A073817 -id:A073817 - cp's OEIS frontend

A074081 Sum of determinants of 3rd-order principal minors of powers of inverse of tetramatrix ((1,1,0,0),(1,0,1,0),(1,0,0,1),(1,0,0,0)).

4, -1, 3, -7, 15, -26, 51, -99, 191, -367, 708, -1365, 2631, -5071, 9775, -18842, 36319, -70007, 134943, -260111, 501380, -966441, 1862875, -3590807, 6921503, -13341626, 25716811, -49570747, 95550687, -184179871, 355018116, -684319421, 1319068095, -2542585503, 4900991135

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 19 2002

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

Cf. A073817.

CoefficientList[Series[(4+3*x-2*x^2+x^3)/(1+x-x^2+x^3-x^4), {x, 0, 40}], x]

a(n) = (-1)^n*T(n), where T(n) are the generalized tetranacci numbers A073817.
a(n) = -a(n-1)+a(n-2)-a(n-3)+a(n-4), a(0)=4, a(1)=-1, a(2)=3, a(3)=-7.
G.f.: (4+3x-2x^2+x^3)/(1+x-x^2+x^3-x^4).

A106280 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has 4 distinct zeros.

Original entry on oeis.org

137, 179, 653, 859, 991, 1279, 1601, 1609, 2089, 2437, 2591, 2693, 2789, 2897, 3701, 3823, 3847, 4451, 4691, 4751, 4919, 5431, 5479, 5807, 5903, 5953, 6203, 6421, 6781, 6917, 7253, 7867, 8317, 9187, 9277, 9533, 9629, 9767, 9907, 9967, 10009, 10079

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. The periods of the sequences A000078(k) mod p and A073817(k) mod p have length less than p.

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

Cf. A106277 (number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n)), A106296 (period of 4-step sequence mod prime(n)).

t=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, 1600}];Prime[Flatten[Position[t, 4]]]

A106283 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has no zeros.

Original entry on oeis.org

2, 5, 11, 13, 31, 43, 53, 79, 83, 89, 97, 103, 109, 131, 139, 151, 197, 199, 229, 233, 239, 251, 257, 271, 283, 313, 317, 347, 359, 367, 379, 389, 433, 443, 461, 479, 487, 521, 569, 571, 577, 593, 599, 601, 617, 631, 641, 643, 647, 659, 673, 677, 719, 769, 797

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.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A106277 (number of distinct zeros of x^4-x^3-x^2-x-1 mod prime(n)), A106296 (period of Lucas 4-step sequence mod prime(n)), A003631 (primes p such that x^2-x-1 is irreducible in mod p).

Res:= NULL: count:= 0: p:= 0:
P:= x^4 - x^3 - x^2 - x - 1:
while count < 100 do
  p:= nextprime(p);
  if [msolve(P,p)] = [] then
    Res:= Res, p; count:= count+1;
  fi
od:
Res; # Robert Israel, Mar 13 2024

t=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, 200}];Prime[Flatten[Position[t, 0]]]

from itertools import islice
from sympy import Poly, nextprime
from sympy.abc import x
def A106283_gen(): # generator of terms
    p = 2
    while True:
        if len(Poly(x*(x*(x*(x-1)-1)-1)-1, x, modulus=p).ground_roots())==0:
            yield p
        p = nextprime(p)
A106283_list = list(islice(A106283_gen(),20)) # Chai Wah Wu, Mar 14 2024

Name corrected by Robert Israel, Mar 13 2024

A127208 Union of all n-step Lucas sequences, that is, all sequences s(1-n) = s(2-n) = ... = s(-1) = -1, s(0) = n and for k > 0, s(k) = s(k-1) + ... + s(k-n).

Original entry on oeis.org

1, 3, 4, 7, 11, 15, 18, 21, 26, 29, 31, 39, 47, 51, 57, 63, 71, 76, 99, 113, 120, 123, 127, 131, 191, 199, 223, 239, 241, 247, 255, 322, 367, 439, 443, 475, 493, 502, 511, 521, 708, 815, 843, 863, 943, 983, 1003, 1013, 1023, 1364, 1365, 1499, 1695, 1871, 1959

Offset: 1

T. D. Noe, Jan 09 2007

nonn

Noe and Post conjectured that the only positive terms that are common to any two distinct n-step Lucas sequences are the Mersenne numbers (A001348) that begin each sequence and 7 and 11 (in 2- and 3-step) and 5071 (in 3- and 4-step). The intersection of this sequence with the union of all the n-step Fibonacci sequences (A124168) appears to consist of 4, 21, 29, the Mersenne numbers 2^n-1 for all n and the infinite set of Eulerian numbers in A127232.

T. D. Noe, Table of n, a(n) for n=1..1000
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. A227885.

LucasSequence[n_,kMax_] := Module[{a,s,lst={}}, a=Join[Table[ -1,{n-1}],{n}]; While[s=Plus@@a; a=RotateLeft[a]; a[[n]]=s; s<=kMax, AppendTo[lst,s]]; lst]; nn=10; t={}; Do[t=Union[t,LucasSequence[n,2^(nn+1)]], {n,2,nn}]; t

Union(A000032, A001644, A073817, A074048, A074584, A104621, A105754, A105755,...)

A247505 Generalized Lucas numbers: square array A(n,k) read by antidiagonals, A(n,k)=(-1)^(k+1)k[x^k](-log((1+sum_{j=1..n}(-1)^(j+1)*x^j)^(-1))), (n>=0, k>=0).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 3, 1, 0, 0, 1, 3, 4, 1, 0, 0, 1, 3, 7, 7, 1, 0, 0, 1, 3, 7, 11, 11, 1, 0, 0, 1, 3, 7, 15, 21, 18, 1, 0, 0, 1, 3, 7, 15, 26, 39, 29, 1, 0, 0, 1, 3, 7, 15, 31, 51, 71, 47, 1, 0, 0, 1, 3, 7, 15, 31, 57, 99, 131, 76, 1, 0

Offset: 0

Peter Luschny, Nov 02 2014

			n\k[0][1][2][3] [4] [5] [6]  [7]  [8]  [9]  [10]  [11]  [12]
[0] 0, 0, 0, 0,  0,  0,  0,   0,   0,   0,    0,    0,    0
[1] 0, 1, 1, 1,  1,  1,  1,   1,   1,   1,    1,    1,    1
[2] 0, 1, 3, 4,  7, 11, 18,  29,  47,  76,  123,  199,  322 [A000032]
[3] 0, 1, 3, 7, 11, 21, 39,  71, 131, 241,  443,  815, 1499 [A001644]
[4] 0, 1, 3, 7, 15, 26, 51,  99, 191, 367,  708, 1365, 2631 [A073817]
[5] 0, 1, 3, 7, 15, 31, 57, 113, 223, 439,  863, 1695, 3333 [A074048]
[6] 0, 1, 3, 7, 15, 31, 63, 120, 239, 475,  943, 1871, 3711 [A074584]
[7] 0, 1, 3, 7, 15, 31, 63, 127, 247, 493,  983, 1959, 3903 [A104621]
[8] 0, 1, 3, 7, 15, 31, 63, 127, 255, 502, 1003, 2003, 3999 [A105754]
[.] .  .  .  .   .   .   .    .    .    .     .     .     .
oo] 0, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095 [A000225]
'
As a triangular array, starts:
0,
0, 0,
0, 1, 0,
0, 1, 1, 0,
0, 1, 3, 1, 0,
0, 1, 3, 4, 1, 0,
0, 1, 3, 7, 7, 1,  0,
0, 1, 3, 7, 11, 11, 1, 0,
0, 1, 3, 7, 15, 21, 18, 1, 0,
0, 1, 3, 7, 15, 26, 39, 29, 1, 0,

Cf. A247506, A000225, A000032, A001644, A073817, A074048, A074584, A104621, A105754.

Cf. A125127.

A := proc(n, k) f := -log((1+add((-1)^(j+1)*x^j, j=1..n))^(-1));
(-1)^(k+1)*k*coeff(series(f,x,k+2),x,k) end:
seq(print(seq(A(n,k), k=0..12)), n=0..8);

A[n_, k_] := Module[{f, x}, f = -Log[(1+Sum[(-1)^(j+1) x^j, {j, 1, n}] )^(-1)]; (-1)^(k+1) k SeriesCoefficient[f, {x, 0, k}]];
Table[A[n-k, k], {n, 0, 12}, {k, 0, n}] (* Jean-François Alcover, Jun 28 2019, from Maple *)

A074193 Sum of determinants of 2nd order principal minors of powers of the matrix ((1,1,0,0),(1,0,1,0),(1,0,0,1),(1,0,0,0)).

Original entry on oeis.org

6, -1, -3, -1, 17, -16, -15, 13, 81, -127, -58, 175, 329, -885, -31, 1424, 833, -5543, 2181, 9233, -2298, -31025, 27893, 49495, -54879, -150416, 245697, 204965, -526887, -570895, 1801670, 407711, -3882303, -946397, 11542929, -3442672, -24121039, 10317745, 64959629, -56727711, -127083514

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 20 2002

From Kai Wang, Oct 21 2020: (Start)
Let f(x) = x^4 - x^3 - x^2 - x - 1 be the characteristic polynomial for Tetranacci numbers (A000078). Let {x1,x2,x3,x4} be the roots of f(x). Then a(n) = (x1*x2)^n + (x1*x3)^n + (x1*x4)^n + (x2*x3)^n + (x2*x4)^n + (x3*x4)^n.
Let g(y) = y^6 + y^5 + 2*y^4 + 2*y^3 - 2*y^2 + y - 1 be the characteristic polynomial for a(n). Let {y1,y2,y3,y4,y5,y6} be the roots of g(y). Then a(n) = y1^n + y2^n + y3^n + y4^n + y5^n + y6^n. (End)

Michael De Vlieger, Table of n, a(n) for n = 0..5052
Kai Wang, Identities, generating functions and Binet formula for generalized k-nacci sequences, 2020.
Index entries for linear recurrences with constant coefficients, signature (-1,-2,-2,2,-1,1).

Cf. A073817, A073937, A000078, A074081.

CoefficientList[Series[(6+5*x+8*x^2+6*x^3-4*x^4+x^5)/(1+x+2*x^2+2*x^3-2*x^4+x^5-x^6), {x, 0, 50}], x]
LinearRecurrence[{-1,-2,-2,2,-1,1},{6,-1,-3,-1,17,-16},50] (* Harvey P. Dale, Sep 06 2025 *)

polsym(x^6 + x^5 + 2*x^4 + 2*x^3 - 2*x^2 + x - 1,44) \\ Joerg Arndt, Oct 22 2020

a(n) = -a(n-1)-2a(n-2)-2a(n-3)+2a(n-4)-a(n-5)+a(n-6).
G.f.: (6+5x+8x^2+6x^3-4x^4+x^5)/(1+x+2x^2+2x^3-2x^4+x^5-x^6).
abs(a(n)) = abs(A074453(n)). - Joerg Arndt, Oct 22 2020

A227885 Primes in the union of all n-step Lucas sequences.

Original entry on oeis.org

2, 3, 7, 11, 29, 31, 47, 71, 113, 127, 131, 191, 199, 223, 239, 241, 367, 439, 443, 521, 863, 983, 1013, 1499, 1871, 2003, 2207, 3571, 6553, 8087, 8191, 9349, 16369, 32647, 32707, 36319, 63487, 65407, 65519, 122401, 126719, 131071, 196331, 260111, 524287

Offset: 1

Robert Price, Oct 25 2013

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000 (first 383 terms from Robert Price)
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. A127208, A000032, A001644, A073817, A074048, A074584, A104621, A105754, A105755.

plst={2}; plimit=10^39; For[n=2, n<=3+Log[2,plimit], n++, llst={}; For[i=1, i

2 and the primes in A127208.

A074453 Sum of determinants of 2nd order principal minors of powers of inverse of the matrix ((1,1,0,0),(1,0,1,0),(1,0,0,1),(1,0,0,0)).

Original entry on oeis.org

6, 1, -3, 1, 17, 16, -15, -13, 81, 127, -58, -175, 329, 885, -31, -1424, 833, 5543, 2181, -9233, -2298, 31025, 27893, -49495, -54879, 150416, 245697, -204965, -526887, 570895, 1801670, -407711, -3882303, 946397, 11542929, 3442672, -24121039, -10317745, 64959629, 56727711, -127083514

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 22 2002

a(n) is the reflected (A074058) sequence of sequence A074193.

Index entries for linear recurrences with constant coefficients, signature (1, -2, 2, 2, 1, 1).

Cf. A073817, A073937, A074058, A074081, A074193.

CoefficientList[Series[(6-5*x+8*x^2-6*x^3-4*x^4-x^5)/(1-x+2*x^2-2*x^3-2*x^4-x^5-x^6), {x, 0, 40}], x]
LinearRecurrence[{1,-2,2,2,1,1},{6,1,-3,1,17,16},50] (* Harvey P. Dale, Mar 16 2012 *)

a(n)=a(n-1)-2a(n-2)+2a(n-3)+2a(n-4)+a(n-5)+a(n-6).
G.f.: (6-5x+8x^2-6x^3-4x^4-x^5)/(1-x+2x^2-2x^3-2x^4-x^5-x^6).
abs(a(n)) = abs(A074193(n)). - Joerg Arndt, Oct 22 2020

A106627 Product L(n)*L_4(n), where L(n) are Lucas numbers and L_4(n) are Lucas 4-step numbers.

Original entry on oeis.org

8, 1, 9, 28, 105, 286, 918, 2871, 8977, 27892, 87084, 271635, 847182, 2641991, 8240325, 25700488, 80156033, 249994997, 779700654, 2431777739, 7584375260

Offset: 0

Jonathan Vos Post, May 11 2005

a(n) is semiprime iff n is an element of A001606 (an index of a prime Lucas number) and an element of A104577 (an index of a prime Lucas 4-step number). The only known such are n = 2, 8, 16, 19, 71, (through 145858).

			a(0) = 8 because L(0) * L_4(0) = 2 * 4.
a(1) = 1 because L(1) * L_4(1) = 1 * 1.
a(2) = 9 because L(2) * L_4(2) = 3 * 3.
a(3) = 28 because L(3) * L_4(3) = 4 * 7.
a(4) = 105 because L(4) * L_4(4) = 7 * 15.
a(5) = 286 because L(5) * L_4(5) = 11 * 26.
a(6) = 918 because L(6) * L_4(6) = 18 * 51.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,5,9,3,-2,1,-1).

Cf. A000032, A000040, A001358, A001606, A073446, A073817, A104576.

a:=[8,1,9,28,105,286,918,2871];; for n in [9..30] do a[n]:=a[n-1] +4*a[n-2]+5*a[n-3]+9*a[n-4]+3*a[n-5]-2*a[n-6]+a[n-7]-a[n-8]; od; a; # G. C. Greubel, Feb 19 2019

I:=[8,1,9,28,105,286,918,2871]; [n le 8 select I[n] else Self(n-1)+4*Self(n-2)+5*Self(n-3)+9*Self(n-4)+3*Self(n-5)-2*Self(n-6) + Self(n-7)-Self(n-8): n in [1..30]]; // G. C. Greubel, Feb 19 2019

LinearRecurrence[{1,4,5,9,3,-2,1,-1}, {8,1,9,28,105,286,918,2871}, 40] (* G. C. Greubel, Feb 19 2019 *)

my(x='x+O('x^40)); Vec((8-7*x-24*x^2-25*x^3-36*x^4-9*x^5+4*x^6 -x^7)/(1-x-4*x^2-5*x^3-9*x^4-3*x^5+2*x^6-x^7+x^8)) \\ G. C. Greubel, Feb 19 2019

((8-7*x-24*x^2-25*x^3-36*x^4-9*x^5+4*x^6-x^7)/(1-x-4*x^2-5*x^3 -9*x^4-3*x^5+2*x^6-x^7+x^8)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Feb 19 2019

a(n) = A000032(n) * A073817(n).
a(n) = +a(n-1) +4*a(n-2) +5*a(n-3) +9*a(n-4) +3*a(n-5) -2*a(n-6) +a(n-7) -a(n-8). - R. J. Mathar, Dec 22 2010
G.f.: (8 -7*x -24*x^2 -25*x^3 -36*x^4 -9*x^5 +4*x^6 -x^7) / (1 -x -4*x^2 -5*x^3 -9*x^4 -3*x^5 +2*x^6 -x^7 +x^8). - Colin Barker, Jun 17 2012

A113294 First differences of Lucas 4-step numbers.

Original entry on oeis.org

1, 3, 4, 8, 11, 12, 19, 22, 23, 25, 36, 44, 47, 48, 73, 84, 92, 95, 96, 140, 165, 176, 184, 187, 188, 268, 316, 341, 352, 360, 363, 364, 517, 609, 657, 682, 693, 701, 704, 705, 998, 1174, 1266, 1314, 1339, 1350, 1358, 1361, 1362, 1923, 2264, 2440, 2532, 2580

Offset: 1

Jonathan Vos Post, Oct 23 2005

Lucas 4-step numbers are also known as "Tetranacci Lucas numbers" or "Tetranacci numbers with different initial conditions" in A073817. Primes in this sequence are A113295. In this sequence are: 13340261 = 11 * 19 * 29 * 31 * 71 is a product of 5 distinct 2-digit primes; 95550683 = 269 * 593 * 599 is a product of 3 distinct 3-digit primes.

			a(0) = 1 because A073817(0)-A001644(2) = 4 - 3 = 1.
a(1) = 3 because A073817(3)-A001644(0) = 7 - 4 = 3.
a(2) = 4 because A073817(3)-A001644(2) = 7 - 3 = 4.
a(3) = 8 because A073817(4)-A001644(3) = 15 - 7 = 8.
a(122) = 70000 because A073817(17)-A001644(3) = 70007 - 7 = 70000.

Eric Weisstein's World of Mathematics, Lucas n-Step Number.
Noe, T. D. and Post, J. V., Primes in Fibonacci n-step and Lucas n-Step Sequences." J. Integer Seq. 8, Article 05.4.4, 2005.

Cf. A000040, A073817, A113188-A113194, A113238, A113239, A113244, A113293, A113295.

{a(n)} = { | A073817(i) - A073817(j) | such that i>=j }

cp's OEIS Frontend

A074081 Sum of determinants of 3rd-order principal minors of powers of inverse of tetramatrix ((1,1,0,0),(1,0,1,0),(1,0,0,1),(1,0,0,0)).

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A106280 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has 4 distinct zeros.

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A106283 Primes p such that the polynomial x^4-x^3-x^2-x-1 mod p has no zeros.

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A127208 Union of all n-step Lucas sequences, that is, all sequences s(1-n) = s(2-n) = ... = s(-1) = -1, s(0) = n and for k > 0, s(k) = s(k-1) + ... + s(k-n).

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A247505 Generalized Lucas numbers: square array A(n,k) read by antidiagonals, A(n,k)=(-1)^(k+1)*k*[x^k](-log((1+sum_{j=1..n}(-1)^(j+1)*x^j)^(-1))), (n>=0, k>=0).

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A074193 Sum of determinants of 2nd order principal minors of powers of the matrix ((1,1,0,0),(1,0,1,0),(1,0,0,1),(1,0,0,0)).

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A227885 Primes in the union of all n-step Lucas sequences.

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A074453 Sum of determinants of 2nd order principal minors of powers of inverse of the matrix ((1,1,0,0),(1,0,1,0),(1,0,0,1),(1,0,0,0)).

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A247505 Generalized Lucas numbers: square array A(n,k) read by antidiagonals, A(n,k)=(-1)^(k+1)k[x^k](-log((1+sum_{j=1..n}(-1)^(j+1)*x^j)^(-1))), (n>=0, k>=0).