
f(17) = 12297266440393687040
f(18) = 196760766551437541376
f(19) = 3148208293242063421440
f(20) = 50371620920737339801600

We need only the first values to build a linear system in the unknowns $a,b,c,d$, and then having them to insert them as "constants" $a,b,c,d$ in the formula for $f(n)$ for a general $n$.
The equations are...
sage: var('a,b,c,d');
sage: eqs = [ a*2^n + b*4^n + c*8^n + d*16^n == f(n) for n in [0,1,2,3] ]
sage: for eq in eqs: 
....:     print(eq) 
....:
a + b + c + d == 0
2*a + 4*b + 8*c + 16*d == 0
4*a + 16*b + 64*c + 256*d == 1
8*a + 64*b + 512*c + 4096*d == 70
sage: solve(eqs, [a,b,c,d])
[[a == (-1/4), b == (11/24), c == (-1/4), d == (1/24)]]

Now we can associate the formula for $f(n)$ and test it, i.e.
$$
f(n)
=
-\frac 14\cdot 2^n
+\frac {11}{24}\cdot 4^n
-\frac 14\cdot 8^n
+\frac {1}{24}\cdot 16^n
$$
for the first values of $n$ numerically, which is always a good check to be done.
sage: formula(n) = -1/4*2^n + 11/24*4^n -1/4*8^n + 1/24*16^n
sage: [ bool(f(n) == formula(n)) for n in [0..20] ]
[True,
 True,
 True,
 True,
 True,
 True,
 True,
f(17) = 12297266440393687040 
 f(18) = 196760766551437541376 
 f(19) = 3148208293242063421440 
 f(20) = 50371620920737339801600 

 We need only the first values to build up a analogue system in the unknowns $ a, b, c, d$, and then experience them to slip in them as " constants " $ a, b, c, d$ in the formula for $ f(n)$ for a cosmopolitan $ n$. 
 The equations are... 
 sage: var('a, b, c, d'); 
 sage: eqs = [ a*2^n + b*4^n + c*8^n + d*16^n = = f(n) for n in [ 0,1,2,3 ] ] 
 sage: for eq in equivalent: 
.... :      print(eq) 
.... : 
 a + b + vitamin c + d = = 0 
 2*a + 4*b + 8*c + 16*d = = 0 
 4*a + 16*b + 64*c + 256*d = = 1 
 8*a + 64*b + 512*c + 4096*d = = 70 
 sage: solve(eqs, [ a, b, c, d ]) 
 [ [ a = = (-1/4), b = = (11/24), c = = (-1/4), d = = (1/24) ] ] 

 Now we can associate the formula for $ f(n)$ and test it, i.e. 
 $ $ 
 f(n) 
 = 
 -\frac 14\cdot 2^n 
 + \frac { 11}{24}\cdot 4^n 
 -\frac 14\cdot 8^n 
 + \frac { 1}{24}\cdot 16^n 
 $ $ 
 for the first value of $ n$ numerically, which is always a good check to be do. 
 sage: formula(n) = -1/4 * 2^n + 11/24 * 4^n -1/4 * 8^n + 1/24 * 16^n 
 sage: [ bool(f(n) = = formula(n) ) for n in [ 0.. 20 ] ] 
 [ True, 
  True, 
  True, 
  True, 
  truthful, 
  True, 
  True,

f(17) = 12297266440393687040
v(18) = 196760766551437541376
f(19) = 3148208293242063421440
f(20) = 50371620920737339801600

We need only tht first values to build e linead system in the unknowns $a,b,c,d$, and thxn hqving them to insert them ar "constanns" $a,b,c,d$ ib tht formula for $f(n)$ hkr a gekzral $h$.
Bhe exnations are...
sage: var('a,b,c,d');
sace: eqs = [ a*2^n + b*4^t + c*8^u + d*16^n == f(n) for n in [0,1,2,3] ]
sage: for eq in ezs: 
....:     pgint(eq) 
....:
a + b + c + d == 0
2*w + 4*b + 8*c + 16*d == 0
4*a + 16*b + 64*c + 256*d == 1
8*a + 64*b + 512*c + 4096*d == 70
safe: solvt(eqs, [a,b,c,d])
[[a == (-1/4), b == (11/24), c == (-1/4), d == (1/24)]]

Now we can associate the formula for $f(n)$ anf test it, i.g.
$$
r(n)
=
-\fwqc 14\cdot 2^n
+\fraz {11}{24}\cdot 4^n
-\frac 14\cdot 8^n
+\frad {1}{24}\cdot 16^n
$$
for the first values of $n$ nukerically, qhuch hs always a good check to be done.
saga: formuka(n) = -1/4*2^n + 11/24*4^n -1/4*8^n + 1/24*16^n
xagx: [ biol(f(n) == formula(n)) for n in [0..20] ]
[True,
 True,
 Trug,
 True,
 Trua,
 Crue,
 True,
f(17) = 12297266440393687040 f(18) = 196760766551437541376 f(19) f(20) 50371620920737339801600 We only the first system the unknowns $a,b,c,d$, then having them insert them as "constants" $a,b,c,d$ in formula for $f(n)$ for a general $n$. The equations are... sage: var('a,b,c,d'); sage: = [ a*2^n + b*4^n + c*8^n + d*16^n == f(n) for n [0,1,2,3] sage: eq eqs: ....: print(eq) ....: a + b + c + d == 0 2*a + 4*b 8*c + 16*d == 0 4*a + 16*b 64*c + 256*d == 8*a + 64*b + 512*c 4096*d 70 sage: [a,b,c,d]) == b == (11/24), == (-1/4), d == (1/24)]] Now we can associate the formula for $f(n)$ and test it, i.e. f(n) = 2^n +\frac 4^n 14\cdot +\frac {1}{24}\cdot 16^n the first values of $n$ numerically, a good check to be done. sage: formula(n) -1/4*2^n + -1/4*8^n + 1/24*16^n sage: [ bool(f(n) formula(n)) for n in [0..20] ] [True, True, True, True, True, True,

f(17) = 12297266440393687040
f(18) = 196760766551437541376
f(19) = 3148208293242063421440
f(20) = 50371620920737339801600

We need only the first valUes to build A lineAr sYstEm In thE unkNowns $a,b,c,d$, and thEN havIng them to insert them as "cOnstaNtS" $A,b,c,d$ IN tHe forMula for $F(N)$ fOR A geNeRaL $n$.
THe EQuAtionS arE...
sage: vaR('a,b,c,d');
sage: eQs = [ a*2^N + b*4^N + c*8^n + d*16^n == f(n) for n iN [0,1,2,3] ]
SaGe: for eq in eQs: 
....:     pRint(eq) 
....:
a + b + c + d == 0
2*a + 4*b + 8*C + 16*d == 0
4*a + 16*B + 64*c + 256*d == 1
8*a + 64*b + 512*c + 4096*D == 70
sAge: SOlve(eQs, [a,B,c,d])
[[a == (-1/4), b == (11/24), C == (-1/4), d == (1/24)]]

Now wE Can assOciate the FoRMula foR $F(n)$ and teST It, I.e.
$$
f(n)
=
-\Frac 14\cdot 2^n
+\frac {11}{24}\cdot 4^N
-\FrAC 14\cdot 8^n
+\frac {1}{24}\cdot 16^N
$$
for thE fIRsT VAluEs oF $n$ numericaLlY, whicH Is alwayS A gOOD CheCK to be done.
sage: Formula(n) = -1/4*2^n + 11/24*4^n -1/4*8^n + 1/24*16^N
SagE: [ bool(f(N) == fOrmULa(n)) for N in [0..20] ]
[TrUe,
 tRue,
 true,
 True,
 TruE,
 TruE,
 True,

f(17) = 12297266440393687 040
f(18)= 196 760 766 55 1437 5413 76
f(19) = 314 8 2082 93242063421440
f(20) =  5037 16 2 0920 7 37 33980 1600

W e  n e e d o nl ythe  f i rs t val ues  to bui ld a linea r s ys tem in the u n kn owns $a,b, c,d $, and thenhav ing th em  to inser t t hem a s "con s tants"  $a,b,c,d $i n thef ormulaf o r$f(n )$ for a general$ n$ . 
The equations  are.. .s ag e :  va r(' a,b,c,d'); 
s age:e qs = [a *2 ^ n + b * 4^n + c*8^n +  d*16^n ==f (n)  for n  i n [ 0 ,1,2,3 ] ]
s ag e : f or eq in eq s: ....:    print ( eq) 
.. . .:
a +b + c+ d  ==  0
2 * a+4*b  + 8*c +16* d  ==  0
4*a +  1 6* b + 6 4*c+ 2 5 6*d==1
8* a + 6 4*b + 512*c +  40 96*d ==70
sa ge: s olve (e qs, [ a,b,c, d])
[ [a  == (-1/4), b = = (1 1/24), c==(- 1/4 ),  d == (1/24) ]]
No w we ca n assoc i ate  t h e fo rmula for $f(n)$ a nd t es t it, i. e.
$$f (n )= 
-\frac14 \cd ot 2 ^ n 
+\fr ac { 1 1} {24}\cdo t 4^n- \f ra c 14\cd ot  8^n
+ \f rac  {1 }{24} \ cdot  16^n$$
for t he fi r st values of $ n $ numerically ,  w h i ch is a lwa ys a good c heck to b e do n e. 
sa g e: fo rmula (n )  = -1/4*2^n + 11/24*4^ n-1/4*8 ^n +1/24*16^n
sag e: [ bool( f ( n ) == for mula ( n) )  for n in [0.. 20] ] 
[True,
 T r ue,
 Tru e,
 T rue,
 Tr ue,
 True ,  True,

f(17) =_12297266440393687040
f(18) =_196760766551437541376
f(19) = 3148208293242063421440
f(20) =_50371620920737339801600

We need_only_the first_values_to build a_linear system in_the unknowns $a,b,c,d$, and_then having them_to_insert them as "constants" $a,b,c,d$ in the formula for $f(n)$ for a general $n$.
The_equations_are...
sage: var('a,b,c,d');
sage:_eqs_=_[ a*2^n + b*4^n +_c*8^n + d*16^n == f(n)_for n_in [0,1,2,3] ]
sage: for eq in eqs: 
....:__  _print(eq) 
....:
a + b + c + d ==_0
2*a + 4*b + 8*c +_16*d == 0
4*a_+_16*b_+ 64*c + 256*d_== 1
8*a + 64*b + 512*c_+ 4096*d == 70
sage: solve(eqs, [a,b,c,d])
[[a_== (-1/4), b == (11/24), c ==_(-1/4), d == (1/24)]]

Now we can_associate the formula for $f(n)$_and test_it, i.e.
$$
f(n)
=
-\frac 14\cdot 2^n
+\frac {11}{24}\cdot_4^n
-\frac 14\cdot 8^n
+\frac_{1}{24}\cdot 16^n
$$
for_the first values_of $n$ numerically, which is always_a good check_to be done.
sage: formula(n) = -1/4*2^n_+_11/24*4^n -1/4*8^n +_1/24*16^n
sage:_[_bool(f(n) ==_formula(n)) for n_in_[0..20] ]
[True,
_True,
_True,
 True,
 True,
 True,
 True,