A054433 -id:A054433 - cp's OEIS frontend

A054432 a(n) = Sum_{1<=k<=n, gcd(k,n)=1} 2^(k-1).

1, 1, 3, 5, 15, 17, 63, 85, 219, 325, 1023, 1105, 4095, 5397, 13515, 21845, 65535, 70737, 262143, 333125, 890523, 1397077, 4194303, 4527185, 16236015, 22365525, 57521883, 88429845, 268435455, 272962625, 1073741823, 1431655765, 3679302363, 5726557525

Offset: 1

Antti Karttunen

nonn

For n>0, numbers formed by interpreting the reduced residue set of n (the rows of triangle A054431) as binary numbers.

			For n=6 we have k = 1 and 5 and then 2^0 + 2^4 = 17 = a(6).

Robert Israel, Table of n, a(n) for n = 1..2658

Cf. A054431, A054433, A001317, A054521.

rrs2bincode := proc(n) local i,z; z := 0; for i from 1 to n-1 do z := z*2; if (1 = igcd(n,i)) then z := z + 1; fi; od; RETURN(z); end;

f[n_] := Sum[2^k, {k, Select[ Range@ n, GCD[#, n] == 1 &] - 1}]; Array[f, 35] (* Robert G. Wilson v, Jul 21 2014 *)

a(n) = sum(k=1, n, if (gcd(k,n)==1, 2^(k-1), 0)); \\ Michel Marcus, Jul 20 2014

a(n) = subst(Polrev(vector(n, i, gcd(n, i)==1)), x, 2); \\ Michel Marcus, Jul 21 2014

M * V, where M = A054521 is an infinite lower triangular matrix and V = [1, 2, 4, 8, ...] is a vector. - Gary W. Adamson, Jan 13 2007
a(4*n) = (2^(2*n) + 1)*a(2*n) [think how the reduced residue set of the numbers of the form 4n are formed].
For all primes p and integers e > 1, A054432(p^e) = A019320(p^e)*(((2^(p^(e-1)))-1)* ((2^(p-1))-1))/((2^p)-1).
a(n-1) = Sum_{k=1..n, gcd(n, k) = 1} 2^(k-1). - Vladeta Jovovic, Aug 15 2002

Edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

More terms from Michel Marcus, Jul 20 2014

A051258 Fibocyclotomic numbers: numbers formed from cyclotomic polynomials and Fibonacci numbers (A000045).

Original entry on oeis.org

1, 1, 1, 2, 1, 7, 0, 20, 3, 10, 1, 143, 2, 376, 4, 11, 21, 2583, 6, 6764, 15, 74, 33, 46367, 18, 7435, 88, 2618, 104, 832039, 25, 2178308, 987, 3399, 609, 20160, 136, 39088168, 1596, 23228, 861, 267914295, 182, 701408732, 4895, 35920, 10945, 4807526975

Offset: 0

Antti Karttunen, Oct 24 1999

For all primes p, a(p) = fib(p+1)-1 and for all n of the form 2^i*p^j (where p is an odd prime and i >= 0 and j >= 2) fib(p)|a(2^i*p^j).

a(0) depends on how the zeroth cyclotomic polynomial is defined.

			a(22) = fib(10)-fib(9)+fib(8)-fib(7)+fib(6)-fib(5)+fib(4)-fib(3)+fib(2)-fib(1) = 33 as the 22nd cyclotomic polynomial is x^10-x^9+x^8-x^7+x^6-x^5+x^4-x^3+x^2-x+1 (The constant term does not affect the result, as fib(0)=0.)

T. D. Noe, Table of n, a(n) for n = 0..500

Cf. A019320, A054433, A063704, A063706, A063708.

get_coefficient := proc(e); if(1 = nops(e)) then if(`integer` = op(0,e)) then RETURN(e); else RETURN(1); fi; else if(2 = nops(e)) then if(`*` = op(0,e)) then RETURN(op(1,e)); else RETURN(1); fi; else RETURN(`Cannot find coefficient!`); fi; fi; end;
get_exponent := proc(e); if((1 = e) or (-1 = e)) then RETURN(0); else if(1 = nops(e)) then RETURN(1); else if(2 = nops(e)) then if(`^` = op(0,e)) then RETURN(op(2,e)); else RETURN(get_exponent(op(2,e))); fi; else RETURN(`Cannot find exponent!`); fi; fi; fi; end;
fibo_cyclotomic := proc(j) local i,p; p := sort(cyclotomic(j,x)); RETURN(add((get_coefficient(op(i,p))*fibonacci(get_exponent(op(i,p)))),i=1..nops(p))); end;

f[n_]:=Module[{cy=CoefficientList[Cyclotomic[n,x],x]},Total[ Times@@@ Thread[ {Fibonacci[ Range[0, Length[cy]- 1]],cy}]]]; Join[{1},Array[f,50]] (* Harvey P. Dale, Oct 02 2011 *)

a(n)=my(P=polcyclo(n));sum(i=1,poldegree(P),polcoeff(P,i)*fibonacci(i)) \\ Charles R Greathouse IV, Jan 05 2013

a(n) = Sum (coefficient_of_term_i_of_cp_n times Fib(exponent_of_term_i_of_cp_n)), i=1..degree of cp_n, where cp_n is the n-th cyclotomic polynomial.

A048757 Sum_{i=0..2n} (C(2n,i) mod 2)Fibonacci(i+2) = Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i+2).

Original entry on oeis.org

1, 4, 9, 33, 56, 203, 441, 1596, 2585, 9353, 20304, 73461, 124033, 448756, 974169, 3524577, 5702888, 20633243, 44791065, 162055596, 273617239, 989956471, 2149017696, 7775219067, 12591974497, 45558191716, 98898651657

Offset: 0

Antti Karttunen, Jul 13 1999

The history of 1-D CA Rule 90 starting from the seed pattern 1 interpreted as Zeckendorffian expansion.

Also, product of distinct terms of A001566 and appropriate Fibonacci or Lucas numbers: a(n) = FL(n+2)Product(L(2^i)^bit(n,i),i=0..) Here L(2^i) = A001566 and FL(n) = n-th Fibonacci number if n even, n-th Lucas number if n odd. bit(n,i) is the i-th digit (0 or 1) in the binary expansion of n, with the least significant digit being bit(n,0).

			1 = Fib(2) = 1;
101 = Fib(4) + Fib(2) = 3 + 1 = 4;
10001 = Fib(6) + Fib(2) = 8 + 1 = 9;
1010101 = Fib(8) + Fib(6) + Fib(4) + Fib(2) = 21 + 8 + 3 + 1 = 33; etc.

Antti Karttunen, On Pascal's Triangle Modulo 2 in Fibonacci Representation, Fibonacci Quarterly, 42 (2004), 38-46.

a(n) = A022290(A038183(n)) = A022290(A048723(5, n)) = A003622(A051656(n)) = A075148(n, 2)*A050613(n). Third row of A050609, third column of A050610.

Cf. A054433.

Table[Sum[Mod[Binomial[2n, i], 2] Fibonacci[i + 2], {i, 0, 2n}], {n, 0, 19}] (* Alonso del Arte, Apr 27 2014 *)

A063683 Integers formed from the reduced residue sets of even numbers and Fibonacci numbers.

Original entry on oeis.org

1, 3, 6, 21, 50, 108, 364, 987, 1938, 6150, 17622, 34776, 121160, 306852, 549000, 2178309, 5701290, 11197764, 39083988, 93031050, 191708244, 697884066, 1836283246, 3605645232, 11442062750, 32888033880, 64700678454

Offset: 1

Antti Karttunen, Jul 31 2001

nonn

a(2n) = L(2n)*a(n), where L(2n) is the 2n-th Lucas number = A000032(2n).

			The reduced residue set of 2*6 = 12 is {1,5,7,11}, thus a(6) = F_1 + F_5 + F_7 + F_11 = 108.

Cf. A054432, A054433, A050611.

A063683 := [seq(A063683_as_sum(2*n), n=1..101)]; A063683_as_sum := proc(n) local i; RETURN(add((one_or_zero(igcd(n,i))*fibonacci(i)),i=1..(n-1))); end; Yours, Antti Karttunen

a(n) = Sum_{i | gcd(i, 2n)=1} Fib(i) (where Fib(i) = A000045[i])

cp's OEIS Frontend

A054432 a(n) = Sum_{1<=k<=n, gcd(k,n)=1} 2^(k-1).

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A051258 Fibocyclotomic numbers: numbers formed from cyclotomic polynomials and Fibonacci numbers (A000045).

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A048757 Sum_{i=0..2n} (C(2n,i) mod 2)*Fibonacci(i+2) = Sum_{i=0..n} (C(n,i) mod 2)*Fibonacci(2i+2).

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A063683 Integers formed from the reduced residue sets of even numbers and Fibonacci numbers.

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Maple

Formula

A048757 Sum_{i=0..2n} (C(2n,i) mod 2)Fibonacci(i+2) = Sum_{i=0..n} (C(n,i) mod 2)Fibonacci(2i+2).