A000129 -id:A000129 - cp's OEIS frontend

A054456 Convolution triangle of A000129(n) (Pell numbers).

1, 2, 1, 5, 4, 1, 12, 14, 6, 1, 29, 44, 27, 8, 1, 70, 131, 104, 44, 10, 1, 169, 376, 366, 200, 65, 12, 1, 408, 1052, 1212, 810, 340, 90, 14, 1, 985, 2888, 3842, 3032, 1555, 532, 119, 16, 1, 2378, 7813, 11784, 10716, 6482, 2709, 784, 152, 18, 1, 5741, 20892, 35223

Offset: 0

Wolfdieter Lang, Apr 27 2000 and May 08 2000

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group.
The G.f. for the row polynomials p(n,x) (increasing powers of x) is Pell(z)/(1-x*z*Pell(z)) with Pell(x)=1/(1-2*x-x^2) = g.f. for A000129(n+1) (Pell numbers without 0).
Column sequences are A000129(n+1), A006645(n+1), A054457(n) for m=0..2.
Riordan array (1/(1-2x-x^2),x/(1-2x-x^2)). - Paul Barry, Mar 15 2005
As a Riordan array, this factors as (1/(1-x^2),x/(1-x^2))*(1/(1-2x),x/(1-2x)), [abs(A049310) times square of A007318, or A038207]. - Paul Barry, Jul 28 2005
Coefficients of polynomials defined by P(x, 0) = 1; P(x, 1) = 2 - x; P(x, n) = (2 - x)*P(x, n - 1) + P(x, n - 2). - Roger L. Bagula, Mar 24 2008
Subtriangle (obtained by dropping the first column) of the triangle given by (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 19 2013
T(n,k) is the number of words of length n over {0,1,2,3} having k letters 3 and avoiding runs of odd length of the letter 0. - Milan Janjic, Jan 14 2017

			Fourth row polynomial (n=3): p(3,x)= 12+14*x+6*x^2+x^3
Triangle begins:
{1},
{2, 1},
{5, 4, 1},
{12, 14, 6, 1},
{29, 44, 27, 8, 1},
{70, 131,104, 44, 10, 1},
{169, 376, 366, 200, 65, 12, 1},
{408, 1052, 1212, 810, 340, 90, 14, 1},
{985, 2888, 3842, 3032, 1555, 532, 119, 16, 1},
{2378, 7813, 11784, 10716, 6482, 2709, 784, 152, 18, 1},
{5741, 20892, 35223, 36248, 25235, 12432, 4396, 1104, 189, 20, 1},
The triangle (0, 2, 1/2, -1/2, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, ...) begins:
1
0, 1
0, 2, 1
0, 5, 4, 1
0, 12, 14, 6, 1
0, 29, 44, 27, 8, 1 - _Philippe Deléham_, Feb 19 2013

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. A000129. Row sums: A006190(n+1).

Cf. A129844.

G := 1/(1-(x+2)*z-z^2): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 15 do seq(coeff(P[n], x, j), j = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Aug 30 2015
T := (n,k) -> `if`(n=0,1,2^(n-k)*binomial(n,k)*hypergeom([(k-n)/2,(k-n+1)/2],[-n], -1)): seq(seq(simplify(T(n,k)),k=0..n),n=0..10); # Peter Luschny, Apr 25 2016
# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, A000129); # Peter Luschny, Oct 19 2022

P[x_, 0] := 1; P[x_, 1] := 2 - x; P[x_, n_] := P[x, n] = (2 - x) P[x, n - 1] + P[x, n - 2]; Table[Abs@ CoefficientList[P[x, n], x], {n, 0, 10}] // Flatten (* Roger L. Bagula, Mar 24 2008, edited by Michael De Vlieger, Apr 25 2018 *)

a(n, m) := ((n-m+1)*a(n, m-1) + (n+m)*a(n-1, m-1))/(4*m), n >= m >= 1, a(n, 0)= P(n+1)= A000129(n+1) (Pell numbers without P(0)), a(n, m) := 0 if n

G.f. for column m: Pell(x)*(x*Pell(x))^m, m >= 0, with Pell(x) G.f. for A000129(n+1).

Number triangle T(n, k) with T(n, 0)=A000129(n), T(1, 1)=1, T(n, k)=0 if k>n, T(n, k)=T(n-1, k-1)+T(n-2, k)+2T(n-1, k) otherwise; T(n, k)=if(k<=n, sum{j=0..floor((n-k)/2), C(n-j, k)C(n-k-j, j)2^(n-2j-k)}; - Paul Barry, Mar 15 2005

Bivariate g.f. G(x,z) = 1/[1 - (2 + x)z - z^2]. G.f. for column k = z^k/(1 - 2z - z^2)^{k+1} (k>=0). - Emeric Deutsch, Aug 30 2015

T(n,k) = 2^(n-k)*C(n,k)*hypergeom([(k-n)/2,(k-n+1)/2],[-n],-1)) for n>=1. - Peter Luschny, Apr 25 2016

A006645 Self-convolution of Pell numbers (A000129).

Original entry on oeis.org

0, 0, 1, 4, 14, 44, 131, 376, 1052, 2888, 7813, 20892, 55338, 145428, 379655, 985520, 2545720, 6547792, 16777993, 42847988, 109099078, 277040572, 701794187, 1773851304, 4474555476, 11266301976, 28318897549, 71070913036, 178106093666, 445740656420, 1114147888655

Offset: 0

N. J. A. Sloane

			G.f. = x^2 + 4*x^3 + 14*x^4 + 44*x^5 + 131*x^6 + 376*x^7 + 1052*x^8 + ...

R. P. Grimaldi, Ternary strings with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149. (The sequences w_n and z_n)

Michael De Vlieger, Table of n, a(n) for n = 0..2605
Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96-106.
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

a(n)= A054456(n-1, 1), n>=1 (second column of triangle), A054457.

a:= n-> (Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [4, -2, -4, -1][i] else 0 fi)^n) [1,3]: seq(a(n), n=0..40); # Alois P. Heinz, Oct 28 2008

pell[n_] := Simplify[ ((1+Sqrt[2])^n - (1-Sqrt[2])^n)/(2*Sqrt[2])]; a[n_] := First[ ListConvolve[ pp = Array[pell, n+1, 0], pp]]; Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Oct 21 2011 *)
Table[(n Fibonacci[n - 1, 2] + (n - 1) Fibonacci[n, 2])/4, {n, 0, 30}] (* Vladimir Reshetnikov, May 08 2016 *)

taylor( mul(x/(1 - 2*x - x^2) for i in range(1,3)),x,0,28) # Zerinvary Lajos, Jun 03 2009

a(n) = Sum_{k=0..n} b(k)*b(n-k) with b(k) := A000129(k).
a(n) = Sum_{k=0..floor((n-2)/2)} 2^(n-2)*(n-k-1)*binomial(n-2-k, k)*(1/4)^k, n >= 2.
From Wolfdieter Lang, Apr 11 2000: (Start)
a(n) = ((n-1)*P(n) + n*P(n-1))/4, P(n)=A000129(n).
G.f.: (x/(1 - 2*x - x^2))^2. (End)
a(n) = F'(n, 2), the derivative of the n-th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

Sum formulas and cross-references added by Wolfdieter Lang, Aug 07 2002

A098616 Product of Pell and Catalan numbers: a(n) = A000129(n+1)*A000108(n).

Original entry on oeis.org

1, 2, 10, 60, 406, 2940, 22308, 175032, 1408550, 11561836, 96425836, 814773960, 6960289532, 60012947800, 521582661000, 4564643261040, 40190674554630, 355772529165900, 3164408450118300, 28266363849505320, 253466716153665300, 2280803103062033160, 20588945107316958840

Offset: 0

Paul D. Hanna, Oct 09 2004

Radius of convergence: r = (sqrt(2)-1)/4, where A(r) = sqrt(2+sqrt(2)).

More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), |b|>0, |c|>0, then Sum_{n>=0} S(n)*Catalan(n)*x^n = sqrt( (1-2*b*x - sqrt(1-4*b*x-16*c*x^2))/(2*b^2+8*c) )/x.

			Sequence begins: [1*1, 2*1, 5*2, 12*5, 29*14, 70*42, 169*132, 408*429,...].

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000129, A000108, A098614, A098617, A098618, A000984, A214377.

With[{nn=30},Times@@@Thread[{LinearRecurrence[{2,1},{1,2},nn], CatalanNumber[ Range[0,nn-1]]}]] (* Harvey P. Dale, Jan 04 2012 *)
a[n_] := Fibonacci[n + 1, 2] * CatalanNumber[n]; Array[a, 25, 0] (* Amiram Eldar, May 05 2023 *)

a(n) = binomial(2*n,n)/(n+1)*round(((1+sqrt(2))^(n+1)-(1-sqrt(2))^(n+1))/(2*sqrt(2)))

G.f.: A(x) = sqrt( (1-4*x - sqrt(1-8*x-16*x^2))/16 )/x.
Run lengths of zeros (mod 10) equal (5^k - (-1)^k)/2 - 1 starting at index (5^k + (-1)^k)/2:
a(n) == 0 (mod 10) for n = (5^k + (-1)^k)/2 through n = 5^k - 1 when k>=1.
a(n) ~ 2^(2*n-3/2) * (1+sqrt(2))^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, May 09 2014
A(-x) = 1/x * series reversion( x*(2*x + sqrt(1 - 4*x^2)) ). Compare with the o.g.f. B(x) of the central binomial numbers A000984, which satisfies B(-x) = 1/x * series reversion( x*(2*x + sqrt(1 + 4*x^2)) ). See also A214377. - Peter Bala, Oct 19 2015
n*(n+1)*a(n) -4*n*(2*n-1)*a(n-1) -4*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Nov 17 2018
Sum_{n>=0} a(n)/16^n = 2*sqrt(3-sqrt(7)). - Amiram Eldar, May 05 2023
G.f. A(x) satisfies A(x) = sqrt( 1 + 4*x*A(x)^2 + 8*x^2*A(x)^4 ). - Paul D. Hanna, Dec 14 2024

A048776 First partial sums of A048739; second partial sums of A000129.

Original entry on oeis.org

1, 4, 12, 32, 81, 200, 488, 1184, 2865, 6924, 16724, 40384, 97505, 235408, 568336, 1372096, 3312545, 7997204, 19306972, 46611168, 112529329, 271669848, 655869048, 1583407968, 3822685009, 9228778012, 22280241060, 53789260160, 129858761409, 313506783008

Offset: 0

Barry E. Williams

Index entries for linear recurrences with constant coefficients, signature (4,-4,0,1).

Cf. A001333, A000129, A048739.

with(combinat):seq((fibonacci(n+3, 2)-n-3)/2, n=0..25); # Zerinvary Lajos, Jun 02 2008

a=b=0;Table[c=2*b+a+n;a=b;b=c,{n,1,40}] (* Vladimir Joseph Stephan Orlovsky, Apr 02 2011*)
LinearRecurrence[{4,-4,0,1},{1,4,12,32},30] (* Harvey P. Dale, Aug 27 2014 *)

a(n) = 2*a(n-1) + a(n-2) + n + 1; a(0)=1, a(1)=4.
a(n) = (((7/2 + (5/2)*sqrt(2))*(1+sqrt(2))^n - (7/2 - (5/2)*sqrt(2))*(1-sqrt(2))^n)/2*sqrt(2)) - (n+3)/2.
a(n) = (A000129(n+3) - (n+3))/2 = Sum_{j} A047662(n-j+1, j+1). - Henry Bottomley, Jul 09 2001
From R. J. Mathar, Feb 06 2010: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-4).
G.f.: -1/((x^2+2*x-1) * (x-1)^2). (End)
Define an array with m(n,1)=1 and m(1,k) = k*(k+1)/2 for n=1,2,3,...  The interior terms are m(n,k) = m(n,k-1) + m(n-1,k-1) + m(n-1,k). The sum of the terms in each antidiagonal=a(n). - J. M. Bergot, Dec 01 2012 [This is A154948 without the first column. The diagonal is m(n,n) = A161731(n-1). R. J. Mathar, Dec 06 2012]
E.g.f.: exp(x)*(10*cosh(sqrt(2)*x) + 7*sqrt(2)*sinh(sqrt(2)*x) - 2*(3 + x))/4. - Stefano Spezia, May 13 2023

More terms from Harvey P. Dale, Aug 27 2014

A265744 a(n) is the number of Pell numbers (A000129) needed to sum to n using the greedy algorithm (A317204).

Original entry on oeis.org

0, 1, 1, 2, 2, 1, 2, 2, 3, 3, 2, 3, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 3, 4, 2, 3, 3, 4, 4, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 3, 4, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 3, 4, 4, 5, 5, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 1, 2, 2, 3, 3, 2, 3, 3, 4, 4, 3, 4, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 3, 4, 4, 5, 5, 2, 3, 3, 4, 4, 3, 4, 4, 5, 5, 4, 5, 3

Offset: 0

Antti Karttunen, Dec 17 2015

a(0) = 0, because no numbers are needed to form an empty sum, which is zero.

It would be nice to know for sure whether this sequence also gives the least number of Pell numbers that add to n, i.e., that there cannot be even better nongreedy solutions.

A. F. Horadam, Zeckendorf representations of positive and negative integers by Pell numbers, Applications of Fibonacci Numbers, Springer, Dordrecht, 1993, pp. 305-316.

Antti Karttunen, Table of n, a(n) for n = 0..13860
L. Carlitz, Richard Scoville, and V. E. Hoggatt, Jr., Pellian Representations, The Fibonacci Quarterly, Vol. 10, No. 5 (1972), pp. 449-488.

Cf. A000129, A007953, A317204.
Cf. also A014420, A053610, A265404, A265743, A265745.
Similar sequences: A007895, A116543, A278043.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; a[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; Plus @@ IntegerDigits[Total[3^(s - 1)], 3]]; Array[a, 100, 0] (* Amiram Eldar, Mar 12 2022 *)

a(n) = A007953(A317204(n)). - Amiram Eldar, Mar 12 2022

A086383 Prime terms in the sequence of Pell numbers, A000129.

Original entry on oeis.org

2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449, 4760981394323203445293052612223893281

Offset: 1

Cino Hilliard, Sep 06 2003; corrected Jul 30 2004

nonn

Previous Name: Primes found among the denominators of the continued fraction rational approximations to sqrt(2).
See A096650 for the indices. - Jon E. Schoenfield, Jan 25 2017
A056869 is essentially the same sequence. - Jianing Song, Jan 02 2019

			a(1) = 2 = A000129(2), a(2) = 5 = A000129(3), a(3) = 29 = A000129(5), etc. - _Zak Seidov_, Oct 21 2013 [Corrected by _Jianing Song_, Jan 02 2019]

Harvey P. Dale, Table of n, a(n) for n = 1..23
J. L. Schiffman, Exploring the Fibonacci sequence of order two with CAS technology, Paper C027, Electronic Proceedings of the Twenty-fourth Annual International Conference on Technology in Collegiate Mathematics, Orlando, Florida, March 22-25, 2012. See p. 262. - _N. J. A. Sloane_, Mar 27 2014

Cf. A000129, A056869, A096650.

f:=[0,1];; for n in [3..100] do f[n]:=2*f[n-1]+f[n-2]; od; a:=Filtered(f,IsPrime);; Print(a); # Muniru A Asiru, Jan 03 2019

Select[Table[ChebyshevU[k,3]-ChebyshevU[k-1,3],{k,0,50}],PrimeQ] (* Ed Pegg Jr, May 10 2007 *)
Select[Denominator[Convergents[Sqrt[2],150]],PrimeQ] (* Harvey P. Dale, Dec 19 2012 *)
Select[LinearRecurrence[{2, 1}, {0, 1}, 16], PrimeQ] (* Zak Seidov, Oct 21 2013 *)

\\ Continued fraction rational approximation of numeric constants f. m=steps.
cfracdenomprime(m,f) = { default(realprecision,3000); cf = vector(m+10); x=f; for(n=0,m, i=floor(x); x=1/(x-i); cf[n+1] = i; ); for(m1=0,m, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n]; ); numer=numerator(r); denom=denominator(r); if(ispseudoprime(denom),print1(denom,",")); ) }

a(n) = A000129(A096650(n)). - Jon E. Schoenfield, Jan 25 2017

a(n) = A056869(n-1), n > 1. - Jianing Song, Jan 02 2019

Name changed (to a Comments entry from Zak Seidov, Oct 21 2013) by Jon E. Schoenfield, Jan 26 2017

A337233 Composite integers m such that P(m)^2 == 1 (mod m), where P(m) is the m-th Pell number A000129(m). Also, odd composite integers m such that U(m)^2 == 1 (mod m) and V(m) == 6 (mod m), where U(m)=A001109(m) and V(m)=A003499(m) are the m-th generalized Lucas and Pell-Lucas numbers of parameters a=6 and b=1, respectively.

Original entry on oeis.org

35, 119, 169, 385, 741, 779, 899, 935, 961, 1105, 1121, 1189, 1443, 1479, 2001, 2419, 2555, 2915, 3059, 3107, 3383, 3605, 3689, 3741, 3781, 3827, 4199, 4795, 4879, 4901, 5719, 6061, 6083, 6215, 6265, 6441, 6479, 6601, 6895, 6929, 6931, 6965, 7055, 7107, 7801, 8119

Offset: 1

Ovidiu Bagdasar, Aug 20 2020

nonn

For a, b integers, the following sequences are defined:
generalized Lucas sequences by U(m+2)=a*U(m+1)-b*U(m) and U(0)=0, U(1)=1,
generalized Pell-Lucas sequences by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a.
In general, one has U^2(p) == 1 and V(p)==a (mod p) whenever p is prime and b=1, -1.
The composite numbers satisfying these congruences may be called weak generalized Lucas-Bruckner pseudoprimes of parameters a and b.
For a=2 and b=-1, U(m) recovers A000129(m) (Pell numbers).
For a=6 and b=1, we have U(m)=A001109(m) and V(m)=A003499(m).
This sequence contains the odd composite integers for which the congruence A000129(m)^2 == 1 (mod m) holds.
This is also the sequence of odd composite numbers satisfying the congruences A001109(m)^2 == 1 and A003499(m)==a (mod m).

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

D. Andrica and O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, preprint for Mediterr. J. Math. 18, 47 (2021).

Cf. A337231 (a=1, odd terms), A337232 (a=1, even terms), A337629 (a=6, b=-1), A337778 (a=4, b=1), A337779 (a=5, b=1).

Select[Range[3, 25000, 2], CompositeQ[#] && Divisible[Fibonacci[#, 2]*Fibonacci[#, 2] - 1, #] &]
Select[Range[3, 20000, 2], CompositeQ[#] && Divisible[2*ChebyshevT[#, 3] - 6, #] && Divisible[ChebyshevU[#-1, 3]*ChebyshevU[#-1, 3] - 1, #] &]

A054457 Pell numbers A000129(n+1) (without P(0)) convoluted twice with itself.

Original entry on oeis.org

1, 6, 27, 104, 366, 1212, 3842, 11784, 35223, 103122, 296805, 842160, 2360780, 6549240, 18004980, 49106992, 132996957, 357948894, 957993823, 2550977112, 6761742234, 17848312884, 46932923478, 122980461816

Offset: 0

Wolfdieter Lang, Apr 27 2000

a(n)= A054456(n+2,2) (third column of Pell convolution triangle).

Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
Index entries for linear recurrences with constant coefficients, signature (6,-9,-4,9,6,1)

Cf. A054456, A000129, A006645.

a(n) = ((10*n^2+39*n+32)*P(n+1)+(n+1)*(4*n+11)*P(n))/32, where P(n)=A000129(n).
G.f.: 1/(1-2*x-x^2)^3.
a(n) = F''(n+3, 2)/2, that is, 1/2 times the 2nd derivative of the (n+3)th Fibonacci polynomial evaluated at x=2. - T. D. Noe, Jan 19 2006

Previous Showing 11-20 of 777 results. Next

T(n, k) are the coefficients of the polynomials p(1, x) = 1, p(2, x) = 2 + 2*x, p(n, x) = u*p(n-1, x) + v*p(n-2, x) for n >= 3, where u = p(2, x), v = 1 - 2*x - x^2.

Because (p(n, x)) is a strong divisibility sequence, for each integer k, the sequence (p(n, k)) is a strong divisibility sequence of integers.

cp's OEIS Frontend

A367211 Triangular array read by rows: T(n, k) = binomial(n, k) * A000129(n - k) for 0 <= k < n.

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A084158 a(n) = A000129(n) * A000129(n+1)/2.

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A054456 Convolution triangle of A000129(n) (Pell numbers).

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A006645 Self-convolution of Pell numbers (A000129).

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A098616 Product of Pell and Catalan numbers: a(n) = A000129(n+1)*A000108(n).

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A048776 First partial sums of A048739; second partial sums of A000129.

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A265744 a(n) is the number of Pell numbers (A000129) needed to sum to n using the greedy algorithm (A317204).

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