Sequence relating to Pell numbers; a new technique

[Gerald McGarvey]

Speaking of the Pell numbers, it appears that the following is true:

for n > 2, Sum i=2..inf A000129[i]/n^i = (2n + 1)/(2*b(n))
where b(n) starts with 3, 14, 35 then b(n) = 3b(n-1) - 3b(n-2) + b(n-3) + 3 
for n>4
so b starts with 3 14 35 69 119 188 279 395 539 714 923 1169 1455 1784

I haven't worked out a proof, is this new, of interest?

Sincerely,
Gerald

At 07:21 AM 10/3/2004, creigh at o2online.de wrote:
>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