A074058 -id:A074058 - 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

A075129 Binomial transform of reflected tetranacci numbers A074058: a(n)=Sum((-1)^k Binomial(n,k)*A074058(k),(k=0,..,n)).

Original entry on oeis.org

4, 5, 5, 5, 13, 50, 155, 390, 861, 1805, 3850, 8640, 20167, 47520, 110780, 254450, 579149, 1316485, 3003095, 6878765, 15790278, 36245235, 83101760, 190322935, 435678591, 997445500, 2284365660, 5233190405, 11989714652, 27467989310

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Sep 03 2002

N. J. A. Sloane, Transforms
Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -3).

Cf. A074058, A075128.

CoefficientList[Series[(4-15*z+20*z^2-10*z^3)/(1-5*z+10*z^2-10*z^3+3*z^4), {z, 0, 30}], z]
LinearRecurrence[{5,-10,10,-3},{4,5,5,5},30] (* Harvey P. Dale, Jan 26 2025 *)

a(n)=5a(n-1)-10a(n-2)+10a(n-3)-3a(n-4), a(0)=4, a(1)=5, a(2)=5, a(3)=5. G.f.: (4 - 15*z + 20*z^2 - 10*z^3)/(1 - 5*z + 10*z^2 - 10*z^3 + 3*z^4).

A057597 a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=0, a(1)=0, a(2)=1.

Original entry on oeis.org

0, 0, 1, -1, 0, 2, -3, 1, 4, -8, 5, 7, -20, 18, 9, -47, 56, 0, -103, 159, -56, -206, 421, -271, -356, 1048, -963, -441, 2452, -2974, 81, 5345, -8400, 3136, 10609, -22145, 14672, 18082, -54899, 51489, 21492, -127880, 157877, -8505, -277252, 443634, -174887, -545999, 1164520, -793408, -917111

Offset: 0

N. J. A. Sloane, Oct 06 2000

Reflected (A074058) tribonacci numbers A000073: A000073(n) = a(1-n).
There is an alternative way to produce this sequence, from A000073, which is 0,0,1,1,2,4,7,13,24,44,... Call this {b(n)}. Taking x1 = (b(2))^2 - b(1)*b(3) = 0; x2 = (b(3))^2 - b(2)*b(4) = 1; x3 = (b(4))^2 - b(3)*b(5) = -1; x4 = 0, x5 = 2, we generate (0),0,1,-1,0,2,-3,1. - John McNamara, Jan 02 2004
Pisano period lengths: 1, 4, 13, 8, 31, 52, 48, 16, 39, 124, 110, 104, 168, 48, 403, 32, 96, 156, 360, 248, ... - R. J. Mathar, Aug 10 2012
The negative powers of the tribonacci constant t = A058265 are t^(-n) = a(n+1)*t^2 + b(n)*t + a(n+2)*1, for n >= 0, with b(n) = A319200(n) = -(a(n+1) - a(n)), for n >= 0. 1/t =  t^2 - t - 1 = A192918. See the example in A319200 for the first powers. - Wolfdieter Lang, Oct 23 2018

Petho Attila, Posting to Number Theory List (NMBRTHRY(AT)LISTSERV.NODAK.EDU), Oct 06 2000.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Milan Janjic, Recurrence Relations and Determinants, arXiv preprint arXiv:1112.2466 [math.CO], 2011.
Milan Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 15 (2012), Article 12.3.5. - From _N. J. A. Sloane_, Sep 16 2012.
Ronald C. King, Generating functions for some series of characters of classical Lie groups, arXiv:2303.00576 [math.CO], 2023, p. 9.
A. G. Shannon, H. M. Srivastava, and József Sàndor, Towards a new generalized Simson's identity, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 479-490. See p. 482.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

Cf. A000073, A058265, A319200. First differences of A077908.

a:=[0,0,1];;  for n in [4..55] do a[n]:=-a[n-1]-a[n-2]+a[n-3]; od; a; # Muniru A Asiru, Oct 23 2018

a057597 n = a057597_list !! n
a057597_list = 0 : 0 : 1 : zipWith3 (\x y z -> - x - y + z)
               (drop 2 a057597_list) (tail a057597_list) a057597_list
-- Reinhard Zumkeller, Oct 07 2012

seq(coeff(series(x^2/(1+x+x^2-x^3),x,n+1), x, n), n = 0 .. 50); # Muniru A Asiru, Oct 23 2018

CoefficientList[Series[x^2/(1+x+x^2-x^3), {x, 0, 50}], x]

{a(n) = polcoeff( if( n<0, x / ( 1 - x - x^2 - x^3), x^2 / ( 1 + x + x^2 - x^3) ) + x*O(x^abs(n)), abs(n))} /* Michael Somos, Sep 03 2007 */

G.f.: x^2/(1+x+x^2-x^3).
G.f.: Q(0)*x^2/2, where Q(k) = 1 + 1/(1 - x*(4*k+1 + x - x^2)/( x*(4*k+3 + x - x^2) - 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 09 2013
G.f. -x*T(1/x), where T is the g.f. of A000073. - Wolfdieter Lang, Oct 26 2018

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A073145 a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=3, a(1)=-1, a(2)=-1.

Original entry on oeis.org

3, -1, -1, 5, -5, -1, 11, -15, 3, 23, -41, 21, 43, -105, 83, 65, -253, 271, 47, -571, 795, -177, -1189, 2161, -1149, -2201, 5511, -4459, -3253, 13223, -14429, -2047, 29699, -42081, 10335, 61445, -113861, 62751, 112555, -289167, 239363, 162359, -690889, 767893, 85355

Offset: 0

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

Previous name was: Sum of the determinants of the principal minors of 2nd order of n-th power of Tribomatrix: first row (1, 1, 0); second row (1, 0, 1); third row (1, 0, 0).
a(n) is related to the generalized Lucas numbers S(n). For instance, 2S(n) = a(n)^2 - a(2n).
a(n) is also the reflected (A074058) sequence of the generalized tribonacci sequence (A001644).

			G.f. = 3 - x - x^2 + 5*x^3 - 5*x^4 - x^5 + 11*x^6 - 15*x^7 + 3*x^8 + 23*x^9 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.
Curtis Cooper, S. Miller, Peter J. C. Moses, M. Sahin, and T. Thanatipanonda, On Identities of Ruggles, Horadam, Howard, and Young, Preprint 2016.
Kai Wang, Identities for generalized enneanacci numbers, Generalized Fibonacci Sequences (2020).
Index entries for linear recurrences with constant coefficients, signature (-1,-1,1).

Cf. A000073, A001644, A057597.

I:=[3,-1,-1]; [n le 3 select I[n] else -Self(n-1)-Self(n-2)+Self(n-3): n in [1..50]]; // Vincenzo Librandi, Aug 17 2013

A = Table[0, {3}, {3}]; A[[1, 1]] = 1; A[[1, 2]] = 1; A[[2, 1]] = 1; A[[2, 3]] = 1; A[[3, 1]] = 1; For[i = 1; t = IdentityMatrix[3], i < 50, i++, t = t.A; Print[t[[2, 2]]*t[[3, 3]] - t[[2, 3]]*t[[3, 2]] + t[[1, 1]]*t[[3, 3]] - t[[1, 3]]*t[[3, 1]] + t[[1, 1]]*t[[2, 2]] - t[[1, 2]]*t[[2, 1]]]]
LinearRecurrence[{-1, -1, 1}, {3, -1, -1}, 50] (* Vincenzo Librandi, Aug 17 2013 *)
nxt[{a_,b_,c_}]:={b,c,a-b-c}; NestList[nxt,{3,-1,-1},50][[;;,1]] (* Harvey P. Dale, Jun 16 2024 *)

{a(n) = if( n<0, polsym(1 + x+ x^2 - x^3, -n)[-n+1], polsym(1 - x - x^2 - x^3, n)[n+1])}; /* Michael Somos, Dec 17 2016 */

a(n)=([0,1,0; 0,0,1; 1,-1,-1]^n*[3;-1;-1])[1,1] \\ Charles R Greathouse IV, Feb 07 2017

((3+2*x+x^2)/(1+x+x^2-x^3)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019

a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=3, a(1)=-1, a(2)=-1.
O.g.f.: (3 + 2*x + x^2)/(1 + x + x^2 - x^3).
a(n) = -T(n)^2 + 2*T(n-1)^2 + 3*T(n-2)^2 - 2*T(n)*T(n-1) + 2*T(n)*T(n-2) + 4*T(n-1)*T(n-2), where T(n) are tribonacci numbers (A000073).
a(n) = 3*A057597(n+2) + 2*A057597(n+1) + A057597(n). - R. J. Mathar, Jun 06 2011
From Peter Bala, Jun 29 2015: (Start)
a(n) = alpha^n + beta^n + gamma^n, where alpha, beta and gamma are the roots of 1 - x - x^2 - x^3 = 0.
x^2*exp( Sum_{n >= 1} a(n)*x^n/n ) = x^2 - x^3 + 2*x*5 - 3*x^6 + x^7 + ... is the o.g.f. for A057597. (End)
a(n) = A001644(-n) for all n in Z. - Michael Somos, Dec 17 2016

Better name by Joerg Arndt, Aug 17 2013
More terms from Vincenzo Librandi, Aug 17 2013
Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

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)

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

A074585 a(n)= Sum_{j=0..floor(n/2)} A073145(2j + q), where q = 2(n/2 - floor(n/2)).

Original entry on oeis.org

3, -1, 2, 4, -3, 3, 8, -12, 11, 11, -30, 32, 13, -73, 96, -8, -157, 263, -110, -308, 685, -485, -504, 1676, -1653, -525, 3858, -4984, 605, 8239, -13824, 6192, 15875, -35889, 26210, 25556, -87651, 88307, 24904, -200860, 264267, -38501, -426622

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Aug 28 2002

a(n) is the convolution of A073145(n) with the sequence (1,0,1,0,1,0, ...).

a(n) is also the sum of the reflected (see A074058) sequence of the generalized tribonacci sequence (A001644).

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

Cf. A073145, A074058, A001644, A074331, A074392, A074475.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (3+2*x+x^2)/(1+x-2*x^3-x^4+x^5) )); // G. C. Greubel, Apr 13 2019

CoefficientList[ Series[(3+2*x+x^2)/(1+x-2*x^3-x^4+x^5), {x, 0, 50}], x]

my(x='x+O('x^50)); Vec((3+2*x+x^2)/(1+x-2*x^3-x^4+x^5)) \\ G. C. Greubel, Apr 13 2019

((3+2*x+x^2)/(1+x-2*x^3-x^4+x^5)).series(x, 50).coefficients(x, sparse=False) # G. C. Greubel, Apr 13 2019

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

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

More terms from Robert G. Wilson v, Aug 29 2002

A331890 a(n) = -a(n-1) - a(n-2) + 2*a(n-3) with a(0)=3, a(1)=-1, a(2)=-1.

Original entry on oeis.org

3, -1, -1, 8, -9, -1, 26, -43, 15, 80, -181, 131, 210, -703, 755, 368, -2529, 3671, -406, -8323, 16071, -8560, -24157, 64859, -57822, -55351, 242891, -303184, -50409, 839375, -1395334, 455141, 2618943, -5864752, 4156091, 6946547, -22832142, 24197777

Offset: 0

Wojciech Florek, Jan 30 2020

a(n) is the reflected sequence (cf. A074058) of the generalized tribonacci sequence b(n) with b(0) = 3 and b(n) = A186575(n-1) for n > 0.

Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math/0210201 [math.CO], 2002.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,2).

Cf. A074058, A186575.

a:=[3,-1,-1]; [n le 3 select a[n] else -Self(n-1)-Self(n-2)+2*Self(n-3):n in [1..30]]; // Marius A. Burtea, Feb 02 2020

LinearRecurrence[{-1,-1,2},{3,-1,-1},38] (* Stefano Spezia, Jan 31 2020 *)

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

a(n) = 3*A077975(n)+2*A077975(n-1)+A077975(n-2). - R. J. Mathar, Feb 28 2020

Definition clarified by N. J. A. Sloane, Apr 23 2020

cp's OEIS Frontend

A074062 Reflected (see A074058) pentanacci numbers A074048.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

Extensions

A075129 Binomial transform of reflected tetranacci numbers A074058: a(n)=Sum((-1)^k Binomial(n,k)*A074058(k),(k=0,..,n)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A057597 a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=0, a(1)=0, a(2)=1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A073145 a(n) = -a(n-1) - a(n-2) + a(n-3), a(0)=3, a(1)=-1, a(2)=-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

Extensions

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

Programs

Mathematica

PARI

Formula

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)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A074585 a(n)= Sum_{j=0..floor(n/2)} A073145(2*j + q), where q = 2*(n/2 - floor(n/2)).

Views

Author

A074585 a(n)= Sum_{j=0..floor(n/2)} A073145(2j + q), where q = 2(n/2 - floor(n/2)).