Original entry on oeis.org
101, 10010, 1000100, 100001000, 10000010000, 1000000100000, 100000001000000, 10000000010000000, 1000000000100000000, 100000000001000000000
Offset: 1
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Table[FromDigits@ IntegerDigits[2^(2 n) + 2^(n - 1), 2], {n, 12}] (* or *)
Rest@ CoefficientList[Series[x (101 - 1100 x)/((1 - 10 x) (1 - 100 x)), {x, 0, 12}], x] (* Michael De Vlieger, Jun 21 2016 *)
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x='x+O('x^50); Vec(x*(101 - 1100*x)/((1 - 10*x)*(1 - 100*x))) \\ G. C. Greubel, Sep 15 2017
A001445
a(n) = (2^n + 2^[ n/2 ] )/2.
Original entry on oeis.org
3, 5, 10, 18, 36, 68, 136, 264, 528, 1040, 2080, 4128, 8256, 16448, 32896, 65664, 131328, 262400, 524800, 1049088, 2098176, 4195328, 8390656, 16779264, 33558528, 67112960, 134225920, 268443648
Offset: 2
G.f. = 3*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 36*x^6 + 68*x^7 + 136*x^8 + ...
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f := n->(2^n+2^floor(n/2))/2;
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Table[(2^n + 2^(Floor[n/2]))/2, {n, 2, 50}] (* G. C. Greubel, Sep 08 2017 *)
LinearRecurrence[{2,2,-4},{3,5,10},30] (* Harvey P. Dale, Sep 12 2021 *)
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for(n=2,50, print1((2^n + 2^(n\2))/2, ", ")) \\ G. C. Greubel, Sep 08 2017
A203477
a(n) = Product_{0 <= i < j <= n-1} (2^i + 2^j).
Original entry on oeis.org
1, 3, 90, 97200, 14276736000, 1107198567383040000, 178601637561927097909248000000, 237856509917156074017606774172522905600000000, 10420480393274493153643458442091600404477248333907230720000000000
Offset: 1
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[(&*[(&*[2^j + 2^k: k in [0..j]])/2^(j+1): j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 28 2023
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a:= n-> mul(mul(2^i+2^j, i=0..j-1), j=1..n-1):
seq(a(n), n=1..10); # Alois P. Heinz, Jul 23 2017
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(* First program *)
f[j_]:= 2^(j-1); z = 13;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}] (* A203477 *)
Table[v[n+1]/v[n], {n,z-1}] (* A203478 *)
Table[v[n]*v[n+2]/(2*v[n+1]^2), {n,22}] (* A164051 *)
(* Second program *)
Table[Product[(2^j^2)*QPochhammer[-1/2^j,2,j], {j,0,n-1}], {n,20}] (* G. C. Greubel, Aug 28 2023 *)
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a(n)=prod(i=0,n-2,prod(j=i+1,n-1,2^i+2^j)) \\ Charles R Greathouse IV, Feb 16 2021
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[product(product(2^j + 2^k for k in range(j)) for j in range(n)) for n in range(1,21)] # G. C. Greubel, Aug 28 2023
A203478
a(n) = v(n+1)/v(n), where v = A203477.
Original entry on oeis.org
3, 30, 1080, 146880, 77552640, 161309491200, 1331771159347200, 43809944057885491200, 5753472333233985788313600, 3019422280481195741706977280000, 6335279362770913356551778761441280000
Offset: 1
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[(&*[2^j + 2^n: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 28 2023
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(* First program *)
f[j_]:= 2^(j-1); z = 13;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}] (* A203477 *)
Table[v[n+1]/v[n], {n,z-1}] (* A203478 *)
Table[v[n]*v[n+2]/(2*v[n+1]^2), {n,22}] (* A164051 *)
(* Second program *)
Table[Product[2^j +2^n, {j,0,n-1}], {n,20}] (* G. C. Greubel, Aug 28 2023 *)
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a(n)=prod(i=0,n-1,2^i+2^n) \\ Charles R Greathouse IV, Feb 16 2021
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[product(2^j + 2^n for j in range(n)) for n in range(1,21)] # G. C. Greubel, Aug 28 2023
Showing 1-4 of 4 results.
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