A001591 -id:A001591 - cp's OEIS frontend

A103814 Pentanacci constant: decimal expansion of limit of A001591(n+1)/A001591(n).

1, 9, 6, 5, 9, 4, 8, 2, 3, 6, 6, 4, 5, 4, 8, 5, 3, 3, 7, 1, 8, 9, 9, 3, 7, 3, 7, 5, 9, 3, 4, 4, 0, 1, 3, 9, 6, 1, 5, 1, 3, 2, 7, 1, 7, 7, 4, 5, 6, 8, 6, 1, 3, 9, 3, 2, 3, 6, 9, 3, 4, 5, 0, 8, 4, 4, 2, 2, 5, 2, 7, 1, 2, 8, 7, 1, 8, 8, 6, 8, 8, 1, 7, 3, 4, 8, 1, 8, 6, 6, 5, 5, 5, 4, 6, 3, 0, 4, 7, 2, 0, 2, 1, 3, 0

Offset: 1

Jonathan Vos Post, Mar 29 2005

The pentanacci constant P is the limit as n -> infinity of the ratio of Pentanacci(n+1)/Pentanacci(n) = A001591(n+1)/A001591(n), which is the principal root of x^5-x^4-x^3-x^2-x-1 = 0. Note that we have: P + P^-5 = 2.
The pentanacci constant corresponds to the Golden Section in a fivepartite division 1 = u_1 + u_2 + u_3 + u_4 + u_5 of a unit line segment, i.e., if 1/u_1 = u_1/u_2 = u_2/u_3 = u_3/u_4 + u_4/u_5 = c, c is the pentanacci constant. - Seppo Mustonen, Apr 19 2005
The other 4 roots of the polynomial 1+x+x^2+x^3+x^4-x^5 are the two complex-conjugated pairs -0.6783507129699967... +- i * 0.458536187273144499.. and 0.1953765946472540452... +- i * 0.848853640546245551858... - R. J. Mathar, Oct 25 2008
The continued fraction expansion is 1, 1, 28, 2, 1, 2, 1, 1, 1, 2, 4, 2, 1, 3, 1, 6, 1, 4, 1, 1, 5, 3, 2, 15, 69, 1, 1, 14, 1, 8, 1, 6,... - R. J. Mathar, Mar 09 2012
For n>=5, round(c^prime(n)) == 1 (mod 2*prime(n)). Proof in Shevelev link. - Vladimir Shevelev, Mar 21 2014
Note that the k-nacci constant approaches 2 when k approaches infinity (Martin Gardner). - Bernard Schott, May 07 2022

			1.965948236645485337189937375934401396151327177456861393236934508442...

Martin Gardner, The Second Scientific American Book Of Mathematical Puzzles and Diversions, "Phi: The Golden Ratio", Chapter 8, p. 101, Simon & Schuster, NY, 1961.

S. Litsyn and Vladimir Shevelev, Irrational Factors Satisfying the Little Fermat Theorem, International Journal of Number Theory, vol.1, no.4 (2005), 499-512.
Vladimir Shevelev, A property of n-bonacci constant, Seqfan (Mar 23 2014)
Eric Weisstein et al., Tetranacci Constant.
Eric Weisstein's World of Mathematics, Pentanacci Constant
Eric Weisstein's World of Mathematics, Pentanacci Number
Index entries for algebraic numbers, degree 5.

Cf. A001591.

k-nacci constants: A001622 (Fibonacci), A058265 (tribonacci), A086088 (tetranacci), this sequence (pentanacci), A118427 (hexanacci), A118428 (heptanacci).

RealDigits[Root[x^5-Total[x^Range[0,4]],1],10,120][[1]] (* Harvey P. Dale, Mar 22 2017 *)

solve(x=1, 2, 1+x+x^2+x^3+x^4-x^5) \\ Michel Marcus, Mar 21 2014

A106303 Period of the Fibonacci 5-step sequence A001591 mod n.

Original entry on oeis.org

1, 6, 104, 12, 781, 312, 2801, 24, 312, 4686, 16105, 312, 30941, 16806, 81224, 48, 88741, 312, 13032, 9372, 291304, 96630, 12166, 312, 3905, 185646, 936, 33612, 70728, 243672, 190861, 96, 1674920, 532446, 2187581, 312, 1926221, 13032

Offset: 1

T. D. Noe, May 02 2005

This sequence can differ from the corresponding Lucas sequence (A106297) only when n is a multiple of 2 or 599 because 9584 is the discriminant of the characteristic polynomial x^5-x^4-x^3-x^2-x-1 and the prime factors of 9584 are 2 and 599. [Corrected by Avery Diep, Aug 25 2025]

Chai Wah Wu, Table of n, a(n) for n = 1..388
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Cf. A001591, A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1), A106297 (period of Lucas 5-step sequence mod n).

n=5; Table[p=i; a=Join[{1}, Table[0, {n-1}]] a=Mod[a, p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i, 50}]

from itertools import count
def A106303(n):
    a = b = (0,)*4+(1 % n,)
    s = 1 % n
    for m in count(1):
        b, s = b[1:] + (s,), (s+s-b[0]) % n
        if a == b:
            return m # Chai Wah Wu, Feb 21-27 2022

Let the prime factorization of n be p1^e1...pk^ek. Then a(n) = lcm(a(p1^e1), ..., a(pk^ek)).

Conjectures: a(5^k) = 781*5^(k-1) for k > 0. If a(p) != a(p^2) for p prime, then a(p^k) = p^(k-1)*a(p) for k > 0. - Chai Wah Wu, Feb 25 2022

A288120 Number of partitions of n into distinct pentanacci numbers (with a single type of 1) (A001591).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Ilya Gutkovskiy, Jun 05 2017

nonn

The first occurrences of 1, 2, 3, 4, 5, ... are at n=0, 31, 912, 1824, 26815, ... - Antti Karttunen, Dec 22 2017

			a(31) = 2 because we have [31] and [16, 8, 4, 2, 1].

Antti Karttunen, Table of n, a(n) for n = 0..52656
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Pentanacci Number
Index entries for related partition-counting sequences

Cf. A001591, A000119, A000121, A003263, A117546, A287656, A296209.

A001591(n) = { if(n<=3,return(0)); my(p0=0,p1=0,p2=0,p3=1,p4=1,old_p0); while(n>5,n--;old_p0=p0;p0=p1;p1=p2;p2=p3;p3=p4;p4=old_p0+p0+p1+p2+p3;); p4; }
v288120nthgen(up_to) = { my(k=6,fk,vec = [1],vec2); while(k<=up_to, fk = A001591(k); k++; vec2 = vector(length(vec)+fk,i,(i==fk)+if(i>fk,vec[i-fk],0)+if(i<=length(vec),vec[i],0)); vec = vec2); vector(fk,i,vec[i]); }
write_to_bfile_with_a0_as_given(a0,vec,bfilename) = { write(bfilename, 0, " ", a0); for(n=1, length(vec), write(bfilename, n, " ", vec[n])); }
write_to_bfile_with_a0_as_given(1,v288120nthgen(21),"b288120.txt"); \\ Antti Karttunen, Dec 22 2017

(define (A288120 n) (let ((s (list 0))) (let fork ((r n) (i 5)) (cond ((zero? r) (set-car! s (+ 1 (car s)))) ((> (A001591 i) r) #f) (else (begin (fork (- r (A001591 i)) (+ 1 i)) (fork r (+ 1 i)))))) (car s)))
;; This one uses memoization-macro definec
(definec (A001591 n) (cond ((<= n 3) 0) ((= 4 n) 1) (else (+ (A001591 (- n 1)) (A001591 (- n 2)) (A001591 (- n 3)) (A001591 (- n 4)) (A001591 (- n 5))))))
;; Antti Karttunen, Dec 22 2017

G.f.: Product_{k>=5} (1 + x^A001591(k)).

More terms from Antti Karttunen, Dec 22 2017

A104412 Number of prime factors, with multiplicity, of the pentanacci numbers A001591.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 1, 1, 5, 3, 5, 6, 2, 4, 6, 5, 9, 6, 2, 2, 6, 4, 6, 7, 1, 3, 3, 6, 4, 7, 4, 2, 7, 3, 9, 7, 3, 2, 2, 8, 9, 8, 2, 4, 11, 7, 9, 7, 7, 2, 4, 5, 6, 7, 3, 2, 7, 7, 13, 8, 4, 5, 5, 6, 10, 5, 4, 7, 11, 7, 8, 9, 4, 5, 4, 8, 7, 8, 4, 5, 6, 5, 10, 12, 2, 4, 5, 9, 12, 11, 7, 4, 13

Offset: 4

Jonathan Vos Post, Mar 05 2005

nonn

Prime pentanacci numbers (A105757) correspond to 1s in the sequence, semiprimes to 2, and so forth.

Amiram Eldar, Table of n, a(n) for n = 4..237
(terms 4..124 from Charles R Greathouse IV) [offset adapted by _Georg Fischer_, Jan 04 2021]

Cf. A001591, A001222, A104411.

PrimeOmega/@Drop[LinearRecurrence[{1,1,1,1,1},{0,0,0,0,1},100],4] (* Harvey P. Dale, Mar 30 2019 *)

a(n) = A001222(A001591(n)). a(n) = bigomega(A001591(n)).

Offset corrected by Joerg Arndt, Dec 19 2020

A105756 Indices of prime Fibonacci 5-step numbers, A001591.

Original entry on oeis.org

3, 7, 8, 25, 146, 169, 182, 751, 812, 1507, 1591, 3157, 3752, 10651, 12740, 71804, 155953

Offset: 1

T. D. Noe, Apr 22 2005

nonn

No other n < 44000.
The Noe and Post reference below utilizes a scheme for its indices which differs by 3 from the indices in A001591. - Robert Price, Oct 13 2014
The sequence is similar to A248757 which uses the indexing scheme starting at 0 as defined by A001591. - Robert Price, Oct 13 2014
a(18) > 2*10^5. - Robert Price, Oct 13 2014

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A105757 (prime Fibonacci 5-step numbers).

a = {1, 0, 0, 0, 0}; lst = {}; Do[s = Plus @@ a; a = RotateLeft[a];
  a[[-1]] = s; If[PrimeQ[s], AppendTo[lst, n]], {n, 44000}]; lst
Flatten[Position[LinearRecurrence[{1,1,1,1,1},{0,0,0,0,1},13000],?PrimeQ]]-4 (* _Harvey P. Dale, Mar 06 2023 *)

Restored original sequence after reconsideration and added a(16)-a(17) by Robert Price, Oct 13 2014

A105757 Prime Fibonacci 5-step numbers, A001591.

Original entry on oeis.org

2, 31, 61, 5976577, 1989179797398599794811479787771439644489521, 11241696329548911284929550459702062135838997526529, 73666302576706758839299120422550197113618385815703101

Offset: 1

T. D. Noe, Apr 22 2005

nonn

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A105756 (indices of prime Fibonacci 5-step numbers).

a={1, 0, 0, 0, 0}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, s]], {n, 1000}]; lst

A106304 Period of the Fibonacci 5-step sequence A001591 mod prime(n).

Original entry on oeis.org

6, 104, 781, 2801, 16105, 30941, 88741, 13032, 12166, 70728, 190861, 1926221, 2896405, 79506, 736, 8042221, 102689, 3720, 20151120, 2863280, 546120, 39449441, 48030024, 3690720, 29509760, 104060400, 37516960, 132316201, 28231632, 6384, 86714880, 2248090, 3128

Offset: 1

T. D. Noe, May 02 2005, Nov 19 2006

nonn

This sequence is the same as the period of the Lucas 5-step sequence (A106298) mod prime(n) except for n=1 and 109, which correspond to the primes 2 and 599, because 9584 is the discriminant of the characteristic polynomial x^5-x^4-x^3-x^2-x-1 and the prime factors of 9584 are 2 and 599. We have a(n) < prime(n) for the primes in A106281.

Chai Wah Wu, Table of n, a(n) for n = 1..86
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A106281 (primes p such that x^5-x^4-x^3-x^2-x-1 mod p has 5 distinct zeros).

n=5; Table[p=Prime[i]; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; k=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; a!=a0]; k, {i, 40}]

from itertools import count
from sympy import prime
def A106304(n):
    a = b = (0,)*4+(1 % (p:= prime(n)),)
    s = 1 % p
    for m in count(1):
        b, s = b[1:] + (s,), (s+s-b[0]) % p
        if a == b:
            return m # Chai Wah Wu, Feb 22-27 2022

a(n) = A106303(prime(n)).

a(31)-a(33) from Chai Wah Wu, Feb 27 2022

A107243 Sum of squares of pentanacci numbers (A001591).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1303, 5024, 19424, 75120, 290416, 1122160, 4337009, 16762634, 64787534, 250400910, 967783566, 3740437902, 14456621263, 55874162432, 215950971648, 834640190272, 3225844698176, 12467736540480

Offset: 0

Jonathan Vos Post, May 19 2005

			a(0) = 0 = 0^2 since F_5(0) = A001591(0) = 0.
a(1) = 0 = 0^2 + 0^2
a(2) = 0 = 0^2 + 0^2 + 0^2
a(3) = 0 = 0^2 + 0^2 + 0^2 + 0^2
a(4) = 1 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2
a(5) = 2 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2
a(6) = 6 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2
a(7) = 22 = 0^2 + 0^2 + 0^2 + 0^2 + 1^2 + 1^2 + 2^2 + 4^2
a(8) = 86 = 8^2 + 22
a(9) = 342 = 16^2 + 86

W. C. Lynch, The t-Fibonacci numbers and polyphase sorting, Fib. Quart., 8 (1970), pp. 6ff.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.
Index entries for linear recurrences with constant coefficients, signature (3, 2, 3, 7, 14, -32, -2, 6, -4, -6, 10, 1, -1, 0, 1, -1).

Cf. A001591, A107239, A107242, A107244, A107245, A107246, A107247, A107248.

Accumulate[LinearRecurrence[{1,1,1,1,1},{0,0,0,0,1},30]^2] (* Harvey P. Dale, Jan 04 2015 *)
LinearRecurrence[{3, 2, 3, 7, 14, -32, -2, 6, -4, -6, 10, 1, -1, 0, 1, -1},{0, 0, 0, 0, 1, 2, 6, 22, 86, 342, 1303, 5024, 19424, 75120, 290416, 1122160},28] (* Ray Chandler, Aug 02 2015 *)

a(n) = F_5(1)^2 + F_5(1)^2 + F_5(2)^2 + ... F_5(n)^2 where F_5(n) = A001591(n). a(0) = 0, a(n+1) = a(n) + A001591(n)^2.
a(n)= 3*a(n-1) +2*a(n-2) +3*a(n-3) +7*a(n-4) +14*a(n-5) -32*a(n-6) -2*a(n-7) +6*a(n-8) -4*a(n-9) -6*a(n-10) +10*a(n-11) +a(n-12) -a(n-13) +a(n-15) -a(n-16). [R. J. Mathar, Aug 11 2009]
G.f.: x^4*(x^10 +x^9 +x^7 +x^6 -6*x^5 -5*x^4 -3*x^3 -2*x^2 -x +1) / ((x -1)*(x^5 +x^4 +x^3 +3*x^2 +3*x -1)*(x^10 -x^9 -x^7 +x^6 -6*x^5 +3*x^4 +3*x^3 +2*x^2 +x +1)). - Colin Barker, May 08 2013

a(26) and a(27) corrected by R. J. Mathar, Aug 11 2009

A248757 Indices of prime Fibonacci 5-step numbers, A001591.

Original entry on oeis.org

6, 10, 11, 28, 149, 172, 185, 754, 815, 1510, 1594, 3160, 3755, 10654, 12743, 71807, 155957

Offset: 1

Robert Price, Oct 13 2014

This sequence is similar to A105756 but uses the indexing scheme defined by A001591 whose indices start with 0.

a(18) > 2*10^5.

Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A001591, A105756.

a = {0, 0, 0, 0, 1}; For[n = 5, n ≤ 1000, n++, sum = Plus @@ a;
  If[PrimeQ[sum], Print[n]]; a = RotateLeft[a]; a[[5]] = sum]

a(n) = A105756 (n) + 3.

A357455 Number of compositions (ordered partitions) of n into pentanacci numbers 1,2,4,8,16,31, ... (A001591).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 18, 31, 56, 98, 174, 306, 542, 956, 1690, 2983, 5272, 9310, 16448, 29050, 51318, 90644, 160118, 282826, 499590, 882468, 1558798, 2753448, 4863696, 8591212, 15175514, 26805984, 47350057, 83639033, 147739853, 260967374, 460972308, 814260589

Offset: 0

Ilya Gutkovskiy, Sep 29 2022

nonn

Cf. A001591, A076739, A288120, A357451, A357453, A357454.

A001591[0] = A001591[1] = A001591[2] = A001591[3] = 0; A001591[4] = 1; A001591[n_] := A001591[n] = A001591[n - 1] + A001591[n - 2] + A001591[n - 3] + A001591[n - 4] + A001591[n - 5]; nmax = 37; CoefficientList[Series[1/(1 - Sum[x^A001591[k], {k, 5, 20}]), {x, 0, nmax}], x]

G.f.: 1 / (1 - Sum_{k>=5} x^A001591(k)).

cp's OEIS Frontend

A103814 Pentanacci constant: decimal expansion of limit of A001591(n+1)/A001591(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A106303 Period of the Fibonacci 5-step sequence A001591 mod n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

A288120 Number of partitions of n into distinct pentanacci numbers (with a single type of 1) (A001591).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Scheme

Formula

Extensions

A104412 Number of prime factors, with multiplicity, of the pentanacci numbers A001591.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A105756 Indices of prime Fibonacci 5-step numbers, A001591.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A105757 Prime Fibonacci 5-step numbers, A001591.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A106304 Period of the Fibonacci 5-step sequence A001591 mod prime(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Formula

Extensions

A107243 Sum of squares of pentanacci numbers (A001591).

Views