"jes", for example, was based on the "symmetry" that squaring any of the 
basis vectors ('i, 'j, ...) involved gives -1. 

Here are two other "symmetries" (FAMP, soon to be updated to ver 1.4, will 
include this):

"em", which sums up over the basis vectors arising from proposition 
"Emmy's Three" from the floretion paper. Surprisingly, "em" often gives 
exactly same sequence as "ves" for a given floretion. 

"chu", which sums up over basis vectors arising from 
"Chu's (countably infinite) group".  

For ex., we can now apply "chu" and "em" to the same floretion "z"  
from above to get 

( chu(z^n) ) = (1.5, 3, 7.5, 18, 43.5, 105, )
and 
( em(z^n) ) = (1, 1, 1, 1, 1, 1, 1, 1, )  // (not similar to "ves" this 
time)

Making use of the following simple formula between "chu" and "ves":
chu + chu* = ves  (where chu* is simply defined by summing up over those 
basis vectors not used in chu ) gives

A048654(n) = 2P(n+2) - 3P(n+1) 

Sincerely, 
Creighton