A002203 -id:A002203 - cp's OEIS frontend

A209446 a(n) = Pell(n)A004016(n) for n >= 1, with a(0)=1, where A004016(n) is the number of integer solutions (x,y) to x^2 + xy + y^2 = n.

1, 6, 0, 30, 72, 0, 0, 2028, 0, 5910, 0, 0, 83160, 401532, 0, 0, 2824992, 0, 0, 79501308, 0, 463367580, 0, 0, 0, 7870428726, 0, 45872220270, 221490672624, 0, 0, 3116610274188, 0, 0, 0, 0, 127800022137480, 617073093431772, 0, 3596565555708780, 0, 0, 0, 122177355889216668

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to the Lambert series of A004016: 1 + 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1 - x^n).

			G.f.: A(x) = 1 + 6*x + 30*x^3 + 72*x^4 + 2028*x^7 + 5910*x^9 + 83160*x^12 + ...
where A(x) = 1 + 1*6*x + 5*6*x^3 + 12*6*x^4 + 169*12*x^7 + 985*6*x^9 + 13860*6*x^12 + ... + Pell(n)*A004016(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1 + 6*( 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) + ...).
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. A004016, A205966, A209445, A209447, A204270, A000129 (Pell), A002203.

A004016[n_]:= If[n < 1, Boole[n == 0], 6 DivisorSum[n, KroneckerSymbol[#, 3] &]]; Join[{1}, Table[Fibonacci[n, 2]*A004016[n], {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 + 6*sum(m=1,n,kronecker(m,3)*Pell(m)*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 + 6*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).

A033539 a(0)=1, a(1)=1, a(2)=1, a(n) = 2*a(n-1) + a(n-2) + 1.

Original entry on oeis.org

1, 1, 1, 4, 10, 25, 61, 148, 358, 865, 2089, 5044, 12178, 29401, 70981, 171364, 413710, 998785, 2411281, 5821348, 14053978, 33929305, 81912589, 197754484, 477421558, 1152597601, 2782616761, 6717831124, 16218279010, 39154389145

Offset: 0

Antti Karttunen

a(n) or a(n+1) gives the number of times certain simple recursive procedures are called to effect a reversal of a sequence of n elements (including both the top-level call and any subsequent recursive calls). See example and program lines.

			See the Python, Erlang (myrev), PARI (rev) and Forth definitions (REV3) given at Program section.
PARI, Python and Erlang functions are called a(n+1) times for the list of length n, while Forth word REV3 is called a(n) times if there are n elements in the parameter stack.

T. D. Noe, Table of n, a(n) for n = 0..300
Antti Karttunen, More information
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Cf. A002203, A033538.

# definition, demonstrating the reversal of the lists:
myrev([]) -> [];
myrev([A]) -> [A];
myrev([X|Y]) ->
   [A|B] = myrev(Y),
   [A|myrev([X|myrev(B)])].

# definition, demonstrating how to turn a parameter stack upside down:
: REV3 DEPTH 0= IF ELSE DEPTH 1 = IF ELSE DEPTH 2 = IF SWAP ELSE >R RECURSE R> SWAP >R >R RECURSE R> RECURSE R> THEN THEN THEN ;
-- Antti Karttunen, Mar 04 2013

Concatenation([1], List([1..30], n-> (3*Lucas(2,-1,n-1)[2] -2)/4 )); # G. C. Greubel, Oct 13 2019

a033539 n = a033539_list !! n
a033539_list =
   1 : 1 : 1 : (map (+ 1) $ zipWith (+) (tail a033539_list)
                                        (map (2 *) $ drop 2 a033539_list))
-- Reinhard Zumkeller, Aug 14 2011

I:=[1,1,4]; [1] cat [n le 3 select I[n] else 3*Self(n-1) - Self(n-2) - Self(n-3): n in [1..30]]; // G. C. Greubel, Oct 13 2019

seq(coeff(series((1 -2*x -x^2 +3*x^3)/((1-x)*(1-2*x-x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 13 2019

Join[{1},RecurrenceTable[{a[0]==a[1]==1,a[n]==2a[n-1]+a[n-2]+1},a,{n,30}]] (* or *) LinearRecurrence[{3,-1,-1},{1,1,1,4},30] (* Harvey P. Dale, Nov 20 2011 *)
Table[If[n==0, 1, (3*LucasL[n-1, 2] -2)/4], {n, 0, 30}] (* G. C. Greubel, Oct 13 2019 *)

/* needs version >= 2.5 */
/* function demonstrating the reversal of the lists and counting the function calls: */
rev( L )={ cnt++; if( #L>1, my(x,y); x=L[#L]; listpop(L); L=rev(L); y=L[#L]; listpop(L); L=rev(L); listput(L,x); L=rev(L); listput(L,y)); L }
for(n=0,50,cnt=0;print(n": rev(",L=List(vector(n,i,i)),")=",rev(L),",  cnt="cnt)) \\ Antti Karttunen, Mar 05 2013, partially based on previous PARI code from Michael Somos, 1999. Edited by M. F. Hasler, Mar 05 2013

concat([1], vector(30, n, (3*sum(k=0,(n-1)\2, binomial(n-1,2*k) * 2^k) - 1)/2 )) \\ G. C. Greubel, Oct 13 2019

rev([],[]).
   rev([A],[A]).
   rev([A|XB],[B|YA]) :-
rev(XB,[B|Y]), rev(Y,X), rev([A|X],YA). % Lewis Baxter, Jan 04 2021

# function, demonstrating the reversal of the lists:
def myrev(lista):
  '''Reverses a list, in cumbersome way.'''
  if(len(lista) < 2): return(lista)
  else:
    tr = myrev(lista[1:])
    return([tr[0]]+myrev([lista[0]]+myrev(tr[1:])))

[1]+[(3*lucas_number2(n-1,2,-1) -2)/4 for n in (1..30)] # G. C. Greubel, Oct 13 2019

a(n) = (3/4)*(1+sqrt(2))^(n-1) + 3/4*(1-sqrt(2))^(n-1) - 1/2 + 3*0^n, with n >= 0. - Jaume Oliver Lafont, Sep 10 2009
G.f.: (1 - 2*x - x^2 + 3*x^3)/((1-x)*(1-2*x-x^2)). - Jaume Oliver Lafont, Sep 09 2009
a(n) = 3*a(n-1) - a(n-2) - a(n-3), a(0)=1, a(1)=1, a(2)=1, a(3)=4. - Harvey P. Dale, Nov 20 2011
a(n) = (3*A001333(n-1) - 1)/2. - R. J. Mathar, Mar 04 2013
a(n) = -1/2 - (3/4)*(1+sqrt(2))^n - (3/4)*sqrt(2)*(1-sqrt(2))^n - (3/4)*(1-sqrt(2))^n + (3/4)*(1+sqrt(2))^n*sqrt(2) for n >= 1. - Alexander R. Povolotsky, Mar 05 2013
E.g.f.: 3 + (1/2)*exp(x)*(-1 - 3*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, Oct 13 2019

A100227 Main diagonal of triangle A100226.

Original entry on oeis.org

1, 1, 5, 13, 33, 81, 197, 477, 1153, 2785, 6725, 16237, 39201, 94641, 228485, 551613, 1331713, 3215041, 7761797, 18738637, 45239073, 109216785, 263672645, 636562077, 1536796801, 3710155681, 8957108165, 21624372013, 52205852193, 126036076401, 304278004997

Offset: 0

Paul D. Hanna, Nov 29 2004

Specify that a triangle has T(n,0) = T(n,n) = (n+1)*(n+2)/2. The interior terms T(r,c) = T(r-1,c) + T(r-1,c-1) + T(r-2,c-1). The difference between the sum of the terms in row(n+1) and those in row(n) is a(n+2). - J. M. Bergot, Mar 15 2013
Starting with offset 1 the sequence is A001333: (1, 3, 7, 17, 41, ...), convolved with (1, 2, 0, 2, 0, 2, ...). - Gary W. Adamson, Aug 10 2016
Number of ways to tile a bracelet of length n with single-color squares, and two colors of k-ominoes for k > 1. Compare to A001333 as mentioned in the previous comment: A001333 can be thought of as the number of ways to tile a strip of length n with single-color squares and two-color k-ominoes for k > 1. - Greg Dresden, Feb 26 2020

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

Cf. A000129, A001333, A002203, A100225, A100226.

LucasL[Range[0,35], 2] - 1 (* G. C. Greubel, Feb 26 2020 *)

a(n):=if n=0 then 1 else n*sum(sum(binomial(k, i)*binomial(n-i-1, k-1),i,0,n-k)/k,k,1,n); /* Vladimir Kruchinin, May 13 2011 */

a(n)=if(n==0,1,n*polcoeff(log((1-x)/(1-2*x-x^2)+x*O(x^n)),n))

a(n)=polcoeff((1-2*x+3*x^2)/(1-3*x+x^2+x^3)+x*O(x^n),n)

a(n) = A002203(n) - 1.
a(n) = 2*a(n-1) + a(n-2) + 2 for n > 1, with a(0)=1, a(1)=1.
G.f.: Sum_{n>=1} a(n)*x^n/n = log((1-x)/(1-2*x-x^2)).
G.f.: (1-2*x+3*x^2)/((1-x)*(1-2*x-x^2)). - Paul D. Hanna, Feb 22 2005
a(n) = n*Sum_{k=1..n} (1/k)*Sum_{i=0..n-k} binomial(k, i)*binomial(n-i-1, k-1), n > 0, a(0)=1. - Vladimir Kruchinin, May 13 2011
a(n) = -1 + (1-sqrt(2))^n + (1+sqrt(2))^n. - Colin Barker, Mar 16 2016
E.g.f.: (2*cosh(sqrt(2)*x) - 1)*exp(x). - Ilya Gutkovskiy, Aug 22 2016
a(n) = A000129(n+1) + A000129(n-1)-1 for n > 0. - Rigoberto Florez, Jul 12 2020
a(n) = 3*a(n-1) - a(n-2) - a(n-3). - Wesley Ivan Hurt, Jul 13 2020

A114697 Expansion of (1+x+x^2)/((1-x^2)(1-2x-x^2)); a Pellian-related sequence.

Original entry on oeis.org

1, 3, 9, 22, 55, 133, 323, 780, 1885, 4551, 10989, 26530, 64051, 154633, 373319, 901272, 2175865, 5253003, 12681873, 30616750, 73915375, 178447501, 430810379, 1040068260, 2510946901, 6061962063, 14634871029, 35331704122, 85298279275, 205928262673

Offset: 0

Creighton Dement, Feb 18 2006

Generating floretion: (- .5'j + .5'k - .5j' + .5k' + 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki')*('i + 'j + i').

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, A116699.

Table[(3*LucasL[n, 2] +10*Fibonacci[n, 2] -3 +(-1)^n)/4, {n,0,30}] (* G. C. Greubel, May 24 2021 *)

Vec((1+x+x^2)/((1-x^2)*(1-2*x-x^2)) + O(x^40)) \\ Colin Barker, Jun 24 2015

[(4*lucas_number1(n+2,2,-1) -2*lucas_number1(n+1,2,-1) -3 +(-1)^n)/4 for n in (0..30)] # G. C. Greubel, May 24 2021

a(n+2) - 2*a(n+1) + a(n) = A111955(n+2).
G.f.: (1+x+x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
From Raphie Frank, Oct 01 2012: (Start)
a(2*n) = A216134(2*n+1).
a(2*n+1) = A006452(2*n+3)-1.
Lim_{n->infinity} a(n+1)/a(n) = A014176. (End)
a(n) = (2*A078343(n+2) - A010694(n))/4. - R. J. Mathar, Oct 02 2012
From Colin Barker, May 26 2016: (Start)
a(n) = ( 2*(-3 +(-1)^n) + (6-5*sqrt(2))*(1-sqrt(2))^n + (1+sqrt(2))^n*(6+5*sqrt(2)) )/8.
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n>3. (End)
a(n) = (3*A002203(n) + 10*A000129(n) - 3 + (-1)^n)/4. - G. C. Greubel, May 24 2021

A129744 a(n) = -(u^n-1)*(v^n-1) with u = 1+sqrt(2), v = 1-sqrt(2).

Original entry on oeis.org

2, 4, 14, 32, 82, 196, 478, 1152, 2786, 6724, 16238, 39200, 94642, 228484, 551614, 1331712, 3215042, 7761796, 18738638, 45239072, 109216786, 263672644, 636562078, 1536796800, 3710155682, 8957108164, 21624372014, 52205852192

Offset: 1

N. J. A. Sloane, May 13 2007

Seiichi Manyama, Table of n, a(n) for n = 1..1000
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (2,2,-2,-1).

Cf. A002203, A000129, A001350, A113224.

u:=1+sqrt(2): v:=1-sqrt(2): a:=n->expand(-(u^n-1)*(v^n-1)): seq(a(n),n=1..33); # Emeric Deutsch, May 13 2007

Table[Simplify[ -((1 + Sqrt[2])^n - 1)*((1 - Sqrt[2])^n - 1)], {n, 1, 30}] (* Stefan Steinerberger, May 15 2007 *)

w = quadgen(8); vector(30, n, -((1+w)^n-1)*((1-w)^n-1)) \\ Michel Marcus, Mar 21 2015

Vec(2*x*(1+x^2)/((x^2+2*x-1)*(-1+x)*(1+x))+O(x^99)) \\ Charles R Greathouse IV, Nov 13 2015

a(2n) = A002203(2n)-2. a(2n+1) = A002203(2n+1). - R. J. Mathar, corrected Dec 05 2007.
G.f.: 2*x*(1+x^2)/((x^2+2*x-1)*(-1+x)*(1+x)).
From Peter Bala, Mar 19 2015: (Start)
a(n) = -det(I - M^n) where I is the 2X2 identity matrix and M = [2, 1; 1, 0]. Cf. A001350.
a(n) = 2*A113224(n-1).
This is divisibility sequence, that is, if n | m then a(n) | a(m).
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 2*Sum_{n >= 1} Pell(n) *x^n. (End)
a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n > 4. - Seiichi Manyama, Jun 07 2018

More terms from Emeric Deutsch and Stefan Steinerberger, May 13 2007

A155464 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 0, a(1) = 51, a(2) = 340.

Original entry on oeis.org

0, 51, 340, 2023, 11832, 69003, 402220, 2344351, 13663920, 79639203, 464171332, 2705388823, 15768161640, 91903581051, 535653324700, 3122016367183, 18196444878432, 106056652903443, 618143472542260, 3602804182350151

Offset: 0

Klaus Brockhaus, Jan 30 2009

lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

First trisection of A118120. Equals 17*A001652.

Cf. A155465, A155466, A156035 (decimal expansion of 3+2*sqrt(2)).

I:=[0,51,340]; [n le 3 select I[n] else 7*Self(n-1) - 7*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Aug 21 2018

LinearRecurrence[{7,-7,1},{0,51,340},30] (* Harvey P. Dale, Jun 10 2013 *)
Table[17*(LucasL[2*n+1,2] - 2)/4, {n, 0, 50}] (* G. C. Greubel, Aug 21 2018 *)

{m=20; v=concat([0, 51, 340], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}

a(n) = 6*a(n-1) - a(n-2) + 34 for n > 1; a(0) = 0, a(1) = 51.
a(n) = ((1+sqrt(2))*(3+2*sqrt(2))^n + (1-sqrt(2))*(3-2*sqrt(2))^n -2)*(17/4).
G.f.: 17*x*(3-x)/((1-x)*(1-6*x+x^2)).
a(n) = 17*(A002203(2*n+1) - 2)/4. - G. C. Greubel, Aug 21 2018

Comment and recursion formula added, cross-references edited by Klaus Brockhaus, Sep 23 2009

A155465 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 7, a(1) = 88, a(2) = 555.

Original entry on oeis.org

7, 88, 555, 3276, 19135, 111568, 650307, 3790308, 22091575, 128759176, 750463515, 4374021948, 25493668207, 148587987328, 866034255795, 5047617547476, 29419671029095, 171470408627128, 999402780733707, 5824946275775148

Offset: 0

Klaus Brockhaus, Jan 30 2009

lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Second trisection of A118120. Cf. A001652.

Cf. A155464, A155466, A156035 (decimal expansion of 3+2*sqrt(2)).

I:=[7,88,555]; [n le 3 select I[n] else 7*Self(n-1) - 7*Self(n-2) + Self(n-3): n in [1..50]]; // G. C. Greubel, Aug 21 2018

LinearRecurrence[{7,-7,1},{7,88,555},30] (* Harvey P. Dale, Apr 29 2012 *)
Table[(3*LucasL[2*n+3,2] + 10*LucasL[2*n+1,2] - 34)/4, {n, 0, 50}] (* G. C. Greubel, Aug 21 2018 *)

{m=20; v=concat([7, 88, 555], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}

a(n) = 6*a(n-1) - a(n-2) + 34 for n > 1; a(0) = 7, a(1) = 88.
a(n) = ((31+25*sqrt(2))*(3+2*sqrt(2))^n + (31-25*sqrt(2))*(3-2*sqrt(2))^n - 34)/4.
G.f.: (7+39*x-12*x^2)/((1-x)*(1-6*x+x^2)).
a(n) = (3*A002203(2*n+3) + 10*A002203(2*n+1) - 34)/4. - G. C. Greubel, Aug 21 2018

Comment and recursion formula added, cross-references edited by Klaus Brockhaus, Sep 23 2009

A155466 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 28, a(1) = 207, a(2) = 1248.

Original entry on oeis.org

28, 207, 1248, 7315, 42676, 248775, 1450008, 8451307, 49257868, 287095935, 1673317776, 9752810755, 56843546788, 331308470007, 1931007273288, 11254735169755, 65597403745276, 382329687301935, 2228380720066368

Offset: 0

Klaus Brockhaus, Jan 30 2009

lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Third trisection of A118120. Cf. A001652.

Cf. A155464, A155465, A156035 (decimal expansion of 3+2*sqrt(2)).

I:=[28, 207, 1248]; [n le 3 select I[n] else 7*Self(n-1) - 7*Self(n-2) + Self(n-3): n in [1..50]]; // G. C. Greubel, Aug 21 2018

Table[(10*LucasL[2*n+3,2] + 3*LucasL[2*n+1, 2] -34)/4, {n, 0, 50}] (* or *) LinearRecurrence[{7,-7,1}, {28, 207, 1248}, 50] (* G. C. Greubel, Aug 21 2018 *)

{m=19; v=concat([28, 207, 1248], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}

a(n) = 6*a(n-1) - a(n-2) + 34 for n > 1; a(0) = 28, a(1) = 207.
a(n) = ((73+53*sqrt(2))*(3+2*sqrt(2))^n + (73-53*sqrt(2))*(3-2*sqrt(2))^n - 34)/4.
G.f.: (28+11*x-5*x^2)/((1-x)*(1-6*x+x^2)).
a(n) = (10*A002203(2*n+3) + 3*A002203(2*n+1) - 34)/4. - G. C. Greubel, Aug 21 2018

Comment and recursion formula added, cross-references edited by Klaus Brockhaus, Sep 23 2009

A204271 a(n) = sigma(n)*Pell(n), where sigma(n) = A000203(n), the sum of divisors of n.

Original entry on oeis.org

1, 6, 20, 84, 174, 840, 1352, 6120, 12805, 42804, 68892, 388080, 468454, 1938768, 4680600, 14595792, 20460402, 107024190, 132502180, 671765976, 1235646880, 3356004888, 5401408344, 32600383200, 40663881751, 133006270404, 305814801800

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

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

			G.f.: A(x) = x + 6*x^2 + 20*x^3 + 84*x^4 + 174*x^5 + 840*x^6 + 1352*x^7 +...
where A(x) = x/(1-2*x-x^2) + 2*2*x^2/(1-6*x^2+x^4) + 3*5*x^3/(1-14*x^3-x^6) + 4*12*x^4/(1-34*x^4+x^8) + 5*29*x^5/(1-82*x^5-x^10) + 6*70*x^6/(1-198*x^6+x^12) +...+ n*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. A000203 (sigma), A000129, A203848, A204270, A204272, A204273, A204274, A204275, A002203, A000045.

Table[DivisorSigma[1, 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)*Pell(n)}
```

{a(n)=polcoeff(sum(m=1,n,m*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*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

A204272 a(n) = sigma_2(n)*Pell(n), where sigma_2(n) = A001157(n), the sum of squares of divisors of n.

Original entry on oeis.org

1, 10, 50, 252, 754, 3500, 8450, 34680, 89635, 309140, 700402, 2910600, 5688370, 20195500, 50706500, 160553712, 329639810, 1248615550, 2398289458, 8732957688, 19306982500, 56865638380, 119281100930, 461838762000, 853941516771

Offset: 1

Paul D. Hanna, Jan 14 2012

nonn

Compare g.f. to the Lambert series identity: Sum_{n>=1} n^2*x^n/(1-x^n) = Sum_{n>=1} sigma_2(n)*x^n.

			G.f.: A(x) = x + 10*x^2 + 50*x^3 + 252*x^4 + 754*x^5 + 3500*x^6 +...
where A(x) = x/(1-2*x-x^2) + 2^2*2*x^2/(1-6*x^2+x^4) + 3^2*5*x^3/(1-14*x^3-x^6) + 4^2*12*x^4/(1-34*x^4+x^8) + 5^2*29*x^5/(1-82*x^5-x^10) + 6^2*70*x^6/(1-198*x^6+x^12) +...+ n^2*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) +...

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A203849, A204270, A204271, A204273, A204274, A204275, A001157 (sigma_2), A002203, A000045.

With[{nn=30},Times@@@Thread[{Rest[LinearRecurrence[{2,1},{0,1},nn+1]], DivisorSigma[ 2,Range[nn]]}]] (* Harvey P. Dale, Oct 21 2015 *)

/* 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,2)*Pell(n)}
```

{a(n)=polcoeff(sum(m=1,n,m^2*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^2*Pell(n)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) = Sum_{n>=1} sigma_2(n)*Pell(n)*x^n, where Pell(n) = A000129(n) and A002203 is the companion Pell numbers.

cp's OEIS Frontend

A209446 a(n) = Pell(n)*A004016(n) for n >= 1, with a(0)=1, where A004016(n) is the number of integer solutions (x,y) to x^2 + x*y + y^2 = n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A033539 a(0)=1, a(1)=1, a(2)=1, a(n) = 2*a(n-1) + a(n-2) + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Erlang

Forth

GAP

Haskell

Magma

Maple

Mathematica

PARI

PARI

Prolog

Python

Sage

Formula

A100227 Main diagonal of triangle A100226.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Maxima

PARI

PARI

Formula

A114697 Expansion of (1+x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A129744 a(n) = -(u^n-1)*(v^n-1) with u = 1+sqrt(2), v = 1-sqrt(2).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A155464 a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 2; a(0) = 0, a(1) = 51, a(2) = 340.

Views

Author

Keywords

Comments

Links

Crossrefs

A209446 a(n) = Pell(n)A004016(n) for n >= 1, with a(0)=1, where A004016(n) is the number of integer solutions (x,y) to x^2 + xy + y^2 = n.

A114697 Expansion of (1+x+x^2)/((1-x^2)(1-2x-x^2)); a Pellian-related sequence.

A155464 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 0, a(1) = 51, a(2) = 340.

A155465 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 7, a(1) = 88, a(2) = 555.

A155466 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 28, a(1) = 207, a(2) = 1248.