A203847
a(n) = tau(n)*Fibonacci(n), where tau(n) = A000005(n), the number of divisors of n.
Original entry on oeis.org
1, 2, 4, 9, 10, 32, 26, 84, 102, 220, 178, 864, 466, 1508, 2440, 4935, 3194, 15504, 8362, 40590, 43784, 70844, 57314, 370944, 225075, 485572, 785672, 1906866, 1028458, 6656320, 2692538, 13069854, 14098312, 22811548, 36909860, 134373168, 48315634, 156352676, 252983944
Offset: 1
G.f.: A(x) = x + 2*x^2 + 4*x^3 + 9*x^4 + 10*x^5 + 32*x^6 + 26*x^7 +...
where A(x) = x/(1-x-x^2) + x^2/(1-3*x^2+x^4) + 2*x^3/(1-4*x^3-x^6) + 3*x^4/(1-7*x^4+x^8) + 5*x^5/(1-11*x^5-x^10) + 8*x^6/(1-18*x^6+x^12) +...+ Fibonacci(n)*x^n/(1 - Lucas(n)*x^n + (-1)^n*x^(2*n)) +...
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Table[DivisorSigma[0, n]*Fibonacci[n], {n, 50}] (* G. C. Greubel, Jul 17 2018 *)
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{a(n)=sigma(n,0)*fibonacci(n)}
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{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,fibonacci(m)*x^m/(1-Lucas(m)*x^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
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a(n) = numdiv(n)*fibonacci(n); \\ Michel Marcus, Jul 18 2018
A205963
a(n) = Fibonacci(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, 24, 64, 72, 240, 768, 832, 504, 3536, 7920, 8544, 13824, 26096, 72384, 117120, 23688, 229968, 806208, 668960, 974160, 2802176, 5100768, 5502144, 4451328, 18606200, 40788048, 62853760, 61019712, 123414960, 479255040, 344644864, 52279416, 1353437952, 2463647184
Offset: 0
G.f.: A(x) = 1 + 8*x + 24*x^2 + 64*x^3 + 72*x^4 + 240*x^5 + 768*x^6 +...
where A(x) = 1 + 1*8*x + 1*24*x^2 + 2*32*x^3 + 3*24*x^4 + 5*48*x^5 + 8*96*x^6 + 13*64*x^7 + 21*24*x^8 +...+ Fibonacci(n)*A000118(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1 + 8*( 1*1*x/(1-x-x^2) + 1*2*x^2/(1+3*x^2+x^4) + 2*3*x^3/(1-4*x^3-x^6) + 3*4*x^4/(1+7*x^4+x^8) + 5*5*x^5/(1-11*x^5-x^10) + 8*6*x^6/(1+18*x^6+x^12) + 13*7*x^7/(1-29*x^7-x^14) +...).
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Join[{1}, Table[Fibonacci[n]*SquaresR[4, n], {n,1,50}]] (* G. C. Greubel, Mar 09 2017 *)
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{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(1+8*sum(m=1,n,fibonacci(m)*m*x^m/(1+Lucas(m)*(-x)^m+(-1)^m*x^(2*m)+x*O(x^n))),n)}
for(n=0,31,print1(a(n),", "))
A205965
a(n) = Fibonacci(n)*A001227(n) for n>=1, where A001227(n) is the number of odd divisors of n.
Original entry on oeis.org
1, 1, 4, 3, 10, 16, 26, 21, 102, 110, 178, 288, 466, 754, 2440, 987, 3194, 7752, 8362, 13530, 43784, 35422, 57314, 92736, 225075, 242786, 785672, 635622, 1028458, 3328160, 2692538, 2178309, 14098312, 11405774, 36909860, 44791056, 48315634, 78176338, 252983944
Offset: 1
G.f.: A(x) = x + x^2 + 4*x^3 + 3*x^4 + 10*x^5 + 16*x^6 + 26*x^7 + 21*x^8 +...
where A(x) = 1*1*x + 1*1*x^2 + 2*2*x^3 + 3*1*x^4 + 5*2*x^5 + 8*2*x^6 + 13*2*x^7 + 21*1*x^8 +...+ Fibonacci(n)*A001227(n)*x^n +...
The g.f. is also given by the identity:
A(x) = 1*x/(1-x-x^2) + 2*x^3/(1-4*x^3-x^6) + 5*x^5/(1-11*x^5-x^10) + 13*x^7/(1-29*x^7-x^14) + 34*x^9/(1-76*x^9-x^18) + 89*x^11/(1-199*x^11-x^22) +...
which involves odd-indexed Fibonacci and Lucas numbers.
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A001227[n_]:= DivisorSum[n, Mod[#, 2] &]; Table[A001227[n]*Fibonacci[n], {n, 1, 50}] (* G. C. Greubel, Jul 17 2018 *)
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{Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
{a(n)=polcoeff(sum(m=1,n,fibonacci(2*m-1)*x^(2*m-1)/(1-Lucas(2*m-1)*x^(2*m-1)-x^(4*m-2)+x*O(x^n))),n)}
for(n=1,40,print1(a(n),", "))
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a(n) = fibonacci(n)*sumdiv(n, d, d%2); \\ Michel Marcus, Jul 18 2018
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
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) +...).
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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 *)
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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+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),", "))
Showing 1-4 of 4 results.
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