A229194 -id:A229194 - cp's OEIS frontend

A229198 Difference between integers nearest to (2^((n-3)/2) + 3^((n-3)/2)) (A229194) and Fibonacci numbers (A000045).

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 8, 19, 42, 89, 183, 366, 718, 1385, 2636, 4961, 9249, 17105, 31416, 57356, 104170, 188331, 339119, 608464, 1088286, 1940994, 3453084, 6129207, 10857097, 19196490, 33884792, 59721438, 105113418, 184774518, 324436647, 569068543, 997205614, 1745923072, 3054338540, 5339361915, 9327547185, 16284517131, 28414038840, 49551994304, 86372825386, 150486363173

Offset: 0

Vladimir Pletser, Sep 15 2013

nonn

The following terms are Fibonacci numbers: a(9) = F(2), a(10)= F(4) , a(11) = F(6), a(14) = F(11); or the algebraic sum of two Fibonacci numbers: a(12) = F(8) - F(3), a(13) = F(10) - F(7), a(14) = F(12) - F(10); or the algebraic sum of three Fibonacci numbers: a(15) = F(12) + F(9) + F(5), a(16) = F(14) - F(6) - F(4), a(18) = F(16) + F(14) + F(8), a(19) = F(18) + F(10) - F(3).

Vladimir Pletser, Table of n, a(n) for n = 0..1000

Cf. A000045, A229194.

with (combinat): seq(round(2^((n-3)/2)+3^((n-3)/2))-fibonacci(n), n=0..50);

a(n) = A229194(n) - A000045(n)

A229255 Integer nearest to (2^(n-1) + 3^(n-1))^(2b(n)) where b(n) = (C1n^(Pi)exp(C2n)cos(C3n+C4) + C5)(C6n^C7 + Pi/2) (see coefficients in comments).

Original entry on oeis.org

4, 25, 168, 1229, 9592, 78488, 664356, 5761311, 50857532, 455110791, 4117706679, 37598394076, 345973354409, 3204537723387, 29847287869987, 279317953220125, 2624541016148480, 24747919106286414, 234089443816438414, 2220530456953251916, 21119025631088169139, 201358809736398135352, 1924434871799161020533, 18434884359943473267194, 176994218822287711757127

Offset: 1

Vladimir Pletser, Sep 17 2013

Coefficients are C1=27829/125000000, C2=-0.591561, C3=441/2500, C4=5, C5=19703973/31250000, C6=5.241804273*10^-3, C7=0.6246728093.
This sequence gives a good approximation of pi(10^n) (A006880); see (A229256).
To obtain this sequence, note first that the square roots of the first values of pi(10^n) (A006880) (see A221205) are equal or close to some values of A229194, i.e., A221205(n) = or ≈ A229194(2n+1) = round(2^(n-1) + 3^(n-1)) for 1 <= n <= 25. Then, values of pi(10^n), A006880(n) = or ≈ (A229194(2n+1))^2 = round((2^(n-1) + 3^(n-1)))^2 for 1 <= n <= 25. Finally, the fit is improved by multiplying the exponent 2 by the sequence b(n) which always has values close to one for 1 <= n <= 25, varying between 0.99382... and 1.01511....

			For n=1, b(1) = (C1*exp(C2)*cos(C3+C4) + C5)*(C6 + Pi/2) = 0.99382..., then a(1) = round(2^(2*0.99382...)) = round(3.96588...) = 4.

Vladimir Pletser, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Prime Counting Function

Cf. A006880, A221205, A229194, A229256.

C1:=27829/125000000: C2:=-5.91561e-01: C3:=441/2500: C4:=5: C5:=19703973/31250000: C6:=5.241804273e-03: C7:=6.246728093e-01: b:=n-> (C1*n^(Pi)*exp(C2*n)*cos(C3*n+C4)+C5)*(C6*n^C7+(Pi/2)):  seq(round((2^(n-1)+3^(n-1))^(2*b(n))), n=1..25);

a(n) = round((2^(n-1) + 3^(n-1))^(2*(C1*n^(Pi)*exp(C2*n)*cos(C3*n+C4) + C5)*(C6*n^C7 + Pi/2))).

cp's OEIS Frontend

A229198 Difference between integers nearest to (2^((n-3)/2) + 3^((n-3)/2)) (A229194) and Fibonacci numbers (A000045).

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A229255 Integer nearest to (2^(n-1) + 3^(n-1))^(2b(n)) where b(n) = (C1n^(Pi)exp(C2n)cos(C3n+C4) + C5)(C6n^C7 + Pi/2) (see coefficients in comments).

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cp's OEIS Frontend

A229198 Difference between integers nearest to (2^((n-3)/2) + 3^((n-3)/2)) (A229194) and Fibonacci numbers (A000045).

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Author

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Comments

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Programs

Maple

Formula

A229255 Integer nearest to (2^(n-1) + 3^(n-1))^(2*b(n)) where b(n) = (C1*n^(Pi)*exp(C2*n)*cos(C3*n+C4) + C5)*(C6*n^C7 + Pi/2) (see coefficients in comments).

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Comments

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Programs

Maple

Formula

A229255 Integer nearest to (2^(n-1) + 3^(n-1))^(2b(n)) where b(n) = (C1n^(Pi)exp(C2n)cos(C3n+C4) + C5)(C6n^C7 + Pi/2) (see coefficients in comments).