A002203 -id:A002203 - cp's OEIS frontend

A174504 Continued fraction expansion for exp( Sum_{n>=1} 1/(n*A002203(n)) ), where A002203(n) = (1+sqrt(2))^n + (1-sqrt(2))^n.

1, 1, 5, 1, 13, 33, 1, 81, 197, 1, 477, 1153, 1, 2785, 6725, 1, 16237, 39201, 1, 94641, 228485, 1, 551613, 1331713, 1, 3215041, 7761797, 1, 18738637, 45239073, 1, 109216785, 263672645, 1, 636562077, 1536796801, 1, 3710155681, 8957108165, 1

Offset: 0

Paul D. Hanna, Mar 21 2010

			Let L = Sum_{n>=1} 1/(n*A002203(n)) or, more explicitly,
L = 1/2 + 1/(2*6) + 1/(3*14) + 1/(4*34) + 1/(5*82) + 1/(6*198) +...
so that L = 0.6182475539420223547415636201969551910173293917288...
then exp(L) = 1.8556732235071087707741415885016794127474675663938...
equals the continued fraction given by this sequence:
exp(L) = [1;1,5,1,13,33,1,81,197,1,477,1153,1,2785,6725,1,...]; i.e.,
exp(L) = 1 + 1/(1 + 1/(5 + 1/(1 + 1/(13 + 1/(33 + 1/(1 +...)))))).
Compare these partial quotients to A002203(n), n=1,2,3,...:
[2,6,14,34,82,198,478,1154,2786,6726,16238,39202,94642,228486,...].

G. C. Greubel, Table of n, a(n) for n = 0..1000
P. Bala, Some simple continued fraction expansions for an infinite product, Part 1
P. Bala, Some simple continued fraction expansions for an infinite product, Part 2
Index entries for linear recurrences with constant coefficients, signature (0,0,7,0,0,-7,0,0,1).

Cf. A002203 (companion Pell numbers), A174500, A174503, A174505.

LinearRecurrence[{0,0,7,0,0,-7,0,0,1},{1,1,5,1,13,33,1,81,197},40] (* Harvey P. Dale, Sep 15 2016 *)

{a(n)=local(L=sum(m=1,2*n+1000,1./(m*round((1+sqrt(2))^m+(1-sqrt(2))^m))));contfrac(exp(L))[n]}

a(3n-3) = 1, a(3n-2) = A002203(2n-1) - 1, a(3n-1) = A002203(2n) - 1, for n>=1 [conjecture].
From Colin Barker, Jan 20 2013: (Start)
a(n) = 7*a(n-3) - 7*a(n-6) + a(n-9).
G.f.: -(x^2-x+1)*(x^6-2*x^5-2*x^4-2*x^3+6*x^2+2*x+1) / ((x-1)*(x^2+x+1)*(x^6-6*x^3+1)). (End)
From Peter Bala, Jan 25 2013: (Start)
The above conjectures are correct. The real number exp( Sum {n>=1} 1/(n*A002203(n)) ) is equal to the infinite product F(x) := product {n >= 0} (1 + x^(4*n+3))/(1 - x^(4*n+1)) evaluated at x = sqrt(2) - 1. Ramanujan has given a continued fraction expansion for the product F(x). Using this we can find the simple continued fraction expansion of the numbers F(1/2*(sqrt(N^2 + 4) - N)), N a positive integer. The present case is when N = 2. See the Bala link for details.
The theory also provides the simple continued fraction expansion of the numbers F({sqrt(2) - 1}^(2*k+1)), k = 1, 2, 3, ...: if [1; c(1), c(2), 1, c(3), c(4), 1, ...] denotes the present sequence then the simple continued fraction expansion of F({sqrt(2) - 1}^(2*k+1)) is given by [1; c(2*k+1), c(2*(2*k+1)), 1, c(3*(2*k+1)), c(4*(2*k+1)), 1, ...].
(End)

A204062 Expansion of g.f.: exp( Sum_{n>=1} A002203(n)^2 * x^n/n ) where A002203 are the companion Pell numbers.

Original entry on oeis.org

1, 4, 26, 148, 867, 5048, 29428, 171512, 999653, 5826396, 33958734, 197925996, 1153597255, 6723657520, 39188347880, 228406429744, 1331250230601, 7759094953844, 45223319492482, 263580822001028, 1536261612513707, 8953988853081192, 52187671505973468

Offset: 0

Paul D. Hanna, Jan 10 2012

			G.f.: A(x) = 1 + 4*x + 26*x^2 + 148*x^3 + 867*x^4 + 5048*x^5 + ...
where
log(A(x)) = 2^2*x + 6^2*x^2/2 + 14^2*x^3/3 + 34^2*x^4/4 + 82^2*x^5/5 + 198^2*x^6/6 + 478^2*x^7/7 + ... + A002203(n)^2*x^n/n + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (4,10,4,-1).

Cf. A000129, A002203, A026933, A204061, A212442.

I:=[1,4,26,148]; [n le 4 select I[n] else 4*Self(n-1) +10*Self(n-2) +4*Self(n-3) -Self(n-4): n in [1..31]]; // G. C. Greubel, May 25 2021

LinearRecurrence[{4,10,4,-1},{1,4,26,148},30] (* Vincenzo Librandi, Feb 12 2012 *)
Table[(Fibonacci[2*n+4, 2] + 2*(-1)^n*(n+2))/16, {n, 0, 30}] (* G. C. Greubel, May 25 2021 *)

{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2*x^k/k)+x*O(x^n)), n)}

[(lucas_number1(2*n+4,2,-1) +2*(-1)^n*(n+2))/16 for n in (0..30)] # G. C. Greubel, May 25 2021

G.f.: 1/((1+x)^2*(1-6*x+x^2)).
Self-convolution of A026933.
Self-convolution 4th power of A204061.
a(n) = Pell(n-1)^2 + a(n-2) where Pell(n) = A000129(n).
a(n) = (1/8)*(A001109(n+2) + (-1)^n*(n+2)). - Bruno Berselli, Jan 10 2012
a(n) = (1/16)*(A000129(2*n+4) + 2*(-1)^n*(n+2)). - G. C. Greubel, May 25 2021

A204061 G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers.

Original entry on oeis.org

1, 1, 5, 21, 101, 501, 2561, 13345, 70561, 377281, 2035285, 11059205, 60454005, 332138405, 1832677185, 10150115201, 56398558081, 314273655745, 1755700634981, 9830544087221, 55155558312901, 310027473436821, 1745567243959105, 9843160519978401, 55582528404717601

Offset: 0

Paul D. Hanna, Jan 10 2012

nonn

a(n) == 1 (mod 5) iff n has no 2's in its base 5 expansion (A023729), otherwise a(n) == 0 (mod 5); this is a conjecture needing proof.

			G.f.: A(x) = 1 + x + 5*x^2 + 21*x^3 + 101*x^4 + 501*x^5 + 2561*x^6 +...
where log(A(x)) = x + 3^2*x^2/2 + 7^2*x^3/3 + 17^2*x^4/4 + 41^2*x^5/5 + 99^2*x^6/6 + 239^2*x^7/7 +...+ A001333(n)^2*x^n/n +...
The last digit of the terms in this sequence seems to be either a '1' or a '5':
by conjecture, a(n) == 0 (mod 5) whenever n has a 2 in its base 5 expansion;
if true, terms a(2*5^k) through a(3*5^k - 1) all end with digit '5' for k>=0.

Paul D. Hanna, Table of n, a(n) for n = 0..625

Cf. A204062, A026933, A001333, A023729.

{A001333(n)=polcoeff((1-x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A001333(k)^2*x^k/k)+x*O(x^n)), n)}

{a(n)=polcoeff(1/(sqrt(1+x+x*O(x^n))*(1-6*x+x^2+x*O(x^n))^(1/4)),n)}

G.f.: 1 / ( sqrt(1+x) * (1-6*x+x^2)^(1/4) ).
Self-convolution yields A026933: Sum_{k=0..n} a(n-k)*a(k) = Sum_{k=0..n} D(n-k,k)^2 where D(n,k) = A008288(n,k) are the Delannoy numbers.
a(n) ~ 2^(1/8) * GAMMA(3/4) * (3+2*sqrt(2))^(n+1/2) / (4 * Pi * n^(3/4)). - Vaclav Kotesovec, Oct 30 2014

A072280 Product representation of the Pell numbers A000129 and A002203.

Original entry on oeis.org

2, 1, 7, 6, 41, 5, 239, 34, 199, 29, 8119, 33, 47321, 169, 961, 1154, 1607521, 197, 9369319, 1121, 32641, 5741, 318281039, 1153, 45245801, 33461, 7761799, 38081, 63018038201, 1345, 367296043199, 1331714, 37667521, 1136689, 1273319041, 39201, 72722761475561

Offset: 1

Miklos Kristof, Jul 10 2002

nonn

Define the silver mean constants h=1+sqrt(2) = A014176, h^2=1+2h = A156035, and 1/h=h-2.
Let Phi(n,x) be the n-th cyclotomic polynomial A013595, so that x^n-1 = Product_{d | n} Phi(d, x). Let g(n) be the order of Phi(n, x), A000010. Then a(n)=(h-2)^g(n)*Phi(n, h^2) if n <> 2.
The Binet representations of the Pell numbers yields:
For even n, A000129(n) = Product_{d|n} a(d).
For odd n, A000129(n)=Product_{ d|n} a(2d).
For odd prime p, a(p)=A002203(p)/2, a(2p)=A000129(p).
a(2^(k+1))=A002203(2^k).
For odd n, A002203(n)=Product_{ d|n} a(d).
For k>0 and odd n, A002203(n*2^k)=Product_{ d | n} a(d*2^(k+1)).

			For even n=12, A000129(12) = a(1)*a(2)*a(3)*a(4)*a(6)*a(12) = 2*1*7*6*5*33 = 13860.
For odd n=9, A000129(9) = a(2)*a(6)*a(18)= 1*5*197 = 985.
For even n=8, A002203(12) = a(8)*a(24)=34*1153 = 39202.
For odd n=21, A002203(21) = a(1)*a(3)*a(7)*a(21) = 2*7*239*32641 = 109216786.

Dan Kalman and Robert Mena, The Fibonacci Numbers: Exposed, Math. Mag. 76 (3) (2003) 167-181.
Index to sequences related to cyclotomic polynomials.

Cf. A000129, A002203.

A072280 := proc(n) if n <= 2 then 3-n ; else g := numtheory[phi](n) ; h := 1+sqrt(2) ; (h-2)^g*numtheory[cyclotomic](n,h^2) ; simplify(expand(%)) ; end if; end proc:
seq(A072280(n),n=1..80) ; # R. J. Mathar, Nov 27 2009

a[n_] := If[n <= 2, 3-n, g = EulerPhi[n]; h = 1 + Sqrt[2]; (h - 2)^g*Cyclotomic[n, h^2] // Expand];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, May 08 2023, after R. J. Mathar *)

Edited and extended by R. J. Mathar, Nov 27 2009

A204275 G.f.: Product_{n>=1} (1 + A002203(n)x^n + (-1)^nx^(2*n)) where A002203 is the companion Pell numbers.

Original entry on oeis.org

1, 2, 5, 26, 57, 222, 698, 2096, 6038, 19730, 58915, 169952, 516024, 1484958, 4397513, 13029558, 37094682, 106442928, 311875984, 879620854, 2522107990, 7229956352, 20398904648, 57543374566, 163053304047, 457604617760, 1283583473614, 3606627675050

Offset: 0

Paul D. Hanna, Jan 13 2012

nonn

Analog to Euler's identity: Product_{n>=1} (1+x^n) = Product_{n>=1} 1/(1-x^(2*n-1)), which is the g.f. for the number of partitions of distinct parts.

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 26*x^3 + 57*x^4 + 222*x^5 + 698*x^6 +...
where A(x) = (1+2*x-x^2) * (1+6*x^2+x^4) * (1+14*x^3-x^6) * (1+34*x^4+x^8) * (1+82*x^5-x^10) * (1+198*x^6+x^12) *...* (1 + A002203(n)*x^n + (-1)^n*x^(2*n)) *...
and 1/A(x) = (1-2*x-x^2) * (1-14*x^3-x^6) * (1-82*x^5-x^10) * (1-478*x^7-x^14) * (1-2786*x^9-x^18) * (1-16238*x^11-x^22) *...* (1 - A002203(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)) *...
Also, the logarithm of the g.f. equals the series:
log(A(x)) = 1*2*x + 1*6*x^2/2 + 4*14*x^3/3 + 1*34*x^4/4 + 6*82*x^5/5 + 4*198*x^6/6 + 8*478*x^7/7 + 1*1154*x^8/8 +...+ A000593(n)*A002203(n)*x^n/n +...
The companion Pell numbers (starting at offset 1) begin:
A002203 = [2,6,14,34,82,198,478,1154,2786,6726,16238,...]
and form the logarithm of a g.f. for Pell numbers:
log(1/(1-2*x-x^2)) = 2*x + 6*x^2/2 + 14*x^3/3 + 34*x^4/4 + 82*x^5/5 +...

Eric Weisstein's World of Mathematics, Euler Identity.

Cf. A203801, A204270, A000129 (Pell), A002203 (companion Pell), A000593.

/* Subroutine used in PARI programs below: */
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

{a(n)=polcoeff(prod(k=1,n,1+A002203(k)*x^k+(-1)^k*x^(2*k) +x*O(x^n)),n)}

{a(n)=polcoeff(1/prod(k=1,n,1-A002203(2*k-1)*x^(2*k-1)-x^(4*k-2) +x*O(x^n)),n)}

/* Exponential form using sum of odd divisors of n: */
{A000593(n)=if(n<1, 0, sumdiv(n, d, (-1)^(d+1)*n/d))}
{a(n)=polcoeff(exp(sum(k=1, n, A000593(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}

G.f.: Product_{n>=1} 1/(1 - A002203(2*n-1)*x^(2*n-1) + (-1)^n*x^(4*n-2)).
G.f.: exp( Sum_{n>=1} A000593(n) * A002203(n) * x^n/n ) where A000593(n) = sum of odd divisors of n.
a(n) = (1/n)*Sum_{k=1..n} A000593(k) * A002203(k)*a(n-k) for n>0, with a(0) = 1.

A165937 G.f.: A(x) = exp( Sum_{n>=1} A002203(n^2)*x^n/n ).

Original entry on oeis.org

1, 2, 19, 964, 334965, 742714950, 10042408885191, 814556580116590856, 393147641272746246076745, 1123539400297807898234860367690, 18948227277012085227250633551784337179, 1881331163508674280605070386666674939623268684

Offset: 0

Paul D. Hanna, Oct 18 2009

nonn

A002203 equals the logarithmic derivative of the Pell numbers (A000129).
Note that A002203(n^2) = (1+sqrt(2))^(n^2) + (1-sqrt(2))^(n^2).
Given g.f. A(x), (1-x)^(1/4) * A(x)^(1/8) is an integer series.

			G.f.: A(x) = 1 + 2*x + 19*x^2 + 964*x^3 + 334965*x^4 + 742714950*x^5 +...
log(A(x)) = 2*x + 34*x^2/2 + 2786*x^3/3 + 1331714*x^4/4 + 3710155682*x^5/5 + 60245508192802*x^6/6 + 5701755387019728962*x^7/7 +...+ A002203(n^2)*x^n/n +...

Cf. A165938, A166879, A002203, A000129; variants: A158843, A166168, A155200.

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^(m^2))),m^2)*x^m/m)+x*O(x^(n^2))),n))}

Logarithmic derivative equals A165938.

Self-convolution of A166879.

A165938 a(n) = A002203(n^2) for n>=1.

Original entry on oeis.org

2, 34, 2786, 1331714, 3710155682, 60245508192802, 5701755387019728962, 3145168096065837266706434, 10111847525912679844192131854786, 189482250299273866835746159841800035874, 20694642381734231604510939638726181796865594402

Offset: 1

Paul D. Hanna, Oct 18 2009

nonn

Paul D. Hanna, Table of n, a(n) for n = 1..40

Cf. A165937, A002203, A000129 (Pell numbers).

Simplify[Table[(1 + Sqrt[2])^(n^2) + (1 - Sqrt[2])^(n^2), {n, 1, 7}]] (* G. C. Greubel, Apr 18 2016 *)

{a(n)=polcoeff(2*(1-x)/(1-2*x-x^2 +x*O(x^(n^2))),n^2)}

a(n) == 2 (mod 32).
a(n) = (1+sqrt(2))^(n^2) + (1-sqrt(2))^(n^2).
Logarithmic derivative of A165937.

A204382 G.f.: Product_{n>=1} (1 - A002203(n)x^n + (-1)^nx^(2*n)) where A002203(n) is the companion Pell numbers.

Original entry on oeis.org

1, -2, -7, -2, 1, 82, 34, 464, 198, -82, -1, 0, -39208, -16238, 6725, -551614, -228486, 95120, 0, 82, 6726, 0, 263673800, 109216786, -45239073, 0, 8957108166, 3706940654, -1536796802, -551614, -1, -109216786, 0, -18738638, -6726, -24954506565518, -10336495061766

Offset: 0

Paul D. Hanna, Jan 14 2012

sign

a(A093519(n)) = 0 where A093519 lists numbers that are not equal to the sum of two generalized pentagonal numbers.

			G.f.: A(x) = 1 - 2*x - 7*x^2 - 2*x^3 + x^4 + 82*x^5 + 34*x^6 + 464*x^7 +...
-log(A(x)) = 1*2*x + 3*6*x^2/2 + 4*14*x^3/3 + 7*34*x^4/4 + 6*82*x^5/5 + 12*198*x^6/6 +...+ sigma(n)*A002203(n)*x^n/n +...
The g.f. equals the product:
A(x) = (1-2*x-x^2) * (1-6*x^2+x^4) * (1-14*x^3-x^6) * (1-34*x^4+x^8) * (1-82*x^5-x^10) * (1-198*x^6+x^12) *...* (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) *...
Positions of zeros form A093519:
[11,18,21,25,32,39,43,46,49,54,60,65,67,68,74,76,81,87,88,90,...].
which are numbers that are not the sum of two generalized pentagonal numbers.

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A203860, A204383, A204384, A203534, A093519, A002203.

/* Subroutine used in PARI programs below: */
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

{a(n)=polcoeff(exp(sum(k=1, n, -sigma(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}

{a(n)=polcoeff(prod(m=1, n, 1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}

G.f.: exp( Sum_{n>=1} -sigma(n) * A002203(n) * x^n/n ).

A204384 G.f.: Product_{n>=1} (1 - A002203(n)x^n + (-x^2)^n) / (1 + A002203(n)x^n + (-x^2)^n) where A002203(n) is the companion Pell numbers.

Original entry on oeis.org

1, -4, -4, 0, 68, 56, 0, 0, 4, -5572, -4616, 0, 0, -328, 0, 0, 2663428, 2206456, -4, 0, 156808, 0, 0, 0, 0, -7420309452, -6147187208, 0, 0, -436867144, 0, 0, 4, 0, -5326856, 0, 120491016385604, 99818026262072, 0, 0, 7093848711176, -11144, 0, 0, 0, 86497488056, 0, 0, 0

Offset: 0

Paul D. Hanna, Jan 14 2012

sign

a(A022544(n)) = 0 where A022544 lists numbers that are not the sum of 2 squares.

Compare to: Product_{n>=1} (1-q^k)/(1+q^k) = 1 + 2*Sum_{n>=1} (-1)^n*q^(n^2), the Jacobi theta_4 function, which has the g.f: exp( Sum_{n>=1} -(sigma(2*k)-sigma(k)) * x^n/n ).

			G.f.: A(x) = 1 - 4*x - 4*x^2 + 68*x^4 + 56*x^5 + 4*x^8 - 5572*x^9 - 4616*x^10 +...
-log(A(x)) = 2*2*x + 4*6*x^2/2 + 8*14*x^3/3 + 8*34*x^4/4 + 12*82*x^5/5 + 16*198*x^6/6 +...+ (sigma(2*n)-sigma(n))*A002203(n)*x^n/n +...
Compare to the logarithm of Jacobi theta4 H(x) = 1 + 2*Sum_{n>=1} (-1)^n*q^(n^2):
-log(H(x)) = 2*x + 4*x^2/2 + 8*x^3/3 + 8*x^4/4 + 12*x^5/5 + 16*x^6/6 + 16*x^7/7 +...+ (sigma(2*n)-sigma(n))*x^n/n +...
The g.f. equals the products:
A(x) = (1-2*x-x^2)/(1+2*x-x^2) * (1-6*x^2+x^4)/(1+6*x^2+x^4) * (1-14*x^3-x^6)/(1+14*x^3-x^6) * (1-34*x^4+x^8)/(1+34*x^4+x^8) * (1-82*x^5-x^10)/(1+82*x^5-x^10) *...* (1 - A002203(n)*x^n + (-x^2)^n)/(1 + A002203(n)*x^n + (-x^2)^n) *...
A(x) = (1-2*x-x^2)^2 * (1-6*x^2+x^4) * (1-14*x^3-x^6)^2 * (1-34*x^4+x^8) * (1-82*x^5-x^10)^2 *(1-198*x^6+x^12) * (1-478*x^7-x^14)^2 * (1-1154*x^8+x^16) *...
Positions of zeros form A022544:
[3,6,7,11,12,14,15,19,21,22,23,24,27,28,30,31,33,35,38,39,42,43,44,...]
which are numbers that are not the sum of 2 squares.

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A203850, A204382, A204383, A022544.

/* Subroutine used in PARI programs below: */
{A002203(n)=polcoeff(2*(1-x)/(1-2*x-x^2+x*O(x^n)), n)}

{a(n)=polcoeff(prod(m=1, n, 1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))/prod(m=1, n, 1 + A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n)), n)}

{a(n)=polcoeff(prod(m=1, n\2+1, (1 - A002203(2*m-1)*x^(2*m-1) - x^(4*m-2))^2*(1 - A002203(2*m)*x^(2*m) + x^(4*m) +x*O(x^n))), n)}

{a(n)=polcoeff(exp(sum(k=1, n,-(sigma(2*k)-sigma(k))*A002203(k)*x^k/k)+x*O(x^n)), n)}

G.f.: Product_{n>=1} (1 - A002203(2*n-1)*x^(2*n-1) - x^(4*n-2))^2 * (1 - A002203(2*n)*x^(2*n) + x^(4*n)).

G.f.: exp( Sum_{n>=1} -(sigma(2*n)-sigma(n)) * A002203(n) * x^n/n ) where A002203(n) is the companion Pell numbers.

A212442 G.f.: exp( Sum_{n>=1} A002203(n)^3 * x^n/n ), where A002203 is the companion Pell numbers.

Original entry on oeis.org

1, 8, 140, 1864, 26602, 373080, 5253564, 73911192, 1040045475, 14634444720, 205922568360, 2897549559600, 40771618763540, 573700205699920, 8072574516567400, 113589743388536528, 1598328982089075749, 22490195492277648120, 316461065874934143252

Offset: 0

Paul D. Hanna, May 17 2012

nonn

More generally, exp(Sum_{k>=1} A002203(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A002203(2*k+1)*x - x^2)^binomial(2*n+1,n-k).

Compare to g.f. exp(Sum_{k>=1} A002203(k) * x^k/k) = 1/(1-2*x-x^2).

			G.f.: A(x) = 1 + 8*x + 140*x^2 + 1864*x^3 + 26602*x^4 + 373080*x^5 + ...
where
log(A(x)) = 2^3*x + 6^3*x^2/2 + 14^3*x^3/3 + 34^3*x^4/4 + 82^3*x^5/5 + 198^3*x^6/6 + 478^3*x^7/7 + 1154^3*x^8/8 + ... + A002203(n)^3*x^n/n + ...
Also, the g.f. equals the infinite product:
A(x) = 1/( (1-2*x-x^2)^4 * (1-6*x^2+x^4)^16 * (1-14*x^3-x^6)^64 * (1-34*x^4+x^8)^280 * (1-82*x^5-x^10)^1344 * (1-198*x^6+x^12)^6496 * ... * (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^A212443(n) * ...).
The exponents in these products begin:
A212443 = [4, 16, 64, 280, 1344, 6496, 32640, 166320, 862400, ...].
The companion Pell numbers begin (at offset 1):
A002203 = [2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...].

G. C. Greubel, Table of n, a(n) for n = 0..865
Index entries for linear recurrences with constant coefficients, signature (8,76,136,-38,-136,76,-8,-1).

Cf. A212443, A203803, A002203, A204062.

CoefficientList[Series[1/((1+2x-x^2)^3(1-14x-x^2)),{x,0,30}],x] (* or *) LinearRecurrence[{8,76,136,-38,-136,76,-8,-1},{1,8,140,1864,26602,373080,5253564,73911192},30] (* Harvey P. Dale, Feb 15 2015 *)

/* Subroutine for the PARI programs that follow: */
{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}

/* G.F. by Definition: */
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^3*x^k/k)+x*O(x^n)), n)}

/* G.F. as a Finite Product: */
{a(n, m=1)=polcoeff(prod(k=0, m, 1/(1 - (-1)^(m-k)*A002203(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1, m-k)), n)}

/* G.F. as an Infinite Product: */
{A212443(n)=(1/n)*sumdiv(n,d, moebius(n/d)*A002203(d)^2)}
{a(n)=polcoeff(1/prod(m=1,n, (1 - A002203(m)*x^m + (-1)^m*x^(2*m) +x*O(x^n))^A212443(m)),n)}
for(n=0,30,print1(a(n),", "))

G.f.: 1 / ( (1+2*x-x^2)^3 * (1-14*x-x^2) ).
G.f.: 1 / Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^A212443(n) where A212443(n) = (1/n)*Sum_{d|n} moebius(n/d)*A002203(d)^2.
a(0)=1, a(1)=8, a(2)=140, a(3)=1864, a(4)=26602, a(5)=373080, a(6)=5253564, a(7)=73911192, a(n) = 8*a(n-1) + 76*a(n-2) + 136*a(n-3) - 38*a(n-4) - 136*a(n-5) + 76*a(n-6) - 8*a(n-7) - a(n-8). - Harvey P. Dale, Feb 15 2015

cp's OEIS Frontend

A174504 Continued fraction expansion for exp( Sum_{n>=1} 1/(n*A002203(n)) ), where A002203(n) = (1+sqrt(2))^n + (1-sqrt(2))^n.

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A204062 Expansion of g.f.: exp( Sum_{n>=1} A002203(n)^2 * x^n/n ) where A002203 are the companion Pell numbers.

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Magma

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Sage

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A204061 G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers.

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A072280 Product representation of the Pell numbers A000129 and A002203.

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Maple

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A204275 G.f.: Product_{n>=1} (1 + A002203(n)*x^n + (-1)^n*x^(2*n)) where A002203 is the companion Pell numbers.

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A165937 G.f.: A(x) = exp( Sum_{n>=1} A002203(n^2)*x^n/n ).

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A165938 a(n) = A002203(n^2) for n>=1.

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A204275 G.f.: Product_{n>=1} (1 + A002203(n)x^n + (-1)^nx^(2*n)) where A002203 is the companion Pell numbers.

A204382 G.f.: Product_{n>=1} (1 - A002203(n)x^n + (-1)^nx^(2*n)) where A002203(n) is the companion Pell numbers.

A204384 G.f.: Product_{n>=1} (1 - A002203(n)x^n + (-x^2)^n) / (1 + A002203(n)x^n + (-x^2)^n) where A002203(n) is the companion Pell numbers.