A073817 -id:A073817 - cp's OEIS frontend

A251656 4-step Fibonacci sequence starting with 1,0,1,0.

1, 0, 1, 0, 2, 3, 6, 11, 22, 42, 81, 156, 301, 580, 1118, 2155, 4154, 8007, 15434, 29750, 57345, 110536, 213065, 410696, 791642, 1525939, 2941342, 5669619, 10928542, 21065442, 40604945, 78268548, 150867477, 290806412, 560547382, 1080489819

Offset: 0

Arie Bos, Dec 06 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251655, A251703, A251704, A251705.

Cf. A000336.

NB. see A251655 for the program and apply it to 1,0,1,0.

LinearRecurrence[Table[1, {4}], {1, 0, 1, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n)+a(n+1)+a(n+2)+a(n+3).
G.f.: (-1+x+2*x^3)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+3)-A000078(n+2)-2*A000078(n). - R. J. Mathar, Mar 28 2025

A251703 4-step Fibonacci sequence starting with 1,1,0,0.

Original entry on oeis.org

1, 1, 0, 0, 2, 3, 5, 10, 20, 38, 73, 141, 272, 524, 1010, 1947, 3753, 7234, 13944, 26878, 51809, 99865, 192496, 371048, 715218, 1378627, 2657389, 5122282, 9873516, 19031814, 36685001, 70712613, 136302944, 262732372, 506432930, 976180859

Offset: 0

Arie Bos, Dec 07 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251655, A251656, A251704, A251705.

NB. see A251655 for the program and apply it to 1,1,0,0.

LinearRecurrence[Table[1, {4}], {1, 1, 0, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: (-1+2*x^2+2*x^3)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+3)-2*A000078(n+1)-2*A000078(n). - R. J. Mathar, Mar 28 2025

A104622 Indices of prime values of heptanacci-Lucas numbers A104621.

Original entry on oeis.org

0, 2, 3, 5, 7, 10, 17, 24, 25, 26, 28, 38, 40, 49, 62, 79, 89, 114, 140, 145, 182, 248, 353, 437, 654, 702, 784, 921, 931, 986, 1206, 2136, 2137, 3351, 5411, 13264, 13757, 16348, 27087, 27160

Offset: 1

Jonathan Vos Post, Mar 17 2005

The 7th-order linear recurrence A104622 (heptanacci-Lucas numbers) is a generalization of the Lucas sequence A000032. T. D. Noe and I have noted that the heptanacci-Lucas numbers have many more primes than the corresponding heptanacci (see A104414) which he found has only the first 3 primes that I identified through the first 5000 values, whereas these heptanacci-Lucas numbers have 17 primes among the first 100 values. For semiprimes in heptanacci-Lucas numbers, see A104623.

			A104621(0) = 7,
A104621(2) = 3,
A104621(3) = 7,
A104621(5) = 31,
A104621(7) = 127,
A104621(10) = 983,
A104621(17) = 122401,
A104621(24) = 15231991.

Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math.CO/0210201 v1, Oct 14 2002
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

Cf. A000032, A001644, A073817, A074048, A074584, A104414, A104576, A104577, A104621, A104623.

a[0] = 7; a[1] = 1; a[2] = 3; a[3] = 7; a[4] = 15; a[5] = 31; a[6] = 63; a[n_] := a[n] = a[n - 1] + a[n - 2] + a[n - 3] + a[n - 4] + a[n - 5] + a[n - 6] + a[n - 7]; Do[ If[ PrimeQ[ a[n]], Print[n]], {n, 5000}] (* Robert G. Wilson v, Mar 17 2005 *)
Flatten[Position[LinearRecurrence[{1,1,1,1,1,1,1},{7,1,3,7,15,31,63},28000],?PrimeQ]]-1 (* _Harvey P. Dale, Jan 02 2016 *)

Prime values of the heptanacci-Lucas numbers, which are defined by: a(0) = 7, a(1) = 1, a(2) = 3, a(3) = 7, a(4) = 15, a(5) = 31, a(6) = 63, for n > 6: a(n) = a(n-1)+a(n-2)+a(n-3)+a(n-4)+a(n-5)+a(n-6)+a(n-7).

More terms from T. D. Noe and Robert G. Wilson v, Mar 17 2005

A251654 4-step Fibonacci sequence starting with 0, 1, 1, 0.

Original entry on oeis.org

0, 1, 1, 0, 2, 4, 7, 13, 26, 50, 96, 185, 357, 688, 1326, 2556, 4927, 9497, 18306, 35286, 68016, 131105, 252713, 487120, 938954, 1809892, 3488679, 6724645, 12962170, 24985386, 48160880, 92833081, 178941517, 344920864, 664856342, 1281551804

Offset: 0

Arie Bos, Dec 06 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251655, A251656, A251672, A251703, A251704, A251705.

NB. see A251655 for the program and apply it to 0,1,1,0.

LinearRecurrence[Table[1, {4}], {0, 1, 1, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: x*(-1+2*x^2)/(-1+x+x^2+x^3+x^4). - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+2)-2*A000078(n). - R. J. Mathar, Mar 28 2025

A251655 4-step Fibonacci sequence starting with 0, 1, 1, 1.

Original entry on oeis.org

0, 1, 1, 1, 3, 6, 11, 21, 41, 79, 152, 293, 565, 1089, 2099, 4046, 7799, 15033, 28977, 55855, 107664, 207529, 400025, 771073, 1486291, 2864918, 5522307, 10644589, 20518105, 39549919, 76234920, 146947533, 283250477, 545982849, 1052415779, 2028596638

Offset: 0

Arie Bos, Dec 06 2014

Tamás Lengyel and Diego Marques, The 2-adic Order of Some Generalized Fibonacci Numbers, INTEGERS, 17, 2017, A5.
Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251656, A251672, A251703, A251704, A251705.

(see www.jsoftware.com) First construct the generating matrix
   [M=: (#.@}: + {:)\"1&.|: <:/~i.4
1 1 1 1
1 2 2 2
2 3 4 4
4 6 7 8
Given that matrix, one can produce the first 4*250 numbers with
, M(+/ . *)^:(i.250) 0 1 1 1x

LinearRecurrence[Table[1, {4}], {0, 1, 1, 1}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: x*(x-1)*(1+x)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025
a(n) = A000078(n+2)-A000078(n). - R. J. Mathar, Mar 28 2025

A251704 4-step Fibonacci sequence starting with 1, 1, 0, 1.

Original entry on oeis.org

1, 1, 0, 1, 3, 5, 9, 18, 35, 67, 129, 249, 480, 925, 1783, 3437, 6625, 12770, 24615, 47447, 91457, 176289, 339808, 655001, 1262555, 2433653, 4691017, 9042226, 17429451, 33596347, 64759041, 124827065, 240611904, 463794357, 893992367, 1723225693

Offset: 0

Arie Bos, Dec 07 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251655, A251656, A251703, A251705.

NB. see A251655 for the program and apply it to 1,1,0,1.

LinearRecurrence[Table[1, {4}], {1, 1, 0, 1}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).
G.f.: (1+x)*(x^2+x-1)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025
a(n) = A001630(n-2)+A001630(n-1), n>2. - R. J. Mathar, Mar 28 2025

A251705 4-step Fibonacci sequence starting with 1, 1, 1, 0.

Original entry on oeis.org

1, 1, 1, 0, 3, 5, 9, 17, 34, 65, 125, 241, 465, 896, 1727, 3329, 6417, 12369, 23842, 45957, 88585, 170753, 329137, 634432, 1222907, 2357229, 4543705, 8758273, 16882114, 32541321, 62725413, 120907121, 233055969, 449229824, 865918327, 1669111241

Offset: 0

Arie Bos, Dec 07 2014

Index entries for linear recurrences with constant coefficients, signature (1,1,1,1).

Other 4-step Fibonacci sequences are A000078, A000288, A001630, A001631, A001648, A073817, A100532, A251654, A251655, A251656, A251703, A251704.

NB. see A251655 for the program and apply it to 1,1,1,0.

LinearRecurrence[Table[1, {4}], {1, 1, 1, 0}, 36] (* Michael De Vlieger, Dec 09 2014 *)

a(n+4) = a(n) + a(n+1) + a(n+2) + a(n+3).

G.f.: (-1+3*x^3+x^2)/(-1+x+x^2+x^3+x^4) . - R. J. Mathar, Mar 28 2025

A073937 a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4), a(0)=4, a(1)=1, a(2)=-1, a(3)=1.

Original entry on oeis.org

4, 1, -1, 1, 7, 6, -1, 1, 15, 19, 4, 1, 31, 53, 27, 6, 63, 137, 107, 39, 132, 337, 351, 185, 303, 806, 1039, 721, 791, 1915, 2884, 2481, 2303, 4621, 7683, 7846, 7087, 11545, 19987, 23375, 22020, 30177, 51519, 66737, 67415, 82374, 133215, 184993, 201567, 232163, 348804

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 13 2002

From Kai Wang, Nov 03 2020: (Start)
Let f(x) = x^4 - x^3 - x^2 - x - 1 and {x1,x2,x3,x4} be the roots of f(x). Then a(n) = (x1*x2*x3)^n + (x1*x2*x4)^n + (x1*x3*x4)^n + (x2*x3*x4)^n.
Let g(y) = y^4 - y^3 + y^2 - y - 1 and {y1,y2,y3,y4} be the roots of g(y). Then a(n) = y1^n + y2^n + y3^n + y4^n. (End)

Michael De Vlieger, Table of n, a(n) for n = 0..9024
Kai Wang, Identities, generating functions and Binet formula for generalized k-nacci sequences, 2020.
A. V. Zarelua, On Matrix Analogs of Fermat's Little Theorem, Mathematical Notes, vol. 79, no. 6, 2006, pp. 783-796. Translated from Matematicheskie Zametki, vol. 79, no. 6, 2006, pp. 840-855.
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,1).

Cf. A000078, A001630, A073817, A074058, A100329.

CoefficientList[Series[(4-3*x+2*x^2-x^3)/(1-x+x^2-x^3-x^4), {x, 0, 50}], x]
LinearRecurrence[{1,-1,1,1},{4,1,-1,1},60] (* Harvey P. Dale, Sep 05 2021 *)

polsym(y^4 - y^3 + y^2 - y - 1, 55) \\ Joerg Arndt, Nov 07 2020

G.f.: (4 - 3*x + 2*x^2 - x^3)/(1 - x + x^2 - x^3 - x^4).
From Kai Wang, Nov 03 2020: (Start)
For n >= 1, a(n) = Sum_{j1,j2,j3,j4>=0; j1+2*j2+3*j3+4*j4=n} (-1)^j2*n*(j1+j2+j3+j4-1)!/(j1!*j2!*j3!*j4!).
For n > 1, a(n) = (-1)^n*(4*A100329(n+1) + 3*A100329(n) + 2*A100329(n-1) + A100329(n-2)). (End)
From Peter Bala, Jan 19 2023: (Start)
a(n) = (-1)^n*A074058(n).
a(n) = trace of M^n, where M is the 4 X 4 matrix [[0, 0, 0, -1], [-1, 0, 0, 1], [0, -1, 0, 1], [0, 0, -1, 1]].
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^k) for positive integers n and r and all primes p. See Zarelua. (End)

A106277 Number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n).

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 1, 1, 2, 0, 2, 2, 0, 1, 0, 1, 1, 1, 1, 2, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 2, 2, 2, 2, 4, 1, 1, 1, 0, 0, 1, 1, 2, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 2, 2, 2, 0, 0, 2, 1, 0, 1, 2, 0, 0, 2, 0, 1, 0, 2, 1, 1, 2, 1, 2, 0, 1, 0, 1, 2, 0, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 1, 2, 1, 3, 0, 0

Offset: 1

T. D. Noe, May 02 2005

nonn

This polynomial is the characteristic polynomial of the Fibonacci and Lucas 4-step sequences, A000078 and A073817. Similar polynomials are treated in Serre's paper. The discriminant of the polynomial is -563 and 563 is the only prime for which the polynomial has 3 distinct zeros. The primes p yielding 4 distinct zeros, A106280, correspond to the periods of the sequences A000078(k) mod p and A073817(k) mod p having length less than p. The Lucas 4-step sequence mod p has one additional prime p for which the period is less than p: the discriminant 563. For this prime, the Fibonacci 4-step sequence mod p has a period of p(p-1).

J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 433.
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A106296 (period of the Lucas 4-step sequences mod prime(n)), A106283 (prime moduli for which the polynomial is irreducible).

Table[p=Prime[n]; cnt=0; Do[If[Mod[x^4-x^3-x^2-x-1, p]==0, cnt++ ], {x, 0, p-1}]; cnt, {n, 150}]

from sympy.abc import x
from sympy import Poly, prime
def A106277(n): return len(Poly(x*(x*(x*(x-1)-1)-1)-1, x, modulus=prime(n)).ground_roots()) # Chai Wah Wu, Mar 29 2024

A192742 Number of matchings in the n-antiprism graph.

Original entry on oeis.org

3, 15, 51, 191, 708, 2631, 9775, 36319, 134943, 501380, 1862875, 6921503, 25716811, 95550687, 355018116, 1319068095, 4900991135, 18209608887, 67657713855, 251381908996, 934008268531, 3470303209839, 12893894812259, 47907203888767, 177998984624708, 661354367518327, 2457258957728079, 9129933787225743, 33922224882718431, 126037862684586116

Offset: 1

Eric W. Weisstein, Jul 09 2011

Antiprism graphs have n >= 3; sequence extended via recurrence to start at n = 1

Eric Weisstein's World of Mathematics, Antiprism Graph
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
Index entries for linear recurrences with constant coefficients, signature (3,3,-1,-1).

Bisection of A073817.

LinearRecurrence[{3, 3, -1, -1}, {3, 15, 51, 191}, 20]
Table[RootSum[1 + # - 3 #^2 - 3 #^3 + #^4 &, #^n &], {n, 20}]
CoefficientList[Series[(3 + 6 x - 3 x^2 - 4 x^3)/(1 - 3 x - 3 x^2 + x^3 + x^4), {x, 0, 20}], x]

a(n) = 3*a(n-1) + 3*a(n-2) - a(n-3) - a(n-4).
G.f.: -x*(-3-6*x+3*x^2+4*x^3)/(1-3*x-3*x^2+x^3+x^4).
a(n) = A073817(2*n). - Greg Dresden, Jan 27 2021

cp's OEIS Frontend

A251656 4-step Fibonacci sequence starting with 1,0,1,0.

Views

Author

Keywords

Links

Crossrefs

Programs

J

Mathematica

Formula

A251703 4-step Fibonacci sequence starting with 1,1,0,0.

Views

Author

Keywords

Links

Crossrefs

Programs

J

Mathematica

Formula

A104622 Indices of prime values of heptanacci-Lucas numbers A104621.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A251654 4-step Fibonacci sequence starting with 0, 1, 1, 0.

Views

Author

Keywords

Links

Crossrefs

Programs

J

Mathematica

Formula

A251655 4-step Fibonacci sequence starting with 0, 1, 1, 1.

Views

Author

Keywords

Links

Crossrefs

Programs

J

Mathematica

Formula

A251704 4-step Fibonacci sequence starting with 1, 1, 0, 1.

Views

Author

Keywords

Links

Crossrefs

Programs

J

Mathematica

Formula

A251705 4-step Fibonacci sequence starting with 1, 1, 1, 0.

Views

Author

Keywords

Links

Crossrefs

Programs

J

Mathematica

Formula

A073937 a(n) = a(n-1) - a(n-2) + a(n-3) + a(n-4), a(0)=4, a(1)=1, a(2)=-1, a(3)=1.

Views

Author

Keywords

Comments

Links

Crossrefs