A002203 -id:A002203 - cp's OEIS frontend

A204273 a(n) = sigma_3(n)*Pell(n), where sigma_3(n) = A001158(n), the sum of cubes of divisors of n.

1, 18, 140, 876, 3654, 17640, 58136, 238680, 745645, 2696652, 7647012, 28329840, 73547278, 250101072, 688048200, 2203964592, 5585689746, 18696302730, 45448247740, 147116748744, 371929710880, 1117549627704, 2738514030408, 8899904613600

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^3*x^n/(1-x^n) = Sum_{n>=1} sigma_3(n)*x^n.

			G.f.: A(x) = x + 18*x^2 + 140*x^3 + 876*x^4 + 3654*x^5 + 17640*x^6 + ...
where A(x) = x/(1-2*x-x^2) + 2^3*2*x^2/(1-6*x^2+x^4) + 3^3*5*x^3/(1-14*x^3-x^6) + 4^3*12*x^4/(1-34*x^4+x^8) + 5^3*29*x^5/(1-82*x^5-x^10) + 6^3*70*x^6/(1-198*x^6+x^12) + ... + n^3*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A203838, A204270, A204271, A204272, A204274, A204275, A001158 (sigma_3), A002203, A000045.

Table[DivisorSigma[3, n] Fibonacci[n, 2], {n, 1, 50}] (* G. C. Greubel, Jan 05 2018 *)

/* Subroutines used in PARI programs below: */
{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)}

PARI
```
{a(n)=sigma(n,3)*Pell(n)}
```

{a(n)=polcoeff(sum(m=1,n,m^3*Pell(m)*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}

G.f.: Sum_{n>=1} n^3*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_3(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

A209447 a(n) = Pell(n)*A008653(n) for n>=1, with a(0)=1, where A008653 is the theta series of direct sum of 2 copies of hexagonal lattice.

Original entry on oeis.org

1, 12, 72, 60, 1008, 2088, 2520, 16224, 73440, 11820, 513648, 826704, 1164240, 5621448, 23265216, 14041800, 175149504, 245524824, 98791560, 1590026160, 8061191712, 3706940640, 40272058656, 64816900128, 97801149600, 487966581012, 1596075244848, 91744440540

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to the Lambert series of A008653: 1 + 12*Sum_{n>=1} Chi(n,3)*n*x^n/(1-x^n).

Here Chi(n,3) = principal Dirichlet character of n modulo 3.

			G.f.: A(x) = 1 + 12*x + 72*x^2 + 60*x^3 + 1008*x^4 + 2088*x^5 + 2520*x^6 +...
where A(x) = 1 + 1*12*x + 2*36*x^2 + 5*12*x^3 + 12*84*x^4 + 29*72*x^5 + 70*36*x^6 +...+ Pell(n)*A008653(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 12*( 1*1*x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 12*4*x^4/(1-34*x^4+x^8) + 29*5*x^5/(1-82*x^5-x^10) + 169*7*x^7/(1-478*x^7-x^14) + 408*8*x^8/(1-1154*x^8-x^16)  +...).
The values of the Dirichlet character Chi(n,3) repeat [1,1,0, ...].

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A008653, A205967, A209446, A209448, A204270, A000129 (Pell), A002203.

A008653[n_]:= If[n < 1, Boole[n == 0], 12*Sum[If[Mod[d, 3] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A008653[n], {n, 1, 1000}]] (* G. C. Greubel, Jan 02 2017 *)

{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(1 + 12*sum(m=1,n,Pell(m)*kronecker(m,3)^2*m*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,50,print1(a(n),", "))

G.f.: 1 + 12*Sum_{n>=1} Pell(n)*Chi(n,3)*n*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A209449 a(n) = Pell(n)*A113973(n) for n>=1, with a(0)=1, where A113973 lists the coefficients in phi(x^3)^3/phi(x) and phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 8, -10, 24, 0, 280, -676, 1632, -1970, 0, 0, 27720, -133844, 646256, 0, 941664, 0, 10976840, -26500436, 0, -154455860, 0, 0, 2173358880, -2623476242, 25334527696, -15290740090, 73830224208, 0, 0, -1038870091396, 2508054264192, 0, 0, 0, 42600007379160

Offset: 0

Paul D. Hanna, Mar 10 2012

sign

Compare g.f. to the Lambert series of A113973:

1 - 2*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - (-x)^n).

			G.f.: A(x) = 1 - 2*x + 8*x^2 - 10*x^3 + 24*x^4 + 280*x^6 - 676*x^7 +...
where A(x) = 1 - 1*2*x + 2*4*x^2 - 5*2*x^3 + 12*2*x^4 + 70*4*x^6 - 169*4*x^7 + 408*4*x^8 +...+ Pell(n)*A113973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 2*( 1*x/(1+2*x-x^2) - 2*x^2/(1-6*x^2+x^4) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 408*x^8/(1-1154*x^8+x^16) +...).
The values of the symbol Kronecker(n,3) repeat [1,-1,0, ...].

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A113973, A205969, A209446, A209448, A209450, A204270, A000129 (Pell), A002203.

A113973:= CoefficientList[Series[EllipticTheta[3, 0, q^3]^3 /EllipticTheta[3, 0, q], {q, 0, 60}], q]; Table[If[n == 0, 1, Fibonacci[n, 2]*A113973[[n + 1]]], {n, 0, 50}] (* G. C. Greubel, Dec 03 2017 *)

{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(1 - 2*sum(m=1,n,Pell(m)*kronecker(m,3)*x^m/(1-A002203(m)*(-x)^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,60,print1(a(n),", "))

G.f.: 1 - 2*Sum_{n>=1} Pell(n)*Kronecker(n,3)*x^n/(1 - A002203(n)*(-x)^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A006668 Exponential self-convolution of Pell numbers (divided by 2).

Original entry on oeis.org

0, 0, 1, 6, 32, 160, 784, 3808, 18432, 89088, 430336, 2078208, 10035200, 48455680, 233967616, 1129701376, 5454692352, 26337607680, 127169265664, 614027624448, 2964787822592, 14315262312448, 69120201588736

Offset: 0

N. J. A. Sloane

Binomial transform of A084150. - Paul Barry, May 16 2003

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-4,-8).

Cf. A006646, A002203.

[Floor(((2+Sqrt(8))^n+(2-Sqrt(8))^n-2^(n+1))/16): n in [0..30] ]; // Vincenzo Librandi, Aug 20 2011

LinearRecurrence[{6,-4,-8},{0,0,1},30] (* Harvey P. Dale, Jul 15 2014 *)
Table[2^(n-4)*(LucasL[n, 2] - 2), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 07 2016 *)

a(n) = ((2+sqrt(8))^n+(2-sqrt(8))^n-2^(n+1))/16; E.g.f. : exp(2x)(sinh(sqrt(2)x))^2/4=(exp(x)sinh(sqrt(2)x)/sqrt(2))^2/2. - Paul Barry, May 16 2003
G.f.: x^2/((1-2*x)*(1-4*x-4*x^2)). - Bruno Berselli, Aug 20 2011
a(n) = A006646(n)/2 = 2^(n-4)*(A002203(n) - 2). - Vladimir Reshetnikov, Oct 07 2016

A054459 A001333(n), n >= 1, convolved with itself.

Original entry on oeis.org

1, 6, 23, 76, 233, 682, 1935, 5368, 14641, 39406, 104935, 276996, 725849, 1890258, 4896415, 12624752, 32419297, 82951766, 211573047, 538086716, 1364974409, 3454480250, 8724052271, 21989264232, 55326056977, 138975010110

Offset: 0

Wolfdieter Lang, Apr 26 2000

a(n) = A054458(n+1,1) (second column of convolution triangle).

R. P. Grimaldi, Ternary strings with no consecutive 0's and no consecutive 1's, Congressus Numerantium, 205 (2011), 129-149. (The sequence t_n, also the sequence lev_{n-1}.)

Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A054458, A001333, A002203.

LinearRecurrence[{4, -2, -4, -1}, {1, 6, 23, 76}, 30] (* Paolo Xausa, Feb 06 2024 *)

a(n) = ((4*n+3)*LP(n)+(2*n+1)*LP(n-1))/4, n >= 1, with LP(n) = A001333(n+1), a(0) = 1.
G.f.: ((1+x)/(1-2*x-x^2))^2.
a(n) = 2*A006645(n+2) - A000129(n+1). - R. J. Mathar, Feb 05 2024

A107769 a(n) = (A001333(n+1) - 2*A005409(floor((n+3)/2)) - 1) / 4.

Original entry on oeis.org

0, 1, 2, 8, 19, 54, 130, 334, 806, 1995, 4816, 11746, 28357, 68748, 165972, 401388, 969036, 2341141, 5652014, 13649228, 32952151, 79563330, 192082870, 463752730, 1119598130, 2703006111, 6525634012, 15754412038, 38034515209, 91823775384, 221682203880, 535188986904, 1292060510616, 3119311948585

Offset: 0

Emeric Deutsch, Jun 12 2005

a(n) is the number of free polyominoes of width 2 and height n+1 which have no symmetry, i.e., rotations by 180 degrees, flips along the short or long axis generate a different free polyomino. The three elements t, g+ and g- of sequences by Tasi et al. represent a domino in the short cross-section where either both, only the "upper" or only the "lower" square of the domino is occupied. E.g., a(3) = 8 represents 3 5-ominoes of shape the 2x4, 3 6-ominoes of shape 2x4, and 2 7-ominoes of shape 2x4. - R. J. Mathar, Jun 17 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 63, eq 25).
Index entries for linear recurrences with constant coefficients, signature (3,1,-7,3,-1,1,1).

Cf. A000129, A001333, A002203, A005409, A126877, A135153.

Table[(LucasL[n+2, 2] -4*Fibonacci[Floor[n/2]+2, 2] +2)/8, {n,0,40}] (* G. C. Greubel, May 24 2021 *)

[(lucas_number2(n+2,2,-1) -4*lucas_number1(2+(n//2),2,-1) +2)/8 for n in (0..40)] # G. C. Greubel, May 24 2021

4*a(n) = Pell(n+3) - Pell(n+2) - 2*Pell(floor((n+4)/2)) + 1, with Pell(n) = A000129(n). - Ralf Stephan, Jun 02 2007
G.f.: x*(1-x+x^2)/((1-x)*(1-2*x-x^2)*(1-2*x^2-x^4)). - Colin Barker, Apr 08 2013
4*a(n) = A001333(n+2) -2*A135153(n+4) +1. - R. J. Mathar, Jun 17 2020
From G. C. Greubel, May 24 2021: (Start)
a(n) = (1/4)*(A001333(n+2) - 2*A000129(floor(n/2)+2) + 1).
a(n) = (1/8)*(A002203(n+2) - 4*A000129(floor(n/2)+2) + 2). (End)

Entry revised by N. J. A. Sloane, Jul 29 2011

A114696 Expansion of (1+4x+x^2)/((1-x^2)(1-2*x-x^2)); a Pellian-related sequence.

Original entry on oeis.org

1, 6, 15, 40, 97, 238, 575, 1392, 3361, 8118, 19599, 47320, 114241, 275806, 665855, 1607520, 3880897, 9369318, 22619535, 54608392, 131836321, 318281038, 768398399, 1855077840, 4478554081, 10812186006, 26102926095, 63018038200, 152139002497, 367296043198

Offset: 0

Creighton Dement, Feb 18 2006

Elements of odd index give match to A065113: Sum of the squares of the n-th and the (n+1)st triangular numbers (A000217) is a perfect square.

Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Cf. A000129, A002203, A005409, A100828, A111954, A113224.

Cf. A114647, A114688, A114689, A114695, A114697.

Q:= proc(n) option remember; # Q=A002203
    if n<2 then 2
  else 2*Q(n-1) + Q(n-2)
    fi; end:
seq((Q(n+2) -3 -(-1)^n)/2, n=0..40); # G. C. Greubel, May 24 2021

CoefficientList[Series[(1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)), {x,0,30}], x] (* or *) LinearRecurrence[{2,2,-2,-1}, {1,6,15,40}, 30] (* Harvey P. Dale, Jan 23 2014 *)

Vec((1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^30)) \\ Colin Barker, May 26 2016

[(lucas_number2(n+2,2,-1) -3 -(-1)^n)/2 for n in (0..30)] # G. C. Greubel, May 24 2021

G.f.: (1 +4*x +x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
a(0)=1, a(1)=6, a(2)=15, a(3)=40, a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4). - Harvey P. Dale, Jan 23 2014
a(n) = (-3 - (-1)^n + (3-2*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(3+2*sqrt(2)))/2. - Colin Barker, May 26 2016
From G. C. Greubel, May 24 2021: (Start)
a(n) = 3*A000129(n+1) + A000129(n) - (3 + (-1)^n)/2.
a(n) = (1/2)*(A002203(n+2) - 3 - (-1)^n). (End)

A117918 Difference row triangle of the Pell sequence.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 2, 4, 7, 12, 4, 6, 10, 17, 29, 4, 8, 14, 24, 41, 70, 8, 12, 20, 34, 58, 99, 169, 8, 16, 28, 48, 82, 140, 239, 408, 16, 24, 40, 68, 116, 198, 338, 577, 985, 16, 32, 56, 96, 164, 280, 478, 816, 1393, 2378, 32, 48, 80, 136, 232, 396, 676, 1154, 1970, 3363, 5741

Offset: 1

Gary W. Adamson, Apr 02 2006

			First few rows of the triangle are:
  1;
  1,  2;
  2,  3,  5;
  2,  4,  7, 12;
  4,  6, 10, 17, 29;
  4,  8, 14, 24, 41, 70;
  8, 12, 20, 34, 58, 99, 169;
  ...

Raymond Lebois, "Le théorème de Pythagore et ses implications", p. 123, Editions PIM, (1979).

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Column 1 is A016116(n-1).

Diagonals include A000129, A001333, A052542, A002203, A371596.

Pell:= func< n | Round(((1+Sqrt(2))^n -(1-Sqrt(2))^n)/(2*Sqrt(2))) >;
T:= func< n,k | (&+[ (-1)^j*Binomial(n-k,j)*Pell(n-j): j in [0..n-k]]) >;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 23 2021

T[n_, k_]:= T[n, k]= If[k==1, 2^Floor[(n-1)/2], T[n, k-1] + T[n-1, k-1]];
Table[T[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 22 2021 *)

def A117918(n,k): return sum( (-1)^j*binomial(n-k, j)*lucas_number1(n-j, 2,-1) for j in (0..n) )
flatten([[A117918(n,k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Oct 23 2021

Difference rows of the Pell sequence A000129 starting (1, 2, 5, 12, ...) become the diagonals of the triangle.
T(n, n) = A000129(n).
T(n, n-1) = A000129(n) - A000129(n-1).
From G. C. Greubel, Oct 23 2021: (Start)
T(n, k) = T(n, k-1) + T(n-1, k-1) with T(n, 1) = 2^floor((n-1)/2).
T(n, k) = Sum_{j=0..n-k} (-1)^j*binomial(n-k, j)*Pell(n-j), where Pell(n) = A000129(n).
Sum_{k=1..n} T(n, k) = Pell(n+1) -2^floor(n/2)*((1 + (-1)^n)/2) - 2^floor((n - 1)/2)*((1 - (-1)^n)/2). (End)

A182143 Number of independent vertex sets in the Moebius ladder graph with 2n nodes (n >= 0).

Original entry on oeis.org

1, 3, 5, 15, 33, 83, 197, 479, 1153, 2787, 6725, 16239, 39201, 94643, 228485, 551615, 1331713, 3215043, 7761797, 18738639, 45239073, 109216787, 263672645, 636562079, 1536796801, 3710155683, 8957108165, 21624372015, 52205852193, 126036076403, 304278004997

Offset: 0

Cesar Bautista, Apr 14 2012

Also the number of vertex covers. - Eric W. Weisstein, Jan 04 2014

Cesar Bautista, Table of n, a(n) for n = 0..1000
C. Bautista-Ramos and C. Guillen-Galvan, Fibonacci numbers of generalized Zykov sums, J. Integer Seq., 15 (2012), Article 12.7.8
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Moebius Ladder
Eric Weisstein's World of Mathematics, Vertex Cover
Index entries for linear recurrences with constant coefficients, signature (1,3,1).

Cf. A000129, A001333, A002203, A098600, A100227.

I:=[1,3,5]; [n le 3 select I[n] else Self(n-1)+3*Self(n-2)+Self(n-3): n in [1..31]]; // Bruno Berselli, Apr 14 2012

Table[(1 + Sqrt[2])^n + (1 - Sqrt[2])^n - (-1)^n, {n, 0, 30}] (* Bruno Berselli, Apr 14 2012 *)
Table[LucasL[n, 2] - (-1)^n, {n, 0, 20}] (* Vladimir Reshetnikov, Sep 15 2016 *)
LinearRecurrence[{1, 3, 1}, {1, 3, 5}, 20] (* Eric W. Weisstein, Mar 31 2017 *)
CoefficientList[Series[(-1 - 2 x + x^2)/(-1 + x + 3 x^2 + x^3), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)

Vec((x^2-2*x-1)/((x+1)*(x^2+2*x-1))+O(x^31)) \\ Bruno Berselli, Apr 14 2012

G.f.: (x^2-2*x-1)/((x+1)*(x^2+2*x-1)).
a(n) = (1+sqrt(2))^n + (1-sqrt(2))^n - (-1)^n = A002203(n) - (-1)^n.
a(n) = a(n-1) + 3*a(n-2) + a(n-3) with a(0)=1, a(1)=3, a(2)=5.
From Peter Bala, Jun 29 2015: (Start)
a(n) = Pell(n-1) + Pell(n+1) - (-1)^n.
a(n) = [x^n] ( (1 + 2*x + sqrt(1 + 8*x + 8*x^2))/2 )^n.
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 7*x^3 + 17*x^4 + 41*x^5 + ... = Sum_{n >= 0} A001333*x^n. Cf. A098600. (End)

A209444 a(n) = Pell(n)*A000143(n) for n>=1 with a(0)=1, where A000143(n) is the number of ways of writing n as a sum of 8 squares.

Original entry on oeis.org

1, 16, 224, 2240, 13632, 58464, 219520, 930176, 3805824, 11930320, 33558336, 122352192, 440858880, 1176756448, 3112368896, 11008771200, 35248366848, 89371035936, 232665100640, 727171963840, 2289378446208, 5950875374080, 13907284255872, 43816224486528

Offset: 0

Paul D. Hanna, Mar 09 2012

nonn

Compare g.f. to the Lambert series of A000143: 1 + 16*Sum_{n>=1} n^3*x^n/(1 - (-x)^n).

			G.f.: A(x) = 1 + 16*x + 112*x^2 + 896*x^3 + 3408*x^4 + 10080*x^5 +...
where A(x) = 1 + 1*16*x + 2*112*x^2 + 5*448*x^3 + 12*1136*x^4 + 29*2016*x^5 + 70*3136*x^6 + 169*5504*x^7 + 408*9328*x^8 +...+ Pell(n)*A000143(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 16*( 1*1*x/(1+2*x-x^2) + 2*8*x^2/(1-6*x^2+x^4) + 5*27*x^3/(1+14*x^3-x^6) + 12*64*x^4/(1-34*x^4+x^8) + 29*125*x^5/(1+82*x^5-x^10) + 70*216*x^6/(1-198*x^6+x^12) + 169*343*x^7/(1+478*x^7-x^14) +...).

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000143, A205964, A205508, A209443, A204270.

A000143:= Table[SquaresR[8, n], {n, 0, 200}]; Join[{1}, Table[Fibonacci[n, 2]*A000143[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2018 *)

{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(1+16*sum(m=1,n,Pell(m)*m^3*x^m/(1-A002203(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
for(n=0,31,print1(a(n),", "))

G.f.: 1 + 16*Sum_{n>=1} Pell(n)*n^3*x^n/(1 - A002203(n)*(-x)^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

cp's OEIS Frontend

A204273 a(n) = sigma_3(n)*Pell(n), where sigma_3(n) = A001158(n), the sum of cubes of divisors of n.

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PARI

PARI

PARI

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A209447 a(n) = Pell(n)*A008653(n) for n>=1, with a(0)=1, where A008653 is the theta series of direct sum of 2 copies of hexagonal lattice.

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A209449 a(n) = Pell(n)*A113973(n) for n>=1, with a(0)=1, where A113973 lists the coefficients in phi(x^3)^3/phi(x) and phi() is a Ramanujan theta function.

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PARI

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A006668 Exponential self-convolution of Pell numbers (divided by 2).

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Magma

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A054459 A001333(n), n >= 1, convolved with itself.

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A107769 a(n) = (A001333(n+1) - 2*A005409(floor((n+3)/2)) - 1) / 4.

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Extensions

A114696 Expansion of (1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

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A114696 Expansion of (1+4x+x^2)/((1-x^2)(1-2*x-x^2)); a Pellian-related sequence.