A002203 -id:A002203 - cp's OEIS frontend

A261332 Expansion of Product_{k>=1} (1+x^k)^(A002203(k)).

1, 2, 7, 26, 83, 278, 894, 2848, 8947, 27844, 85774, 262090, 794802, 2393874, 7165622, 21327412, 63146545, 186063052, 545783103, 1594268778, 4638773567, 13447773510, 38850645513, 111874844146, 321166890522, 919314145044, 2624198013317, 7471158542418

Offset: 0

Vaclav Kotesovec, Aug 15 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotics of the Euler transform of Fibonacci numbers, arXiv:1508.01796 [math.CO], Aug 07 2015
Eric Weisstein's World of Mathematics, Pell Number
Wikipedia, Pell number

Cf. A002203, A261329, A261330, A261331, A261050, A261051.

nmax=40; cPell[0]=2; cPell[1]=2; cPell[n_]:=cPell[n] = 2*cPell[n-1] + cPell[n-2]; CoefficientList[Series[Product[(1+x^k)^cPell[k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (1+sqrt(2))^n * exp(-1 + 2^(-3/2) + 2*sqrt(n) + s) / (2 * sqrt(Pi) * n^(3/4)), where s = Sum_{k>=2} = 2*(-1)^(k+1)/(((1+sqrt(2))^k + 2/(1 + (1+sqrt(2))^k) - 3)*k) = -0.2731939535370496116124191192900280854879921353977...

A166879 G.f.: A(x) = exp( Sum_{n>=1} A002203(n^2)/2*x^n/n ).

Original entry on oeis.org

1, 1, 9, 473, 166969, 371186249, 5020831641761, 407273265807001089, 196573413317730320842177, 561769503571822735164882969633, 9474113076734769687535254457293566857, 940665572280219007549184269220597591870817337

Offset: 0

Paul D. Hanna, Oct 22 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).

			G.f.: A(x) = 1 + x + 9*x^2 + 473*x^3 + 166969*x^4 + 371186249*x^5 +...
log(A(x)) = x + 17*x^2/2 + 1393*x^3/3 + 665857*x^4/4 + 1855077841*x^5/5 + 30122754096401*x^6/6 + 2850877693509864481*x^7/7 +...+ A002203(n^2)/2*x^n/n +...

Paul D. Hanna, Table of n, a(n) for n = 0..40

Cf. A165937, A165938, A002203, A000129.

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

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

a(n) == 1 (mod 8).
a(n) = (1/n)*Sum_{k=1..n} A002203(k^2)/2*a(n-k) for n>0 with a(0)=1.
Self-convolution yields A165937.

A204383 G.f.: Product_{n>=1} (1 - A002203(n)x^n + (-1)^nx^(2*n))^3 where A002203(n) is the companion Pell numbers.

Original entry on oeis.org

1, -6, -9, 70, 90, 0, -1411, -1722, 0, 490, 60534, 75222, 49, -21510, 0, -6067754, -7542180, 0, 2156110, 0, 81, 1420032740, 1764323886, 0, -504516870, -8118, 0, -50196874, -783087782910, -973096740630, -121, 278263575996, 0, 0, 27685627830, 0, 1024173639305948

Offset: 0

Paul D. Hanna, Jan 14 2012

sign

a(A020757(n)) = 0 where A020757 lists numbers that are not the sum of two triangular numbers.

			G.f.: A(x) = 1 - 6*x - 9*x^2 + 70*x^3 + 90*x^4 - 1411*x^6 - 1722*x^7 +...
-log(A(x))/3 = 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)^3 * (1-6*x^2+x^4)^3 * (1-14*x^3-x^6)^3 * (1-34*x^4+x^8)^3 * (1-82*x^5-x^10)^3 * (1-198*x^6+x^12)^3 *...* (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^3 *...
Positions of zeros form A020757:
[5,8,14,17,19,23,26,32,33,35,40,41,44,47,50,52,53,54,59,62,63,...].

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

Cf. A203861, A204382, A204384, A020757.

/* 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, -3*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))^3, n)}

G.f.: exp( Sum_{n>=1} -3 * sigma(n) * A002203(n) * x^n/n ).

A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)x + (-1)^kx^2).

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 6, 5, 4, 0, 1, 14, 35, 12, 5, 0, 1, 34, 197, 204, 29, 6, 0, 1, 82, 1155, 2772, 1189, 70, 7, 0, 1, 198, 6725, 39236, 39005, 6930, 169, 8, 0, 1, 478, 39203, 551532, 1332869, 548842, 40391, 408, 9, 0, 1, 1154, 228485, 7761996, 45232349, 45278310, 7722793, 235416, 985, 10

Offset: 0

Seiichi Manyama, May 07 2025

			Square array begins:
  0,  0,    0,     0,       0,        0, ...
  1,  1,    1,     1,       1,        1, ...
  2,  2,    6,    14,      34,       82, ...
  3,  5,   35,   197,    1155,     6725, ...
  4, 12,  204,  2772,   39236,   551532, ...
  5, 29, 1189, 39005, 1332869, 45232349, ...

Columns k=0..6 give A001477, A000129, A001109, A041085(n-1), A091761, A292423, A097731(n-1).
Rows n=0..5 give A000004, A000012, A002203, A383720, A383740, A383741.
Main diagonal gives A380083.
Cf. A028412.

A[n_, k_] := Fibonacci[k*n, 2]/Fibonacci[k, 2]; A[n_, 0] := n; Table[A[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 08 2025 *)

pell(n) = ([2, 1; 1, 0]^n)[2, 1];
a(n, k) = if(k==0, n, pell(k*n)/pell(k));

A(0,k) = 0, A(1,k) = 1; A(n,k) = A002203(k) * A(n-1,k) - (-1)^k * A(n-2,k) for n > 1.

A(n,k) = Pell(k*n)/Pell(k) for k > 0.

A206141 G.f.: Sum_{n>=0} x^n/Product_{k=1..n} (1 - A002203(k)x^k + (-1)^kx^(2*k)), where A002203 is the companion Pell numbers.

Original entry on oeis.org

1, 1, 3, 8, 26, 67, 216, 555, 1704, 4538, 13320, 35376, 103863, 273792, 783694, 2101835, 5905044, 15745360, 44132278, 117267422, 325136638, 868034994, 2379074541, 6337238658, 17347580484, 46039358056, 125056019725, 332678989816, 898361151760, 2382959919616

Offset: 0

Paul D. Hanna, Feb 04 2012

nonn

Compare to the g.f. of partitions: Sum_{n>=0} x^n/Product_{k=1..n} (1-x^k).

As an analog to the identity: (1-x^n) = Product_{k=0..n-1} (1 - u^k*x), where u=exp(2*Pi*I/n), we have (1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Product_{k=0..n-1} (1 - 2*u^k*x - (u^k*x)^2).

			G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 26*x^4 + 67*x^5 + 216*x^6 + 555*x^7 +...
where
A(x) = 1 + x/(1-2*x-x^2) + x^2/((1-2*x-x^2)*(1-6*x^2+x^4)) + x^3/((1-2*x-x^2)*(1-6*x^2+x^4)*(1-14*x^3-x^6)) + x^4/((1-2*x-x^2)*(1-6*x^2+x^4)*(1-14*x^3-x^6)*(1-34*x^4+x^8)) + x^5/((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)) +...
The companion Pell numbers begin:
A002203 = [2, 6, 14, 34, 82, 198, 478, 1154, 2786, 6726, 16238, ...].

Cf. A002203 (Co.Pell), A206141.

{Pell(n)=polcoeff(x/(1-2*x-x^2+x*O(x^n)),n)}
{A002203(n)=Pell(n-1)+Pell(n+1)}
{a(n)=polcoeff(sum(m=0,n,x^m/prod(k=1,m,1-A002203(k)*x^k+(-1)^k*x^(2*k)+x*O(x^n))),n)}
for(n=0,51,print1(a(n),", "))

A335668 Even composites m such that A002203(m) == 2 (mod m).

Original entry on oeis.org

4, 8, 16, 24, 32, 48, 64, 72, 96, 120, 128, 144, 168, 192, 216, 240, 256, 264, 272, 288, 336, 360, 384, 432, 480, 504, 512, 528, 544, 576, 600, 648, 672, 720, 768, 792, 816, 840, 864, 960, 1008, 1024, 1056, 1080, 1088, 1152, 1176, 1200, 1296, 1320, 1344, 1440, 1512

Offset: 1

Ovidiu Bagdasar, Jun 17 2020

nonn

If p is a prime, then A002203(p)==2 (mod p).
Even composites for which the congruence holds.
Even composites m for which the sum of the Pell numbers A000129(0) + ... + A000129(m-1) is divisible by m.

			4 is the first composite number m for which A002203(m)==2 (mod m) since A002203(4)=34==2 (mod 4), so a(1)=4.
The next even composite for which the congruence holds is m = 8 since A002203(8)=1154==2 (mod 8), so a(2)=8.

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer (to appear, 2020).

Amiram Eldar, Table of n, a(n) for n = 1..10000
D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Cf. A270342 (all positive integers), A270345 (all composites), A330276 (odd composites),

Select[Range[4, 2000, 2], Divisible[LucasL[#, 2] - 2, #] &] (* Amiram Eldar, Jun 18 2020 *)

A085293 Product of Lucas (A000204) and a Pell Companion series (A002203).

Original entry on oeis.org

2, 18, 56, 238, 902, 3564, 13862, 54238, 211736, 827298, 3231362, 12623044, 49308482, 192613698, 752401496, 2939092798, 11480914982, 44847668844, 175187526662, 684331472398, 2673190054136, 10442227799538, 40790261396162, 159338166024964, 622419427368002

Offset: 1

Gary W. Adamson, Jun 24 2003

Convergent a(n+1)/a(n) = ((1+sqrt(5))/2)*(1+sqrt(2)) = (1.618...)*(2.414213...) = 3.9062796... = (1 + sqrt(2) + sqrt(5) + sqrt(10))/2.

Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).

Cf. A000204, A002203, A085292.

I:=[2,18,56,238]; [n le 4 select I[n] else 2*Self(n-1) + 7*Self(n-2) + 2*Self(n-3) - Self(n-4):n in [1..30]]; // Marius A. Burtea, Aug 25 2019

a(n) = A000204(n) * A002203(n), n > 0.
a(n) = 2*A085292(n).
a(n) = (((1+sqrt(5))/2)^n + ((1-sqrt(5))/2)^n) * ((1+sqrt(2))^n + (1-sqrt(2))^n).
From Colin Barker, Oct 15 2013: (Start)
a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).
G.f.: -2*x*(2*x^3 - 3*x^2 - 7*x - 1) / (x^4 - 2*x^3 - 7*x^2 - 2*x + 1). (End)
E.g.f.: 4*(exp(x/2)*(cosh(x/sqrt(2))*cosh(sqrt(5/2)*x)*cosh(sqrt(5)*x/2)+sinh(x/sqrt(2))*sinh(sqrt(5/2)*x)*sinh(sqrt(5)*x/2))-1). - Stefano Spezia, Aug 25 2019

More terms from David Wasserman, Jan 31 2005

More terms from Colin Barker, Oct 16 2013

A203534 G.f.: exp( Sum_{n>=1} sigma(n)A002203(n)x^n/n ) where A002203 is the companion Pell numbers.

Original entry on oeis.org

1, 2, 11, 38, 156, 516, 1991, 6434, 23065, 75132, 255335, 816480, 2724245, 8570794, 27763807, 87057596, 276339126, 855374534, 2681503010, 8218321006, 25421912010, 77383062314, 236519199902, 714226056554, 2165295121179, 6490447624984, 19503550719297, 58127246438024

Offset: 0

Paul D. Hanna, Jan 02 2012

nonn

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), and to the g.f. of Pell numbers: exp( Sum_{n>=1} A002203(n)*x^n/n ).

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 38*x^3 + 156*x^4 + 516*x^5 + 1991*x^6 +...
where
A(x) = 1/((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)) *...).
The companion Pell numbers (starting at offset 1) begin:
A002203 = [2,6,14,34,82,198,478,1154,2786,6726,16238,...].

Cf. A156234, A000129 (Pell), A002203 (companion Pell), A000203 (sigma).

/* Subroutine used in PARI programs below: */
{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, sigma(k)*A002203(k)*x^k/k)+x*O(x^n)), n)}

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

{a(n)=if(n<0,0,if(n==0,1,(1/n)*sum(k=1,n,sigma(k)*A002203(k)*a(n-k))))}

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

a(n) = (1/n)*Sum_{k=1..n} sigma(k)*A002203(k)*a(n-k) for n>0, with a(0) = 1.

A204385 G.f.: Sum_{n>=1} moebius(n)x^n/(1 - A002203(n)x^n + (-1)^nx^(2n)), where A002203 is the companion Pell numbers.

Original entry on oeis.org

1, 1, 4, 6, 28, 22, 168, 204, 788, 1108, 5740, 4356, 33460, 39914, 149296, 235416, 1136688, 862466, 6625108, 7452408, 30662688, 46594942, 225058680, 170763912, 1266505772, 1583313340, 6116296036, 9119790204, 44560482148, 30146578648, 259717522848, 313506783024

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

Compare g.f. to the identity: x = Sum_{n>=1} moebius(n)*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)).

			G.f.: A(x) = x + x^2 + 4*x^3 + 6*x^4 + 28*x^5 + 22*x^6 + 168*x^7 + 204*x^8 +...
where A(x) = x/(1-2*x-x^2) - x^2/(1-6*x^2+x^4) - x^3/(1-14*x^3-x^6) - x^5/(1-82*x^5-x^10) + x^6/(1-198*x^6+x^12) +...+ moebius(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

Paul D. Hanna, Table of n, a(n) for n = 1..256

Cf. A001109, A204291, A204270.

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

a(2^n) = A001109(2^(n-1)) for n>=1, where the g.f. of A001109 is x/(1-6*x+x^2).

A212443 a(n) = (1/n) * Sum_{d|n} moebius(n/d) * A002203(d)^2, where A002203 is the companion Pell numbers.

Original entry on oeis.org

4, 16, 64, 280, 1344, 6496, 32640, 166320, 862400, 4523232, 23970240, 128063040, 689008320, 3728973120, 20285199872, 110841302880, 608029121280, 3346972244000, 18480871268160, 102328688556864, 568014587806720, 3160148362953120, 17617881702072960

Offset: 1

Paul D. Hanna, May 17 2012

nonn

Cf. A008683, A212442, A203853, A002203.

a[n_] := DivisorSum[n, MoebiusMu[n/#] * LucasL[#, 2]^2 &] / n; Array[a, 25] (* Amiram Eldar, Aug 22 2023 *)

{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=if(n<1, 0, sumdiv(n, d, moebius(n/d)*A002203(d)^2)/n)}
for(n=1,30,print1(a(n),","))

G.f.: 1/Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^a(n) = exp(Sum_{n>=1} A002203(n)^3 * x^n/n), which equals the g.f. of A212442.

cp's OEIS Frontend

A261332 Expansion of Product_{k>=1} (1+x^k)^(A002203(k)).

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A166879 G.f.: A(x) = exp( Sum_{n>=1} A002203(n^2)/2*x^n/n ).

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A204383 G.f.: Product_{n>=1} (1 - A002203(n)*x^n + (-1)^n*x^(2*n))^3 where A002203(n) is the companion Pell numbers.

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A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)*x + (-1)^k*x^2).

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A206141 G.f.: Sum_{n>=0} x^n/Product_{k=1..n} (1 - A002203(k)*x^k + (-1)^k*x^(2*k)), where A002203 is the companion Pell numbers.

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A335668 Even composites m such that A002203(m) == 2 (mod m).

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Mathematica

A085293 Product of Lucas (A000204) and a Pell Companion series (A002203).

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A203534 G.f.: exp( Sum_{n>=1} sigma(n)*A002203(n)*x^n/n ) where A002203 is the companion Pell numbers.

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A204383 G.f.: Product_{n>=1} (1 - A002203(n)x^n + (-1)^nx^(2*n))^3 where A002203(n) is the companion Pell numbers.

A383742 Square array A(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of g.f. x/(1 - A002203(k)x + (-1)^kx^2).

A206141 G.f.: Sum_{n>=0} x^n/Product_{k=1..n} (1 - A002203(k)x^k + (-1)^kx^(2*k)), where A002203 is the companion Pell numbers.

A203534 G.f.: exp( Sum_{n>=1} sigma(n)A002203(n)x^n/n ) where A002203 is the companion Pell numbers.

A204385 G.f.: Sum_{n>=1} moebius(n)x^n/(1 - A002203(n)x^n + (-1)^nx^(2n)), where A002203 is the companion Pell numbers.