A014017 -id:A014017 - cp's OEIS frontend

A191497 a(n+1) = 2*a(n) + A014017(n+5), a(0) = 0.

0, 0, 0, 0, 1, 2, 4, 8, 15, 30, 60, 120, 241, 482, 964, 1928, 3855, 7710, 15420, 30840, 61681, 123362, 246724, 493448, 986895, 1973790, 3947580, 7895160, 15790321, 31580642, 63161284, 126322568, 252645135

Offset: 0

Paul Curtz, Jun 03 2011

Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1,2).

Cf. A014017, A133145.

A191497 := proc(n): if n=0 then 0 else A191497(n) := 2*A191497(n-1) + A014017(n+4) fi: end: A014017 := proc(n): (1/8)*(-(n mod 8)-((n+3) mod 8)+((n+4) mod 8)+((n+7) mod 8)) end: seq(A191497(n),n=0..32); # Johannes W. Meijer, Jun 28 2011

LinearRecurrence[{2,0,0,-1,2},{0,0,0,0,1},40] (* Harvey P. Dale, Apr 19 2013 *)

a(n+4) = 2^n - a(n).
a(n) = 2*a(n-1) - a(n-4) + 2*a(n-5).
a(4*n+4) = 16*a(4*n) + (-1)^n.
From R. J. Mathar, Jun 23 2011: (Start)
G.f.: -x^4 / ((2*x-1)*(x^4+1)).
a(n) = (2^n - (-1)^floor(n/4)*A133145(n))/17. (End)

A301712 Coordination sequence for node of type V1 in "usm" 2-D tiling (or net).

Original entry on oeis.org

1, 5, 10, 16, 22, 27, 33, 38, 43, 49, 53, 59, 65, 70, 77, 81, 86, 92, 96, 103, 108, 113, 120, 124, 130, 135, 139, 146, 151, 157, 163, 167, 173, 178, 183, 189, 194, 200, 206, 211, 216, 221, 226, 232, 238, 243, 249, 254, 259, 265, 269, 275, 281, 286, 293, 297, 302, 308, 312, 319, 324, 329, 336, 340

Offset: 0

N. J. A. Sloane, Mar 26 2018

Linear recurrence and g.f. confirmed by Shutov/Maleev link. - Ray Chandler, Aug 30 2023

Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987. See Table 2.2.1, page 67, 2nd row, 2nd tiling.

Ray Chandler, Table of n, a(n) for n = 0..1000
Brian Galebach, Collection of n-Uniform Tilings. See Number 16 from the list of 20 2-uniform tilings.
Brian Galebach, k-uniform tilings (k <= 6) and their A-numbers
Reticular Chemistry Structure Resource (RCSR), The usm tiling (or net)
Anton Shutov and Andrey Maleev, Coordination sequences of 2-uniform graphs, Z. Kristallogr., 235 (2020), 157-166. See supplementary material, krb, vertex u_1.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,-1,2,-1,0,0,1,-1).

Cf. A301714.

Coordination sequences for the 20 2-uniform tilings in the order in which they appear in the Galebach catalog, together with their names in the RCSR database (two sequences per tiling): #1 krt A265035, A265036; #2 cph A301287, A301289; #3 krm A301291, A301293; #4 krl A301298, A298024; #5 krq A301299, A301301; #6 krs A301674, A301676; #7 krr A301670, A301672; #8 krk A301291, A301293; #9 krn A301678, A301680; #10 krg A301682, A301684; #11 bew A008574, A296910; #12 krh A301686, A301688; #13 krf A301690, A301692; #14 krd A301694, A219529; #15 krc A301708, A301710; #16 usm A301712, A301714; #17 krj A219529, A301697; #18 kre A301716, A301718; #19 krb A301720, A301722; #20 kra A301724, A301726.

LinearRecurrence[{1,0,0,-1,2,-1,0,0,1,-1},{1,5,10,16,22,27,33,38,43,49,53},100] (* Paolo Xausa, Nov 16 2023 *)

G.f.: -(-x^10-4*x^9-5*x^8-6*x^7-7*x^6-8*x^5-7*x^4-6*x^3-5*x^2-4*x-1)/(x^10-x^9+x^6-2*x^5+x^4-x+1). - N. J. A. Sloane, Mar 29 2018

Equivalent conjecture: 5*a(n) = 27*n -b(n) -5*A014017(n-2) for n>0, where b(n) = 2,-1,1,-2,0 (5-periodic) for n>=1. - R. J. Mathar, Mar 30 2018

a(11)-a(100) from Davide M. Proserpio, Mar 28 2018

A188510 Expansion of x*(1 + x^2)/(1 + x^4) in powers of x.

Original entry on oeis.org

0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0

Offset: 0

Michael Somos, Apr 10 2011

			G.f. = x + x^3 - x^5 - x^7 + x^9 + x^11 - x^13 - x^15 + x^17 + x^19 - x^21 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Rational Function Multiplicative Coefficients
Eric Weisstein's World of Mathematics, Kronecker Symbol.
Wikipedia, Kronecker Symbol.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Cf. A002325, A057077, A091337, A101455, A257170.

m:=60; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x^2)/(1+x^4))); // G. C. Greubel, Aug 02 2018

Table[KroneckerSymbol[-2, n], {n, 0, 104}] (* Wolfdieter Lang, May 30 2013 *)
a[ n_] := Mod[n, 2] (-1)^Quotient[ n, 4]; (* Michael Somos, Apr 17 2015 *)
CoefficientList[Series[x*(1+x^2)/(1+x^4), {x, 0, 60}], x] (* G. C. Greubel, Aug 02 2018 *)
LinearRecurrence[{0,0,0,-1},{0,1,0,1},120] (* or *) PadRight[{},120,{0,1,0,1,0,-1,0,-1}] (* Harvey P. Dale, Jan 25 2023 *)

PARI
```
{a(n) = (n%2) * (-1)^(n\4)};
```

x='x+O('x^60); concat([0], Vec(x*(1+x^2)/(1+x^4))) \\ G. C. Greubel, Aug 02 2018

Euler transform of length 8 sequence [0, 1, 0, -2, 0, 0, 0, 1].
a(n) is multiplicative with a(2^e) = 0^e, a(p^e) = 1 if p == 1 or 3 (mod 8), a(p^e) = (-1)^e if p == 5 or 7 (mod 8).
G.f.: x * (1 - x^4)^2/((1 - x^2)*(1 - x^8)) = (x + x^3)/(1 + x^4).
a(-n) = -a(n) = a(n+4).
a(n+2) = A091337(n).
a(2*n) = 0, a(2*n+1) = A057077(n).
G.f.: x/(1 - x^2/(1 + 2*x^2/(1 - x^2))). - Michael Somos, Jan 03 2013
a(n) = ((-2)/n), where (k/n) is the Kronecker symbol. Period 8. See the Eric Weisstein link. - Wolfdieter Lang, May 29 2013
a(n) = A257170(n) unless n = 0.
From Jianing Song, Nov 14 2018: (Start)
a(n) = sqrt(2)*sin(Pi*n/2)*cos(Pi*n/4).
E.g.f.: sqrt(2)*sin(x/sqrt(2))*cosh(x/sqrt(2)).
Moebius transform of A002325.
a(n) = A091337(n)*A101455(n).
a(n) = ((-2)^(2*i+1)/n) for all integers i >= 0, where (k/n) is the Kronecker symbol. (End)
a(n) = A014017(n-1)+A014017(n-3). - R. J. Mathar, Dec 17 2024

A107849 Expansion of (1 + x)^2 / ((1 - x^2 - 2x^3)(1 + x^4)).

Original entry on oeis.org

1, 2, 2, 4, 5, 6, 12, 16, 25, 42, 58, 92, 141, 206, 324, 488, 737, 1138, 1714, 2612, 3989, 6038, 9212, 14016, 21289, 32442, 49322, 75020, 114205, 173662, 264244, 402072, 611569, 930562, 1415714, 2153700, 3276837, 4985126, 7584236, 11538800

Offset: 0

Creighton Dement, May 25 2005

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

Cf. A107850, A107851, A107852.

CoefficientList[Series[(1 + x)^2 / ((1 - x^2 - 2*x^3)*(1 + x^4)),{x,0,39}],x] (* James C. McMahon, Feb 19 2024 *)

Vec((1 + x)^2 / ((1 - x^2 - 2*x^3)*(1 + x^4)) + O(x^45)) \\ Colin Barker, Apr 30 2019

a(n) = A052947(n+2) + A014017(n+6). - Ralf Stephan, Nov 30 2010

a(n) = a(n-2) + 2*a(n-3) - a(n-4) + a(n-6) + 2*a(n-7) for n>6. - Colin Barker, Apr 30 2019

A128130 Expansion of (1-x)/(1+x^4); period 8: repeat [1,-1,0,0,-1,1,0,0].

Original entry on oeis.org

1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0, 1, -1, 0, 0, -1, 1, 0, 0

Offset: 0

Paul Barry, Feb 15 2007

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Cf. A014017, A133872.

A128130 := proc(n)
    local m ;
    m := modp(n,8) ;
    op(1+m,[1,-1,0,0,-1,1,0,0]) ;
end proc: # R. J. Mathar, Feb 24 2015

CoefficientList[Series[(1-x)/(1+x^4),{x,0,100}],x]  (* Harvey P. Dale, Mar 28 2011 *)

(define (A128130 n) (list-ref '(1 -1 0 0 -1 1 0 0) (modulo n 8))) ;; Antti Karttunen, Aug 12 2017

a(n) = (sqrt(2)/4 + 1/2)*cos(3*Pi*n/4) - sqrt(2)*sin(3*Pi*n/4)/4 + (1/2 - sqrt(2)/4)*cos(Pi*n/4) - sqrt(2)*sin(Pi*n/4)/4; a(n) = Im(Sum_{k=0..n} i^(n-k+1)), i=sqrt(-1).
abs(a(n)) = A133872(n). - Wesley Ivan Hurt, Feb 23 2015
a(n) = A014017(n) - A014017(n-1). - R. J. Mathar, Feb 24 2015

More terms from Antti Karttunen, Aug 12 2017

A102905 a(n) = A113655(Fibonacci(n+1)).

Original entry on oeis.org

3, 3, 2, 1, 5, 8, 15, 19, 36, 57, 89, 142, 233, 377, 612, 985, 1599, 2586, 4181, 6763, 10946, 17711, 28659, 46366, 75027, 121395, 196418, 317809, 514229, 832040, 1346271, 2178307, 3524580, 5702889, 9227465, 14930350, 24157817, 39088169

Offset: 0

Roger L. Bagula, Mar 16 2005

nonn

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

Cf. A000045, A014017, A113655.

A113655:= func< n | 6*Floor((n+2)/3) -(n+2) >;
A102905:= func< n | A113655(Fibonacci(n+1)) >;
[A102905(n): n in [0..50]]; // G. C. Greubel, Dec 09 2022

f[n_]:= If[Mod[n,3]==0, n-2, If[Mod[n,3]==1, n+2, n]]; (* f=A113655 *)
Table[f[Fibonacci[n+1]], {n,0,50}]

def A113655(n): return 6*((n+2)//3) -(n+2)
def A102905(n): return A113655(fibonacci(n+1))
[A102905(n) for n in range(51)] # G. C. Greubel, Dec 09 2022

a(n) = f(Fibonacci(n+1)), where f(n) = n-2 if (n mod 3) = 0, f(n) = n+2 if (n mod 3) = 1, otherwise f(n) = n.
a(n) = A113655(Fibonacci(n+1)).
G.f.: (3-4*x^2-4*x^3+2*x^4+2*x^5+2*x^6-4*x^7-x^8+2*x^9) / ((1-x)*(1+x)*(1+x^2)*(1-x-x^2)*(1+x^4)). - Colin Barker, Dec 11 2012
a(n) = (1 + 3*(-1)^n)/4 + Fibonacci(n+1) + (3/2)*(-1)^floor(n/2) * (n mod  2) + A014017(n) + A014017(n-1) - A014017(n-2). - G. C. Greubel, Dec 09 2022

Edited by G. C. Greubel, Dec 09 2022

A107854 G.f. x(x^2+1)(x^3-x-1)/((2x^3+x^2-1)(x^4+1)).

Original entry on oeis.org

0, 1, 1, 2, 3, 3, 5, 8, 11, 19, 29, 42, 67, 99, 149, 232, 347, 531, 813, 1226, 1875, 2851, 4325, 6600, 10027, 15251, 23229, 35306, 53731, 81763, 124341, 189224, 287867, 437907, 666317, 1013642, 1542131, 2346275, 3569413, 5430536, 8261963, 12569363

Offset: 0

Creighton Dement, May 25 2005

The sequence A078028 is given by 1em[I* ]forzapseq and is from the same "batch" (i.e., corresponding to the same floretion and symmetry settings) as A107849, A107850, A107851, A107852, A107853 and (a(n)).

Floretion Algebra Multiplication Program, FAMP Code: 1dia[I]forzapseq[(.5i' + .5j' + .5'ki' + .5'kj')*(.5'i + .5'j + .5'ik' + .5'jk')], 1vesforzap = A000004

Index entries for linear recurrences with constant coefficients, signature (0,1,2,-1,0,1,2).

Cf. A107849, A107850, A107851, A107852, A107853, A078028.

CoefficientList[Series[x(x^2+1)(x^3-x-1)/((2x^3+x^2-1)(x^4+1)),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,2,-1,0,1,2},{0,1,1,2,3,3,5},50] (* Harvey P. Dale, Jun 21 2022 *)

a(n)=([0,1,0,0,0,0,0; 0,0,1,0,0,0,0; 0,0,0,1,0,0,0; 0,0,0,0,1,0,0; 0,0,0,0,0,1,0; 0,0,0,0,0,0,1; 2,1,0,-1,2,1,0]^n*[0;1;1;2;3;3;5])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

a(n) = A159284(n) + A014017(n+5).

A122056 Expansion of g.f. x^2/((1 - x)^4(1 + x)(1 + x^2)*(1 + x^4)).

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 46, 58, 72, 88, 106, 126, 148, 172, 199, 229, 262, 298, 337, 379, 424, 472, 524, 580, 640, 704, 772, 844, 920, 1000, 1085, 1175, 1270, 1370, 1475, 1585, 1700, 1820, 1946, 2078, 2216, 2360, 2510, 2666, 2828, 2996, 3171, 3353, 3542, 3738

Offset: 0

Roger L. Bagula, Sep 13 2006

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. N. W. Hone, Algebraic curves, integer sequences and a discrete Painlevé transcendent, arXiv:0807.2538 [nlin.SI], 2008; Proceedings of SIDE 6, Helsinki, Finland, 2004. [Set a(n)=d(n+3) on p. 8]
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,1,-3,3,-1).

Cf. A006731, A014017, A014125.

R:=PowerSeriesRing(Integers(), 72); [0,0] cat Coefficients(R!( x^2/((1-x)^4*(1+x)*(1+x^2)*(1+x^4)) )); // G. C. Greubel, Dec 29 2022

p[n_]:= p[n] = If[n<0, 1, Cancel[Simplify[(x^(n-1)*p[n-1]*p[n-8] + p[n-4]*p[n-5])/p[n-9]]]]; Table[Exponent[p[n], x], {n,0,30}]
LinearRecurrence[{3,-3,1,0,0,0,0,1,-3,3,-1}, {0,0,1,3,6,10,15,21,28,36, 46,58,72}, 61] (* G. C. Greubel, Dec 29 2022 *)

def A122056_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^2/((1-x)^4*(1+x)*(1+x^2)*(1+x^4)) ).list()
A122056_list(70) # G. C. Greubel, Dec 29 2022

a(n) = degree(p(n)) with p(n) = (x^(n-1)*p(n-1)*p(n-8) + p(n-4)*p(n-5))/p(n-9).
From Colin Barker, Oct 08 2019: (Start)
G.f.: x^2 / ((1-x)^4*(1+x)*(1+x^2)*(1+x^4)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-8) - 3*a(n-9) + 3*a(n-10) - a(n-11) for n > 10. (End)
a(n) = (1/192)*(4*n^3 +42*n^2 +80*n -63 +3*(-1)^n) + (1/32)*(i^n*(1 + (-1)^n) + i^(n+1)*(1-(-1)^n)) + (1/4)*(b(n) -b(n-1) -2*b(n-2) -2*b(n -3)), where b(n) = A014017(n). - G. C. Greubel, Dec 29 2022

Edited by G. C. Greubel, Dec 29 2022

A186187 Period 8 sequence [ 2, 2, 1, 2, 4, 2, 1, 2, ...] except a(0) = 1.

Original entry on oeis.org

1, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2, 2, 1, 2, 4, 2, 1, 2, 2

Offset: 0

Michael Somos, Feb 14 2011

Also continued fraction expansion of sqrt(2717)/38. - Bruno Berselli, Mar 07 2011

			1 + 2*x + x^2 + 2*x^3 + 4*x^4 + 2*x^5 + x^6 + 2*x^7 + 2*x^8 + 2*x^9 + ...

Michael Somos, Rational Function Multiplicative Coefficients
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1,-1,1).

[1] cat &cat[ [2, 1, 2, 4, 2, 1, 2, 2]: n in [1..13]];  // Bruno Berselli, Mar 07 2011

PadRight[{1},108,{2,2,1,2,4,2,1,2}] (* Harvey P. Dale, Mar 22 2012 *)

{a(n) = - (n==0) + [ 2, 2, 1, 2, 4, 2, 1, 2] [n%8 + 1]}

{a(n) = polcoeff( (1 + x)^4 * (1 - x + x^2)^2 / (1 - x^8) + x * O(x^abs(n)), abs(n))}

Euler transform of length 8 sequence [ 2, -2, 2, 0, 0, -2, 0, 1].
Moebius transform is length 8 sequence [ 2, -1, 0, 3, 0, 0, 0, -2].
a(n) = 2 * b(n) where b() is multiplicative with b(2) = 1/2, b(4) = 2, b(2^e) = 1 if e>2, b(p^e) = 1 if p>2.
G.f.: (1 + x)^4 * (1 - x + x^2)^2 / (1 - x^8) = (1-x+x^2)^2*(1+x)^3 / ((1-x) *(1+x^2) *(1+x^4)). a(-n) = a(n). a(2*n + 1) = 2, a(4*n + 2) = 1, a(8*n + 4) = 4, a(8*n) = 2 except a(0) = 1.
a(n) = A056594(n)-A014017(n)+2 for n>0. - Bruno Berselli, Feb 15 2011

cp's OEIS Frontend

A191497 a(n+1) = 2*a(n) + A014017(n+5), a(0) = 0.

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A301712 Coordination sequence for node of type V1 in "usm" 2-D tiling (or net).

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A188510 Expansion of x*(1 + x^2)/(1 + x^4) in powers of x.

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A107849 Expansion of (1 + x)^2 / ((1 - x^2 - 2*x^3)*(1 + x^4)).

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A128130 Expansion of (1-x)/(1+x^4); period 8: repeat [1,-1,0,0,-1,1,0,0].

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A102905 a(n) = A113655(Fibonacci(n+1)).

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A107854 G.f. x*(x^2+1)*(x^3-x-1)/((2*x^3+x^2-1)*(x^4+1)).

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A107849 Expansion of (1 + x)^2 / ((1 - x^2 - 2x^3)(1 + x^4)).

A107854 G.f. x(x^2+1)(x^3-x-1)/((2x^3+x^2-1)(x^4+1)).

A122056 Expansion of g.f. x^2/((1 - x)^4(1 + x)(1 + x^2)*(1 + x^4)).