A205964
a(n) = Fibonacci(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, 112, 896, 3408, 10080, 25088, 71552, 195888, 411808, 776160, 1896768, 4580352, 8194144, 14525056, 34433280, 73890768, 125562528, 219081856, 458906560, 968315040, 1686909952, 2642197824, 5579174016, 12110579712, 18907500400, 29884043168, 64236542720
Offset: 0
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 + 1*112*x^2 + 2*448*x^3 + 3*1136*x^4 + 5*2016*x^5 + 8*3136*x^6 + 13*5504*x^7 + 21*9328*x^8 +...+ Fibonacci(n)*A000143(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 16*( 1*1*x/(1+x-x^2) + 1*8*x^2/(1-3*x^2+x^4) + 2*27*x^3/(1+4*x^3-x^6) + 3*64*x^4/(1-7*x^4+x^8) + 5*125*x^5/(1+11*x^5-x^10) + 8*216*x^6/(1-18*x^6+x^12) + 13*343*x^7/(1+29*x^7-x^14) +...).
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Join[{1}, Table[Fibonacci[n]*SquaresR[8, n], {n,1,30}]] (* G. C. Greubel, Mar 05 2017 *)
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{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1+16*sum(m=1,n,fibonacci(m)*m^3*x^m/(1-Lucas(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
for(n=0,31,print1(a(n),", "))
A209443
a(n) = Pell(n)*A000118(n) for n>=1 with a(0)=1, where A000118(n) is the number of ways of writing n as a sum of 4 squares.
Original entry on oeis.org
1, 8, 48, 160, 288, 1392, 6720, 10816, 9792, 102440, 342432, 551136, 1330560, 3747632, 15510144, 37444800, 11299968, 163683216, 856193520, 1060017440, 2303197632, 9885175040, 26848039104, 43211266752, 52160613120, 325311054008, 1064050163232, 2446518414400
Offset: 0
G.f.: A(x) = 1 + 8*x + 48*x^2 + 160*x^3 + 288*x^4 + 1392*x^5 + 6720*x^6 +...
where A(x) = 1 + 1*8*x + 2*24*x^2 + 5*32*x^3 + 12*24*x^4 + 29*48*x^5 + 70*96*x^6 + 169*64*x^7 + 408*24*x^8 +...+ Pell(n)*A000118(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 8*( 1*1*x/(1-2*x-x^2) + 2*2*x^2/(1+6*x^2+x^4) + 5*3*x^3/(1-14*x^3-x^6) + 12*4*x^4/(1+34*x^4+x^8) + 29*5*x^5/(1-82*x^5-x^10) + 70*6*x^6/(1+198*x^6+x^12) + 169*7*x^7/(1-478*x^7-x^14) +...).
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A000118[n_]:= If[n < 1, Boole[n == 0], 8*Sum[If[Mod[d, 4] > 0, d, 0], {d, Divisors@n}]]; Join[{1}, Table[Fibonacci[n, 2]*A000118[n], {n, 1, 50}]] (* G. C. Greubel, Jan 02 2018 *)
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{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+8*sum(m=1,n,Pell(m)*m*x^m/(1+A002203(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
for(n=0,30,print1(a(n),", "))
A209445
a(n) = Pell(n)*A001227(n) for n >= 1, where A001227(n) is the number of odd divisors of n.
Original entry on oeis.org
1, 2, 10, 12, 58, 140, 338, 408, 2955, 4756, 11482, 27720, 66922, 161564, 780100, 470832, 2273378, 8232630, 13250218, 31988856, 154455860, 186444716, 450117362, 1086679440, 3935214363, 6333631924, 30581480180, 36915112104, 89120964298, 430314081400, 519435045698
Offset: 1
G.f.: A(x) = x + 2*x^2 + 10*x^3 + 12*x^4 + 58*x^5 + 140*x^6 + 338*x^7 + ...
where A(x) = 1*1*x + 2*1*x^2 + 5*2*x^3 + 12*1*x^4 + 29*2*x^5 + 70*2*x^6 + 169*2*x^7 + 408*1*x^8 + ... + Pell(n)*A001227(n)*x^n + ...
The g.f. is also given by the identity:
A(x) = 1*x/(1-2*x-x^2) + 5*x^3/(1-14*x^3-x^6) + 29*x^5/(1-82*x^5-x^10) + 169*x^7/(1-478*x^7-x^14) + 985*x^9/(1-2786*x^9-x^18) + 5741*x^11/(1-16238*x^11-x^22) + ...
which involves odd-indexed Pell and A002203 numbers.
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A001227[n_]:= Sum[Mod[d, 2], {d, Divisors[n]}]; Table[Fibonacci[n, 2]*A001227[n], {n, 1, 1000}] (* G. C. Greubel, Jan 02 2018 *)
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{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=1,n,Pell(2*m-1)*x^(2*m-1)/(1-A002203(2*m-1)*x^(2*m-1)-x^(4*m-2)+x*O(x^n))),n)}
for(n=1,40,print1(a(n),", "))
Showing 1-3 of 3 results.
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