A074048 -id:A074048 - cp's OEIS frontend

A074062 Reflected (see A074058) pentanacci numbers A074048.

5, -1, -1, -1, -1, 9, -7, -1, -1, -1, 19, -23, 5, -1, -1, 39, -65, 33, -7, -1, 79, -169, 131, -47, 5, 159, -417, 431, -225, 57, 313, -993, 1279, -881, 339, 569, -2299, 3551, -3041, 1559, 799, -5167, 9401, -9633, 6159, 39, -11133, 23969, -28667, 21951, -6081, -22305

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 17 2002

a(n) is also the trace of A^(-n), where A is the matrix ( (1,1,0,0,0), (1,0,1,0,0), (1,0,0,1,0), (1,0,0,0,1), (1,0,0,0,0) ).

a(n) is also the sum of determinants of 4th-order principal minors of A^n.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1,1).

Cf. A061084, A074058, A074048.

I:=[5,-1,-1,-1,-1]; [n le 5 select I[n] else (-1)*(Self(n-1) +Self(n-2) +Self(n-3) +Self(n-4)) + Self(n-5): n in [1..61]]; // G. C. Greubel, Jul 05 2021

CoefficientList[Series[(5+4*x+3*x^2+2*x^3+x^4)/(1+x+x^2+x^3+x^4-x^5), {x, 0, 60}], x]

Vec((5+4*x+3*x^2+2*x^3+x^4)/(1+x+x^2+x^3+x^4-x^5) + O(x^60)) \\ Michel Marcus, Sep 14 2020

def A074062_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (5+4*x+3*x^2+2*x^3+x^4)/(1+x+x^2+x^3+x^4-x^5) ).list()
A074062_list(60) # G. C. Greubel, Jul 05 2021

a(n) = -a(n-1) -a(n-2) -a(n-3) -a(n-4) +a(n-5), a(0)=5, a(1)=-1, a(2)=-1, a(3)=-1, a(4)=-1.

G.f.: (5 +4*x +3*x^2 +2*x^3 +x^4)/(1 +x +x^2 +x^3 +x^4 -x^5).

More terms from Michel Marcus, Sep 14 2020

A106298 Period of the Lucas 5-step sequence A074048 mod prime(n).

Original entry on oeis.org

1, 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

nonn

This sequence is the same as the period of Fibonacci 5-step sequence (A106304) 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 2, 599 and A106281.

Chai Wah Wu, Table of n, a(n) for n = 1..82
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), A106297.

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 A106298(n):
    a = b = (5%(p:=prime(n)),1%p,7%p,3%p,15%p)
    s = sum(b) % 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) = A106297(prime(n)).

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

A106297 Period of the Lucas 5-step sequence A074048 mod n.

Original entry on oeis.org

1, 1, 104, 6, 781, 104, 2801, 12, 312, 781, 16105, 312, 30941, 2801, 81224, 24, 88741, 312, 13032, 4686, 291304, 16105, 12166, 312, 3905, 30941, 936, 16806, 70728, 81224, 190861, 48, 1674920, 88741, 2187581, 312, 1926221, 13032, 3217864, 9372, 2896405

Offset: 1

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

This sequence can differ from the corresponding Fibonacci sequence (A106303) 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]

a(n) divides A106303(n). - Avery Diep, Aug 25 2025

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

Cf. A074048, A106303 (period of Fibonacci 5-step sequence mod n), A106273 (discriminant of the polynomial x^n-x^(n-1)-...-x-1).

n=5; Table[p=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, 50}]

from itertools import count
def A106297(n):
    a = b = (5%n,1%n,7%n,3%n,15%n)
    s = sum(b) % 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 22-27 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. a(2^k) = 3*2^(k-1) for k > 1. If a(p) != a(p^2) for prime p > 2, then a(p^k) = p^(k-1)*a(p) for k > 0. - Chai Wah Wu, Feb 25 2022

A105289 Indices of Lucas 5-step numbers A074048 which have a nontrivial divisor in common with index.

Original entry on oeis.org

6, 12, 18, 20, 21, 33, 36, 48, 51, 54, 55, 60, 75, 78, 87, 99, 100, 108, 110, 112, 114, 120, 129, 132, 133, 144, 147, 153, 154, 155, 159, 162, 165, 174, 177, 180, 182, 183, 185, 195, 210, 219, 225, 228, 230, 234, 237, 245, 261, 267, 270, 275, 285, 290, 297, 310

Offset: 1

Jonathan Vos Post, Apr 25 2005

Extension by T. D. Noe. Wanted: closed-form formula for this as exists for Fibonacci and Lucas numbers. See also A105764 (indices of prime Lucas 5-step numbers).

			gcd(6, A074048(6)) = gcd(6,57) = 3,
gcd(20, A074048(20)) = gcd(20,743775) = 5.
gcd(21, A074048(21)) = gcd(21,1462223) = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A074048, A104714, A105764.

m=300; s = LinearRecurrence[{1, 1, 1, 1, 1}, {5, 1, 3, 7, 15}, m+1]; Select[Range[m], !CoprimeQ[#, s[[#+1]]] &] (* Amiram Eldar, Sep 05 2019 *)

gcd(a(n), A074048(a(n))) > 1.

More terms from Amiram Eldar, Sep 05 2019

A105764 Indices of prime Lucas 5-step numbers, A074048.

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 10, 13, 30, 35, 77, 98, 126, 160, 192, 810, 1086, 1999, 2021, 3157, 3426, 3471, 15066, 24115, 29782, 29941

Offset: 1

T. D. Noe, Apr 22 2005

nonn

No other n < 30000.

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, Lucas n-Step Number

Cf. A105765 (prime Lucas 5-step numbers).

a={-1, -1, -1, -1, 5}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, n]], {n, 30000}]; lst

A106301 Primes that do not divide any term of the Lucas 5-step sequence A074048.

Original entry on oeis.org

2, 691, 3163, 4259, 5419, 6637, 6733, 14923, 25111, 27947, 29339, 34123, 34421, 34757, 42859, 55207, 57529, 59693, 61643, 68897, 70249, 75991, 82763, 83177, 85607, 86441, 87103, 93169, 93283, 98573, 106121, 106433, 114847, 129589, 132313

Offset: 1

T. D. Noe, May 02 2005

nonn

If a prime p divides a term a(k) of this sequence, then k must be less than the period of the sequence mod p. Hence these primes are found by computing A074048(k) mod p for increasing k and stopping when either A074048(k) mod p = 0 or the end of the period is reached. Interestingly, for all of these primes, the period of the sequence A074048(k) mod p appears to be (p-1)/d, where d is a small integer.

Eric Weisstein's World of Mathematics, Fibonacci n-Step Number.

Cf. A053028 (primes not dividing any Lucas number), A106299 (primes not dividing any Lucas 3-step number), A106300 (primes not dividing any Lucas 4-step number).

n=5; lst={}; Table[p=Prime[i]; a=Join[Table[ -1, {n-1}], {n}]; a=Mod[a, p]; a0=a; While[s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[n]]=s; !(a==a0 || s==0)]; If[s>0, AppendTo[lst, p]], {i, 10000}]; lst

A105765 Prime Lucas 5-step numbers, A074048.

Original entry on oeis.org

3, 7, 31, 113, 223, 439, 863, 6553, 641449337, 18837477823, 40276345611255837298559, 58893004546665606516457357571, 9774215601155945008361439560567878777

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, Lucas n-Step Number

Cf. A105764 (indices of prime Lucas 5-step numbers).

a={-1, -1, -1, -1, 5}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, s]], {n, 1000}]; lst

A075156 Binomial transform of pentanacci numbers A074048: a(n) = Sum_{k=0..n} binomial(n,k)*A074048(k).

Original entry on oeis.org

5, 6, 10, 24, 70, 216, 664, 2008, 5998, 17808, 52770, 156360, 463492, 1374392, 4076222, 12090144, 35859742, 106359928, 315460168, 935639768, 2775057510, 8230670416, 24411730298, 72403913480, 214746249796, 636926269816

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 07 2002

N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (6,-13,14,-7,2).

Cf. A074048.

M := Matrix(5, (i,j)-> if (i=j-1) then 1 elif j>1 then 0 else [6,-13,14,-7,2][i] fi); a := n -> (Matrix([[70,24,10,6,5]]).M^(n))[1,5]; seq (a(n), n=0..50); # Alois P. Heinz, Jul 25 2008

CoefficientList[Series[(5-24*x+39*x^2-28*x^3+7*x^4)/(1-6*x+13*x^2-14*x^3+7*x^4-2*x^5), {x, 0, 25}], x]
LinearRecurrence[{6,-13,14,-7,2},{5,6,10,24,70},30] (* Harvey P. Dale, Mar 10 2019 *)

a(n) = 6a(n-1) - 13a(n-2) + 14a(n-3) - 7a(n-4) + 2a(n-5), a(0)=5, a(1)=6, a(2)=10, a(3)=24, a(4)=70.
G.f.: (5 - 24*x + 39*x^2 - 28*x^3 + 7*x^4)/(1 - 6*x + 13*x^2 - 14*x^3 + 7*x^4 - 2*x^5).
a(n) = term (1,5) in the 1 X 5 matrix [70,24,10,6,5]. [6,1,0,0,0; -13,0,1,0,0; 14,0,0,1,0; -7,0,0,0,1; 2,0,0,0,0]^n. - Alois P. Heinz, Jul 25 2008

A075194 Binomial transform of pentanacci numbers A074048: a(n)=Sum((-1)^kBinomial(n,k)A074048(k),(k=0,..,n)).

Original entry on oeis.org

5, 4, 6, 4, 6, 4, 0, -24, -82, -212, -454, -876, -1548, -2544, -3858, -5276, -6050, -4348, 3744, 25768, 75206, 174444, 357858, 673076, 1175972, 1909904, 2851270, 3789508, 4089238, 2255044, -4809280, -22969880, -62544962, -140412180, -281990486, -521513324, -896946156, -1432099056

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 08 2002

N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (4, -5, 0, 5, -4).

Cf. A074048.

CoefficientList[Series[(5-16x+15x^2-5x^4)/(1-4x+5x^2-5x^4+4x^5), {x, 0, 40}], x]

a(n)=4a(n-1)-5a(n-2)+5a(n-4)-4a(n-5), a(0)=5, a(1)=4, a(2)=6, a(3)=4, a(4)=6. G.f.: (5-16x+15x^2-5x^4)/(1-4x+5x^2-5x^4+4x^5).

A001591 Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5), a(0)=a(1)=a(2)=a(3)=0, a(4)=1.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641, 23099186, 45411804, 89277256, 175514464, 345052351, 678355061, 1333610936, 2621810068

Offset: 0

N. J. A. Sloane

Number of permutations satisfying -k <= p(i) - i <= r, i=1..n-4, with k=1, r=4. - Vladimir Baltic, Jan 17 2005
a(n) is the number of compositions of n-4 with no part greater than 5. - Vladimir Baltic, Jan 17 2005
The pentanomial (A035343(n)) transform of a(n) is a(5n+4), n >= 0. - Bob Selcoe, Jun 10 2014
a(n) is the number of ways to tile a strip of length n-4 with squares, dominoes, trominoes (of length 3), and rectangles with length 4 (tetraminoes) and length 5 (pentaminoes). - Wajdi Maaloul, Jun 21 2022

			n=2: a(14) = (1*1 + 2*1 + 3*2 + 4*4 + 5*8 + 4*16 + 3*31 + 2*61 + 1*120) = 464. - _Bob Selcoe_, Jun 10 2014
G.f. = x^4 + x^5 + 2*x^6 + 4*x^7 + 8*x^8 + 16*x^9 + 31*x^10 + 120*x^11 + ...

Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..200
Abdullah Açikel, Amrouche Said, Hacene Belbachir, and Nurettin Irmak, On k-generalized Lucas sequence with its triangle, Turkish J. Math. (2023) Vol. 47, No. 4, Art. 6, 1129-1143. See p. 1130.
Tomás Aguilar-Fraga, Jennifer Elder, Rebecca E. Garcia, Kimberly P. Hadaway, Pamela E. Harris, Kimberly J. Harry, Imhotep B. Hogan, Jakeyl Johnson, Jan Kretschmann, Kobe Lawson-Chavanu, J. Carlos Martínez Mori, Casandra D. Monroe, Daniel Quiñonez, Dirk Tolson III, and Dwight Anderson Williams II, Interval and L-interval Rational Parking Functions, arXiv:2311.14055 [math.CO], 2023. See p. 14.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 307-309.
Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics Vol. 4, No 1 (April, 2010), 119-135.
Paul Barry, The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1804.05027 [math.CO], 2018.
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Ömür Deveci, Yesim Akuzum, Erdal Karaduman, and Ozgur Erdag, The Cyclic Groups via Bezout Matrices, Journal of Mathematics Research, Vol. 7, No. 2, 2015, pp. 34-41.
Ömür Deveci, Zafer Adıgüzel, and Taha Doğan, On the Generalized Fibonacci-circulant-Hurwitz numbers, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 1, 179-190.
G. P. B. Dresden and Z. Du, A Simplified Binet Formula for k-Generalized Fibonacci Numbers, J. Int. Seq. 17 (2014) # 14.4.7.
I. Flores, k-Generalized Fibonacci numbers, Fib. Quart., 5 (1967), 258-266.
Taras Goy and Mark Shattuck, Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.8.
Tian-Xiao He, Impulse Response Sequences and Construction of Number Sequence Identities, J. Int. Seq. 16 (2013) #13.8.2.
V. E. Hoggatt, Jr. and M. Bicknell, Diagonal sums of generalized Pascal triangles, Fib. Quart., 7 (1969), 341-358, 393.
F. T. Howard and Curtis Cooper, Some identities for r-Fibonacci numbers, Fibonacci Quart. 49 (2011), no. 3, 231-243.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 12
Omar Khadir, László Németh, and László Szalay, Tiling of dominoes with ranked colors, Results in Math. (2024) Vol. 79, Art. No. 253. See p. 2.
Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Michel Mollard, p-th order generalized Fibonacci cubes and maximal cubes in Fibonacci p-cubes, arXiv:2507.16387 [math.CO], 2025. See p. 3.
Elisa Heinrich Mora and Noah A. Rosenberg, An nth-cousin mating model and the n-anacci numbers, arXiv:2506.16577 [q-bio.PE], 2025.
László Németh and László Szalay, Explicit solution of system of two higher-order recurrences, arXiv:2408.12196 [math.NT], 2024. See p. 10.
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.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Helmut Prodinger, Counting Palindromes According to r-Runs of Ones Using Generating Functions, J. Int. Seq. 17 (2014) # 14.6.2, odd length middle 0, r=4.
B. Sivakumar and V. James, A Notes on Matrix Sequence of Pentanacci Numbers and Pentanacci Cubes, Communications in Mathematics and Applications (2022) Vol. 13, Iss. 2, 603-611.
Yüksel Soykan, On A Generalized Pentanacci Sequence, Asian Research Journal of Mathematics (2019) Vol. 14, No. 3, 1-9.
Yüksel Soykan, Sum Formulas for Generalized Fifth-Order Linear Recurrence Sequences, Journal of Advances in Mathematics and Computer Science (2019) Vol. 34, No. 5, 1-14.
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number
Eric Weisstein's World of Mathematics, Pentanacci Number
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1,1).

Row 5 of arrays A048887 and A092921 (k-generalized Fibonacci numbers).
Cf. A106303 (Pisano period lengths).
Cf. A035343 (pentanomial coefficients).
Cf. A074048, A123127, A123126, A074062.

a:=[0,0,0,0,1]; [n le 5 select a[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-4) + Self(n-5): n in [1..40]]; // Marius A. Burtea, Oct 03 2019

g:=1/(1-z-z^2-z^3-z^4-z^5): gser:=series(g, z=0, 49): seq((coeff(gser, z, n)), n=-4..32); # Zerinvary Lajos, Apr 17 2009
# second Maple program:
a:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>, <0|0|0|0|1>, <1|1|1|1|1>>^n)[1, 5]:
seq(a(n), n=0..44);  # Alois P. Heinz, Apr 09 2021

CoefficientList[Series[x^4/(1 - x - x^2 - x^3 - x^4 - x^5), {x, 0, 50}], x]
a[0] = a[1] = a[2] = a[3] = 0; a[4] = a[5] = 1; a[n_] := a[n] = 2 a[n - 1] - a[n - 6]; Array[a, 37, 0]
LinearRecurrence[{1, 1, 1, 1, 1}, {0, 0, 0, 0, 1}, 50] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)

a(n):=mod(floor(10^((n-4)*(n+1))*10^(5*(n+1))*(10^(n+1)-1)/(10^(6*(n+1))-2*10^(5*(n+1))+1)),10^n); /* Tani Akinari, Apr 10 2014 */

a=vector(100);a[4]=a[5]=1;for(n=6,#a,a[n]=a[n-1]+a[n-2]+a[n-3]+a[n-4]+a[n-5]);concat(0, a) \\ Charles R Greathouse IV, Jul 15 2011

A001591(n,m=5)=(matrix(m,m,i,j,i==j-1||i==m)^n)[1,m] \\ M. F. Hasler, Apr 20 2018

a(n)= {my(x='x, p=polrecip(1 - x - x^2 - x^3 - x^4 - x^5)); polcoef(lift(Mod(x, p)^n), 4); }
vector(41, n, a(n-1)) \\ Joerg Arndt, May 16 2021

def pentanacci():
    a, b, c, d, e = 0, 0, 0, 0, 1
    while True:
        yield a
        a, b, c, d, e = b, c, d, e, a + b + c + d + e
f = pentanacci()
print([next(f) for  in range(100)]) # _Reza K Ghazi Apr 09 2021

G.f.: x^4/(1 - x - x^2 - x^3 - x^4 - x^5). - Simon Plouffe in his 1992 dissertation.
G.f.: Sum_{n >= 0} x^(n+4) * (Product_{k = 1..n} (k + k*x + k*x^2 + k*x^3 + x^4)/(1 + k*x + k*x^2 + k*x^3 + k*x^4)). - Peter Bala, Jan 04 2015
Another form of the g.f.: f(z) = (z^4-z^5)/(1-2*z+z^6); then a(n) = Sum_{i=0..floor((n-4)/6)} ((-1)^i*binomial(n-4-5*i,i)*2^(n-4-6*i)) - Sum_{i=0..floor((n-5)/6)} ((-1)^i*binomial(n-5-5*i,i)*2^(n-5-6*i)) with convention Sum_{i=m..n} alpha(i) = 0 for m > n. - Richard Choulet, Feb 22 2010
a(n) = Sum_{k=1..n} (Sum_{r=0..k} (binomial(k,r) * Sum_{m=0..r} (binomial(r,m) * Sum_{j=0..m} (binomial(m,j)*binomial(j,n-m-k-j-r))))), n > 0. - Vladimir Kruchinin, Aug 30 2010
Sum_{k=0..4*n} a(k+b)*A035343(n,k) = a(5*n+b), b >= 0.
a(n) = 2*a(n-1) - a(n-6). - Vincenzo Librandi, Dec 19 2010
a(n) = (Sum_{i=0..n-1} a(i)*A074048(n-i))/(n-4) for n > 4. - Greg Dresden and Advika Srivastava, Oct 01 2019
For k>0 and n>0, a(n+5*k) = A074048(k)*a(n+4*k) - A123127(k-1)*a(n+3*k) + A123126(k-1)*a(n+2*k) - A074062(k)*a(n+k) + a(n). - Kai Wang, Sep 06 2020
lim n->oo a(n)/a(n-1) = A103814. - R. J. Mathar, Mar 11 2024

cp's OEIS Frontend

A074062 Reflected (see A074058) pentanacci numbers A074048.

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A106298 Period of the Lucas 5-step sequence A074048 mod prime(n).

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A106297 Period of the Lucas 5-step sequence A074048 mod n.

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A105289 Indices of Lucas 5-step numbers A074048 which have a nontrivial divisor in common with index.

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A105764 Indices of prime Lucas 5-step numbers, A074048.

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A106301 Primes that do not divide any term of the Lucas 5-step sequence A074048.

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A105765 Prime Lucas 5-step numbers, A074048.

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A075156 Binomial transform of pentanacci numbers A074048: a(n) = Sum_{k=0..n} binomial(n,k)*A074048(k).

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A075194 Binomial transform of pentanacci numbers A074048: a(n)=Sum((-1)^kBinomial(n,k)A074048(k),(k=0,..,n)).