A059878 -id:A059878 - cp's OEIS frontend

A059879 Those n for which the absolute value of A059878[n] is prime.

1, 2, 3, 4, 5, 6, 7, 9, 10, 14, 16, 18, 20, 21, 24, 27, 31, 34, 37, 39, 41, 43, 46, 47, 48, 50, 51, 53, 54, 55, 57, 60, 61, 65, 67, 68, 71, 78, 83, 84, 86, 87, 101, 103, 105, 106, 109, 112, 114, 117, 126, 127, 128, 129, 141, 143, 145, 154, 155, 158, 161, 168, 179, 181

Offset: 1

Antti Karttunen, Feb 05 2001

nonn

Of the first 511 terms of A059878, 128 are primes.

Cf. A059877.

positions(true,map(isprime,map(abs, A059878))); # positions function given in A059649.

A014437 Odd Fibonacci numbers.

Original entry on oeis.org

1, 1, 3, 5, 13, 21, 55, 89, 233, 377, 987, 1597, 4181, 6765, 17711, 28657, 75025, 121393, 317811, 514229, 1346269, 2178309, 5702887, 9227465, 24157817, 39088169, 102334155, 165580141, 433494437, 701408733, 1836311903, 2971215073, 7778742049, 12586269025

Offset: 0

Mohammad K. Azarian

Nathaniel Johnston, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (0,4,0,1).

Cf. A001651, A059878, A000045.

Cf. A360957 (sum of reciprocals).

[Fibonacci((3*Floor((n+1)/2)) + (-1)^n): n in [0..50]]; // Vincenzo Librandi, Apr 18 2011

with(combinat):A014437:=proc(n)return fibonacci((3*floor((n+1)/2)) + (-1)^n):end:
seq(A014437(n),n=0..31); # Nathaniel Johnston, Apr 18 2011
# second Maple program:
a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|0|4|0>>^n.<<1,1,3,5>>)[1,1]:
seq(a(n), n=0..33);  # Alois P. Heinz, May 22 2025

RecurrenceTable[{a[n] == 4*a[n-2] + a[n-4], a[0]==1, a[1]==1, a[2]==3, a[3]==5},a,{n,0,500}] (* G. C. Greubel, Oct 30 2015 *)
Table[ SeriesCoefficient[(-1 - x + x^2 - x^3)/(-1 + 4*x^2 + x^4), {x, 0, n}], {n, 0, 20}] (* Nikolaos Pantelidis, Feb 01 2023 *)
Select[Fibonacci[Range[50]],OddQ] (* Harvey P. Dale, Sep 01 2023 *)

Vec((-1-x+x^2-x^3)/(-1+4*x^2+x^4) + O(x^200)) \\ Altug Alkan, Oct 31 2015

apply( A014437(n)=fibonacci(n\/2*3+(-1)^n), [0..30]) \\ M. F. Hasler, Nov 18 2018

Fibonacci(3n+1) union Fibonacci(3n+2).
a(n) = Fibonacci(3*floor((n+1)/2) + (-1)^n). - Antti Karttunen, Feb 05 2001
G.f.: ( -1-x+x^2-x^3 ) / ( -1+4*x^2+x^4 ). - R. J. Mathar, Feb 16 2011
a(2n) = v-w, a(2n+1) = v+w, with v=A001076(n+1), w=A001076(n). Therefore, a(2n)+a(2n+1) = 2*A001076(n+1). - Ralf Stephan, Aug 31 2013
From Vladimir Reshetnikov, Oct 30 2015: (Start)
a(n) = ((cos(Pi*n/2)-sqrt(phi)*sin(Pi*n/2))/phi^((3*n+2)/2) + (sqrt(phi)*cos(Pi*n/2)^2+sin(Pi*n/2)^2)*phi^((3*n+1)/2))/sqrt(5), where phi=(1+sqrt(5))/2.
E.g.f.: (cos(x/phi^(3/2))/phi - sin(x/phi^(3/2))/sqrt(phi) + cosh(x*phi^(3/2))*phi + sinh(x*phi^(3/2))*sqrt(phi))/sqrt(5).
(End)

a(30)-a(31) from Vincenzo Librandi, Apr 18 2011

A059876 a(n) = bin_prime_sum(n).

Original entry on oeis.org

2, 1, 3, 3, 5, 7, 9, -1, 1, 3, 5, 5, 7, 9, 11, 3, 5, 7, 9, 9, 11, 13, 15, 13, 15, 17, 19, 19, 21, 23, 25, -7, -5, -3, -1, -1, 1, 3, 5, 3, 5, 7, 9, 9, 11, 13, 15, 7, 9, 11, 13, 13, 15, 17, 19, 17, 19, 21, 23, 23, 25, 27, 29, -3, -1, 1, 3, 3, 5, 7, 9, 7, 9, 11, 13, 13, 15, 17, 19, 11, 13, 15, 17, 17, 19, 21, 23, 21, 23, 25, 27, 27, 29, 31, 33, 19, 21, 23, 25, 25, 27, 29, 31, 29, 31

Offset: 1

Antti Karttunen, Feb 05 2001

sign

From R. J. Mathar, Nov 12 2011: (Start)
The function bin_prime_sum of an argument n is a sum of three numbers. Let s = A000523(n) be the exponent of the largest power of 2 less than or equal to n and prime=A000040. Then the three terms are:
i) (-1)^(n+1);
ii) sum_{i=1..s} prime(i) * (1 + (-1)^[n/2^i] ); where [..] is the floor bracket;
iii) 1 (if n=1), otherwise prime(s) (if s even) or 0 (if s odd). (End)

Cf. A059871, A059872, A059873, A059877, A059878, A059880.

with(numtheory); bin_prime_sum := proc(n) local i,s; s := floor_log_2(n); RETURN(((-1)^(n+1)) + add( (((-1)^(floor(n/(2^i))+1))*ithprime(i)),i=1..s) + (`if`((1 = n),1,((`mod`((s+1),2))*ithprime(s)))) ); end;

a[n_] := With[{s = Floor[Log[2, n]]}, (-1)^(n+1) + Sum[(-1)^(Floor[n/2^i] + 1)*Prime[i], {i, 1, s}] + If[1 == n, 1, Mod[s+1, 2]*Prime[s]]]; Array[a, 105] (* Jean-François Alcover, Mar 07 2016, adapted from Maple *)

a(A059873(n)) = A000040(n).

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A059879 Those n for which the absolute value of A059878[n] is prime.

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A014437 Odd Fibonacci numbers.

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A059876 a(n) = bin_prime_sum(n).

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