A002203 -id:A002203 - cp's OEIS frontend

A209448 a(n) = Pell(n)A008655(n) for n>=1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)theta_3(3x)+theta_2(x)theta_2(3*x))^4.

1, 24, 432, 4440, 21024, 87696, 559440, 1395264, 5728320, 23852760, 64719648, 183528288, 898460640, 1765134672, 6002425728, 21820957200, 52895150208, 134056553904, 598084104240, 1090757945760, 3530801969856, 11795485116480, 26821191064896, 65724336729792

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to the Lambert series of A008655:

1 + Sum_{n>=1} 24*n^3*x^n/(1-x^n) + 8*(3*n)^3*x^(3*n)/(1-x^(3*n)).

			G.f.: A(x) = 1 + 24*x + 432*x^2 + 4440*x^3 + 21024*x^4 + 87696*x^5 +...
where A(x) = 1 + 1*24*x + 2*216*x^2 + 5*888*x^3 + 12*1752*x^4 + 29*3024*x^5 +...+ Pell(n)*A008655(n)*x^n +...

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

Cf. A008655, A205968, A209447, A209449, A204270, A000129 (Pell), A002203.

A008655[n_]:= SeriesCoefficient[((EllipticTheta[3, 0, q]^3 + EllipticTheta[3, Pi/3, q]^3 + EllipticTheta[3, 2 Pi/3, q]^3)^4/(3* EllipticTheta[3, 0, q^3])^4), {q, 0, n}]; b:= Table[A008655[n], {n, 0, 102}][[1 ;; ;; 2]]; Join[{1}, Table[Fibonacci[n, 2]*b[[n + 1]], {n, 1, 50}]] (* G. C. Greubel, Jan 26 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 + sum(m=1,n, 24*Pell(m)*m^3*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n)) + 8*Pell(3*m)*(3*m)^3*x^(3*m)/(1-A002203(3*m)*x^(3*m)+(-1)^m*x^(6*m) +x*O(x^n))  ),n)}
for(n=0,40,print1(a(n),", "))

G.f.: 1 + Sum_{n>=1} 24*Pell(n)*n^3*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)) + 8*Pell(3*n)*(3*n)^3*x^(3*n)/(1 - A002203(3*n)*x^(3*n) + (-1)^n*x^(6*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A209450 a(n) = Pell(n)*A132973(n) for n>=1, with a(0)=1, where A132973 lists the coefficients in psi(-q)^3/psi(-q^3) and where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 6, -15, 36, 0, 210, -1014, 1224, -2955, 0, 0, 41580, -200766, 484692, 0, 1412496, 0, 8232630, -39750654, 0, -231683790, 0, 0, 1630019160, -3935214363, 19000895772, -22936110135, 110745336312, 0, 0, -1558305137094, 1881040698144, 0, 0, 0, 63900011068740

Offset: 0

Paul D. Hanna, Mar 10 2012

sign

Compare g.f. to the Lambert series of A132973: 1 - 3*Sum_{n>=0} x^(6*n+1)/(1+x^(6*n+1)) - x^(6*n+5)/(1+x^(6*n+5)).

			G.f.: A(x) = 1 - 3*x + 6*x^2 - 15*x^3 + 36*x^4 + 210*x^6 - 1014*x^7 +...
where A(x) = 1 - 1*3*x + 2*3*x^2 - 5*3*x^3 + 12*3*x^4 + 70*3*x^6 - 169*6*x^7 + 408*3*x^8 +...+ Pell(n)*A132973(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 - 3*( 1*x/(1+2*x-x^2) - 29*x^5/(1+82*x^5-x^10) + 169*x^7/(1+478*x^7-x^14) - 5741*x^11/(1+16238*x^11-x^22) + 33461*x^13/(1+94642*x^13-x^26) - 1136689*x^17/(1+3215042*x^17-x^34) +...).

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

Cf. A132973, A205970, A209449, A209451, A204270, A000129 (Pell), A002203.

A132973[n_]:= SeriesCoefficient[EllipticTheta[2, Pi/4, q^(1/2)]^3/EllipticTheta[2, Pi/4, q^(3/2)]/2, {q, 0, n}]; Join[{1}, Table[ Fibonacci[n, 2]*A132973[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 - 3*sum(m=0,n, Pell(6*m+1)*x^(6*m+1)/(1+A002203(6*m+1)*x^(6*m+1)-x^(12*m+2) +x*O(x^n)) - Pell(6*m+5)*x^(6*m+5)/(1+A002203(6*m+5)*x^(6*m+5)-x^(12*m+10) +x*O(x^n)) ),n)}
for(n=0,61,print1(a(n),", "))

G.f.: 1 - 3*Sum_{n>=0} Pell(6*n+1)*x^(6*n+1)/(1+A002203(6*n+1)*x^(6*n+1)-x^(12*n+2)) - Pell(6*n+5)*x^(6*n+5)/(1+A002203(6*n+5)*x^(6*n+5)-x^(12*n+10)), where A002203(n) = Pell(n-1) + Pell(n+1).

A209451 a(n) = Pell(n)A034896(n) for n >= 1, with a(0)=1, where A034896 lists the number of solutions to a^2 + b^2 + 3c^2 + 3*d^2 = n.

Original entry on oeis.org

1, 4, 8, 20, 240, 696, 280, 5408, 21216, 3940, 57072, 275568, 277200, 1873816, 2585024, 4680600, 54616512, 81841608, 10976840, 530008720, 1919331360, 1235646880, 4474673184, 21605633376, 28253665440, 162655527004, 177341693872, 30581480180, 2953208968320

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to the Lambert series of A034896:
1 + 4*Sum_{n>=1} Chi(n,3)*n*x^n/(1 - (-x)^n).
Here Chi(n,3) = principal Dirichlet character modulo 3.

			G.f.: A(x) = 1 + 4*x + 8*x^2 + 20*x^3 + 240*x^4 + 696*x^5 + 280*x^6 + ...
where A(x) = 1 + 1*4*x + 2*4*x^2 + 5*4*x^3 + 12*20*x^4 + 29*24*x^5 + 70*4*x^6 + ... + Pell(n)*A034896(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1 + 4*( 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. A034896, A205971, A205884, A209447, A209450, A209452, A204270, A000129 (Pell), A002203.

A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A034896[n], {n, 1, 50}]] (* G. C. Greubel, Dec 24 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 + 4*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=0,61,print1(a(n),", "))

G.f.: 1 + 4*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).

A209452 a(n) = Pell(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 24, -30, -72, 0, 840, -2028, 4896, -5910, 0, 0, -83160, -401532, 1938768, 0, -2824992, 0, 32930520, -79501308, 0, -463367580, 0, 0, 6520076640, -7870428726, 76003583088, -45872220270, -221490672624, 0, 0, -3116610274188, 7524162792576, 0, 0, 0, -127800022137480

Offset: 0

Paul D. Hanna, Mar 10 2012

sign

Compare the g.f. to the Lambert series of A122859: 1 - 6*Sum_{n>=1} Kronecker(n,3)*x^n/(1+x^n).

			G.f.: A(x) = 1 - 6*x + 24*x^2 - 30*x^3 - 72*x^4 + 840*x^6 - 2028*x^7 + ...
where A(x) = 1 - 1*6*x + 2*12*x^2 - 5*6*x^3 - 12*6*x^4 + 70*12*x^6 - 169*12*x^7 + 408*12*x^8 - 985*6*x^9 + ... + Pell(n)*A122859(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) - 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. A122859, A205972, A209451, A209453, A209446, A209449, A204270, A000129 (Pell), A002203.

A122859[n_]:= SeriesCoefficient[EllipticTheta[4, 0, q]^3/EllipticTheta[4, 0, q^3], {q, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A122859[n], {n, 1, 50}]] (* 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 - 6*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=0,40,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).

A209453 a(n) = Pell(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.

Original entry on oeis.org

1, -9, 54, -45, -1404, 6264, 1890, -76050, 187272, -8865, -1540944, 6200280, -1621620, -51195330, 109055700, 42125400, -868685040, 2946297888, 74093670, -21584605122, 44912353824, -17376284250, -302040439920, 1069478852112, 249392931480, -7095191496489

Offset: 0

Paul D. Hanna, Mar 10 2012

sign

Compare the g.f. to the Lambert series of A109041:

1 - 9*Sum_{n>=1} Kronecker(n,3)*n^2*x^n/(1-x^n).

			G.f.: A(x) = 1 - 9*x + 54*x^2 - 45*x^3 - 1404*x^4 + 6264*x^5 + 1890*x^6 +...
where A(x) = 1 - 1*9*x + 2*27*x^2 - 5*9*x^3 - 12*117*x^4 + 29*216*x^5 + 70*27*x^6 - 169*450*x^7 + 408*459*x^8 +...+ Pell(n)*A109041(n)*^n +...
The g.f. is also given by the identity:
A(x) = 1 - 9*( 1*1*x/(1-2*x-x^2) - 2*4*x^2/(1-6*x^2+x^4) + 12*16*x^4/(1-34*x^4+x^8) - 29*25*x^5/(1-82*x^5-x^10) + 169*49*x^7/(1-478*x^7-x^14) - 408*64*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. A109041, A205973, A209452, A209454, A204270, A000129 (Pell), A002203.

A109041[n_]:= If[n < 1, Boole[n == 0], -9 DivisorSum[n, #^2 KroneckerSymbol[-3, #] &]]; Join[{1}, Table[Fibonacci[n, 2]*A109041[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 - 9*sum(m=1,n,Pell(m)*kronecker(m,3)*m^2*x^m/(1-A002203(m)*x^m+(-1)^m*x^(2*m) +x*O(x^n))),n)}
for(n=0,40,print1(a(n),", "))

G.f.: 1 - 9*Sum_{n>=1} Pell(n)*Kronecker(n,3)*n^2*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A209454 a(n) = Pell(n)A033719(n) for n>=1, with a(0)=1, where A033719 lists the coefficients in theta_3(q)theta_3(q^7).

Original entry on oeis.org

1, 2, 0, 0, 24, 0, 0, 338, 1632, 1970, 0, 22964, 0, 0, 0, 0, 2824992, 0, 0, 0, 0, 0, 0, 900234724, 0, 2623476242, 0, 0, 36915112104, 178241928596, 0, 0, 5016108528384, 0, 0, 0, 42600007379160, 205691031143924, 0, 0, 0, 0, 0, 40725785296405556, 98320743200877072, 0, 0, 0, 0

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to 1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-(-x)^n) (the Lambert series of A033719).

			G.f.: A(x) = 1 + 2*x + 24*x^4 + 338*x^7 + 1632*x^8 + 1970*x^9 + 22964*x^11 +...
where A(x) = 1 + 1*2*x + 12*2*x^4 + 169*2*x^7 + 408*4*x^8 + 985*2*x^9 + 5741*4*x^11 + 470832*6*x^16 + 225058681*4*x^23 +...+ Pell(n)*A033719(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) - 5*x^3/(1+14*x^3-x^6) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1+82*x^5-x^10) - 70*x^6/(1-198*x^6+x^12) + 0*169*13*x^7/(1+478*x^7-x^14) +...).
The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].

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

Cf. A033719, A205974, A209453, A209455, A204270, A000129 (Pell), A002203.

A033719[n_]:= SeriesCoefficient[EllipticTheta[3, 0, x] EllipticTheta[3, 0, x^7], {x, 0, n}]; Join[{1}, Table[Fibonacci[n, 2]*A033719[n], {n, 1, 50}]] (* 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 + 2*sum(m=1,n,Pell(m)*kronecker(m,7)*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 + 2*Sum_{n>=1} Pell(n)*Kronecker(n,7)*x^n/(1 - A002203(n)*(-x)^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

A209455 a(n) = Pell(n)*A002652(n) for n>=1, with a(0)=1, where A002652 lists the coefficients in theta series of Kleinian lattice Z[(-1+sqrt(-7))/2].

Original entry on oeis.org

1, 2, 8, 0, 72, 0, 0, 338, 3264, 1970, 0, 22964, 0, 0, 323128, 0, 4708320, 0, 10976840, 0, 0, 0, 745778864, 900234724, 0, 2623476242, 0, 0, 110745336312, 178241928596, 0, 0, 7524162792576, 0, 0, 0, 127800022137480, 205691031143924, 0, 0, 0, 0, 0, 40725785296405556

Offset: 0

Paul D. Hanna, Mar 10 2012

nonn

Compare g.f. to the Lambert series of A002652: 1 + 2*Sum_{n>=1} Kronecker(n,7)*x^n/(1-x^n).

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 72*x^4 + 338*x^7 + 3264*x^8 + 1970*x^9 +...
where A(x) = 1 + 1*2*x + 2*4*x^2 + 12*6*x^4 + 169*2*x^7 + 408*8*x^8 + 985*2*x^9 + 5741*4*x^11 + 80782*4*x^14 + 470832*10*x^16 +...+ Pell(n)*A002652(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) - 5*x^3/(1-14*x^3-x^6) + 12*x^4/(1-34*x^4+x^8) - 29*x^5/(1-82*x^5-x^10) - 70*x^6/(1-198*x^6+x^12) + 0*169*13*x^7/(1+478*x^7-x^14) +...).
The values of the symbol Kronecker(n,7) repeat [1,1,-1,1,-1,-1,0, ...].

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

Cf. A002652, A205975, A209454, A209456, A204270, A000129 (Pell), A002203.

terms = 44; s = 1 + 2 Sum[x^n*Fibonacci[n, 2]*KroneckerSymbol[n, 7]/(1 + (-1)^n*x^(2*n) - x^n*(Fibonacci[n - 1, 2] + Fibonacci[n + 1, 2])), {n, 1, terms}] + O[x]^terms; CoefficientList[s, x] (* Jean-François Alcover, Jul 05 2017 *)
A002652[n_]:= If[n < 1, Boole[n == 0], 2*Sum[KroneckerSymbol[-7, d], {d, Divisors[n]}]]; Join[{1}, Table[Fibonacci[n, 2]*A002652[n], {n,1,50}]] (* G. C. Greubel, Jan 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,7)*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,7)*x^n/(1 - A002203(n)*x^n + (-1)^n*x^(2*n)), where A002203(n) = Pell(n-1) + Pell(n+1).

G.f.: 1 + 2*Sum_{n>=1} F(n,2)*Kronecker(n,7)*x^n/(1 + (-1)^n*x^(2*n)-x^n* (F(n-1,2)+F(n+1,2))), where F is the Fibonacci polynomial. - Jean-François Alcover, Jul 05 2017

A255496 3rd diagonal of triangle in A255494.

Original entry on oeis.org

1, 38, 1106, 26544, 567203, 11179686, 207768576, 3692419776, 63361188037, 1057109514902, 17235551954894, 275697361933728, 4339725043253447, 67384965236252310, 1034147721558836220, 15711425790758327952, 236612932874975360809, 3536182524466029241958, 52494462902614684280330

Offset: 0

N. J. A. Sloane, Mar 06 2015

nonn

G. C. Greubel, Table of n, a(n) for n = 0..850
S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675.
Index entries for linear recurrences with constant coefficients, signature (44,-649,2770,11885,-65240,-215431,-67286,139956,-23560,-2400).

Cf. A000129, A002203, A255494, A255495.

a[n_]:= (12)^(n+4) -(-2)^(n+1) -2^n*LucasL[2*n+9, 2] -5^(n+4)*Fibonacci[n+5, 2] +(1/10)*Fibonacci[n+4, 2]*(Fibonacci[n+4, 2]^2 +(-1)^n);
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Sep 20 2021 *)

def P(n): return lucas_number1(n, 2, -1)
def Q(n): return lucas_number2(n, 2, -1)
def a(n): return (12)^(n+4) - (-2)^(n+1) - 2^n*Q(2*n+9) - 5^(n+4)*P(n+5) + (1/10)*P(n+4)*(P(n+4)^2 + (-1)^n)
[a(n) for n in (0..30)] # G. C. Greubel, Sep 20 2021

From G. C. Greubel, Sep 20 2021: (Start)
a(n) = 12*a(n-1) + P(n+1)*A255495(n), where P(n) = A000129(n).
a(n) = (12)^(n+4) - (-2)^(n+1) - 2^n*Q(2*n+9) - 5^(n+4)*P(n+5) + (1/10)*P(n+4)*(P(n+4)^2 + (-1)^n), where P(n) = A000129(n), Q(n) = A002203(n).
G.f.: (1 -6*x +83*x^2 -228*x^3 -84*x^4 -200*x^5)/((1+2*x)*(1-12*x)*(1 +2*x -x^2)*(1 -10*x -25*x^2)*(1 -12*x +4*x^2)*(1 -14*x -x^2)). (End)

3 more terms. - R. J. Mathar, Jun 14 2015

Terms a(12) onward added by G. C. Greubel, Sep 20 2021

A261330 Euler transform of Pell-Lucas numbers.

Original entry on oeis.org

1, 2, 9, 30, 106, 348, 1153, 3698, 11798, 37034, 115294, 355202, 1086080, 3294912, 9931019, 29745296, 88597104, 262508288, 774073787, 2272321666, 6642701371, 19342768210, 56117550874, 162247236638, 467563212923, 1343273262184, 3847866714452, 10991864363660

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, A261331, A261332, A166861, A261031.

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

G.f.: Product_{k>=1} 1/(1-x^k)^(A002203(k)).

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+sqrt(2))^k + 2/(1 + (1+sqrt(2))^k) - 3)*k) = 0.40371233206538058741995064489690066306587648488344483...

A266504 a(n) = 2*a(n - 2) + a(n - 4) with a(0) = a(1) = 2, a(2) = 1, a(3) = 3.

Original entry on oeis.org

2, 2, 1, 3, 4, 8, 9, 19, 22, 46, 53, 111, 128, 268, 309, 647, 746, 1562, 1801, 3771, 4348, 9104, 10497, 21979, 25342, 53062, 61181, 128103, 147704, 309268, 356589, 746639, 860882, 1802546, 2078353, 4351731, 5017588, 10506008, 12113529, 25363747, 29244646, 61233502

Offset: 0

Raphie Frank, Dec 30 2015

This sequence gives all x in N | 2*x^2 - 7(-1)^x = y^2. The companion sequence to this sequence, giving y values, is A266505.
A266505(n)/a(n) converges to sqrt(2).
Alternatively, 1/4*(3*A002203(floor[n/2]) - A002203(n-(-1)^n)), where A002203 gives the Companion Pell numbers, or, in Lucas sequence notation, V_n(2, -1).
Alternatively, bisection of A266506.
Alternatively, A048654(n -1) and A078343(n + 1) interlaced.
Alternatively, A100525(n-1), A266507(n), A038761(n) and A253811(n) interlaced.
Let b(n) = (a(n) - a(n)(mod 2))/2, that is b(n) = {1, 1, 0, 1, 2, 4, 4, 9, 11, 23, 26, 55, 64, ...}. Then:
A006452(n) = {b(4n+0) U b(4n+1)} gives n in N such that n^2 - 1 is triangular;
A216134(n) = {b(4n+2) U b(4n+3)} gives n in N such that n^2 + n + 1 is triangular (indices of Sophie Germain triangular numbers);
A216162(n) = {b(4n+0) U b(4n+2) U b(4n+1) U b(4n+3)}, sequences A006452 and A216134 interlaced.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1).

Cf. A000129, A001333, A002203, A002965, A006451, A006452, A002965, A038761, A038762, A048654, A048655, A054490, A078343, A098586, A098790, A100525, A101386, A135532, A216134, A216162, A253811, A255236, A266504, A266505, A266507.

I:=[2,2,1,3]; [n le 4 select I[n] else 2*Self(n-2)+Self(n-4): n in [1..70]]; // Vincenzo Librandi, Dec 31 2015

LinearRecurrence[{0, 2, 0, 1}, {2, 2, 1, 3}, 70] (* Vincenzo Librandi, Dec 31 2015 *)
Table[SeriesCoefficient[(1 - x) (2 + 4 x + x^2)/(1 - 2 x^2 - x^4), {x, 0, n}], {n, 0, 41}] (* Michael De Vlieger, Dec 31 2015 *)

Vec((1-x)*(2+4*x+x^2)/(1-2*x^2-x^4) + O(x^50)) \\ Colin Barker, Dec 31 2015

a(n) = 1/sqrt(8)*(+sqrt(2)*(1+sqrt(2))^(floor(n/2)-(-1)^n)*(-1)^n - 3*(1-sqrt(2))^(floor(n/2)-(-1)^n) + sqrt(2)*(1-sqrt(2))^(floor(n/2)-(-1)^n)*(-1)^n + 3*(1+sqrt(2))^(floor(n/2)-(-1)^n)).
a(n) = 1/4*((3*((1+sqrt(2))^floor(n/2)+(1-sqrt(2))^floor(n/2))) - (-1)^n*((1+sqrt(2))^(floor(n/2)-(-1)^n)+(1-sqrt(2))^(floor(n/2)-(-1)^n))).
a(2n) = (+sqrt(2)*(1+sqrt(2))^(n-1) - 3 *(1-sqrt(2))^(n-1) + sqrt(2)*(1-sqrt(2))^(n-1) + 3*(1 + sqrt(2))^(n-1))/sqrt(8) = A048654(n -1).
a(2n) = 1/4*((3*((1+sqrt(2))^n+(1-sqrt(2))^n)) - ((1+sqrt(2))^(n-1)+(1-sqrt(2))^(n-1))) = A048654(n -1).
a(2n + 1) = (-sqrt(2)*(1+sqrt(2))^(n+1) - 3 *(1-sqrt(2))^(n+1) - sqrt(2)*(1-sqrt(2))^(n+1) + 3*(1+sqrt(2))^(n+1))/sqrt(8) = A078343(n + 1).
a(2n + 1) =1/4*((3*((1+sqrt(2))^n+(1-sqrt(2))^n)) + ((1+sqrt(2))^(n+1)+(1-sqrt(2))^(n+1))) =  A078343(n + 1).
a(4n + 0) = 6*a(4n - 4) - a(4n - 8) = A100525(n-1).
a(4n + 1) = 6*a(4n - 3) - a(4n - 7) = A266507(n).
a(4n + 2) = 6*a(4n - 2) - a(4n - 6) = A038761(n).
a(4n + 3) = 6*a(4n - 1) - a(4n - 5) = A253811(n).
sqrt(2*a(n)^2 - 7(-1)^a(n))*sgn(2*n - 1) = A266505(n).
(a(2n + 1) + a(2n))/2 = A002203(n), where A002203 gives the companion Pell numbers.
(a(2n + 1) - a(2n))/2 = A000129(n), where A000129 gives the Pell numbers.
(a(2n+2) + a(2n+1))*2 = A002203(n+2)
(a(2n+2) - a(2n+1))*2 = A002203(n-1).
G.f.: (1-x)*(2+4*x+x^2) / (1-2*x^2-x^4). - Colin Barker, Dec 31 2015

cp's OEIS Frontend

A209448 a(n) = Pell(n)*A008655(n) for n>=1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)*theta_3(3*x)+theta_2(x)*theta_2(3*x))^4.

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A209450 a(n) = Pell(n)*A132973(n) for n>=1, with a(0)=1, where A132973 lists the coefficients in psi(-q)^3/psi(-q^3) and where psi() is a Ramanujan theta function.

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A209451 a(n) = Pell(n)*A034896(n) for n >= 1, with a(0)=1, where A034896 lists the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.

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A209452 a(n) = Pell(n)*A122859(n) for n>=1, with a(0)=1, where A122859 lists the coefficients in phi(-q)^3/phi(-q^3) and phi() is a Ramanujan theta function.

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A209453 a(n) = Pell(n)*A109041(n) for n>=1, with a(0)=1, where A109041 lists the coefficients in eta(q)^9/eta(q^3)^3.

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A209454 a(n) = Pell(n)*A033719(n) for n>=1, with a(0)=1, where A033719 lists the coefficients in theta_3(q)*theta_3(q^7).

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A209455 a(n) = Pell(n)*A002652(n) for n>=1, with a(0)=1, where A002652 lists the coefficients in theta series of Kleinian lattice Z[(-1+sqrt(-7))/2].

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A209448 a(n) = Pell(n)A008655(n) for n>=1, with a(0)=1, where A008655 lists the coefficients in (theta_3(x)theta_3(3x)+theta_2(x)theta_2(3*x))^4.

A209451 a(n) = Pell(n)A034896(n) for n >= 1, with a(0)=1, where A034896 lists the number of solutions to a^2 + b^2 + 3c^2 + 3*d^2 = n.

A209454 a(n) = Pell(n)A033719(n) for n>=1, with a(0)=1, where A033719 lists the coefficients in theta_3(q)theta_3(q^7).