A057078 -id:A057078 - cp's OEIS frontend

A062000 a(n) = a(n-1)^2 - a(n-2)^2 with a(0) = 0, a(1) = 2.

0, 2, 4, 12, 128, 16240, 263721216, 69548879504781056, 4837046640370554355727482727956480, 23397020201120067002755280700388456275000098577861376610994277515264

Offset: 0

Henry Bottomley, May 29 2001

nonn

			a(3) = 4^2 - 2^2 = 12.

Harry J. Smith, Table of n, a(n) for n = 0..12
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A001042 and A057078 have the same recurrence.

Cf. A061999.

t = {0, 2}; Do[AppendTo[t, t[[-2]]^2 - t[[-1]]^2], {n, 8}]; Abs[t] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
RecurrenceTable[{a[0]==0, a[1]==2, a[n]==a[n-1]^2 - a[n-2]^2}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 17 2014 *)

{ for (n=0, 12, if (n>1, a=a1^2 - a2^2; a2=a1; a1=a, if (n==0, a=a2=0, a=a1=2)); write("b062000.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 29 2009

def a(n): # a = A062000
    if (n<2): return 2*n
    else: return a(n-1)^2 - a(n-2)^2
[a(n) for n in (0..14)] # G. C. Greubel, May 01 2022

a(n) = 2*A061999(n).

a(n) ~ c^(2^n), where c = 1.35388068260888709216374860554901303232201699191445590979673901150215855854... . - Vaclav Kotesovec, Dec 17 2014

First term corrected by Harry J. Smith, Jul 29 2009

A122918 Expansion of (1+x)^2/(1+x+x^2)^2.

Original entry on oeis.org

1, 0, -2, 2, 1, -4, 3, 2, -6, 4, 3, -8, 5, 4, -10, 6, 5, -12, 7, 6, -14, 8, 7, -16, 9, 8, -18, 10, 9, -20, 11, 10, -22, 12, 11, -24, 13, 12, -26, 14, 13, -28, 15, 14, -30, 16, 15, -32, 17, 16, -34, 18, 17, -36, 19, 18, -38

Offset: 0

Paul Barry, Sep 19 2006

Row sums of Riordan array (1/(1+x+x^2), x/(1+x)^2), A122917.

For n>=1, a(n) equals (-1)^(n+1) times the second immanant of the n X n matrix with 1's along the main diagonal, superdiagonal, and subdiagonal, and 0's everywhere else. The second immanant of an n X n matrix A is the immanant of A given by the partition (2, 1^(n-2)). - John M. Campbell, Apr 12 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (-2,-3,-2,-1).

Cf. A187430 (series reversion, with offset 1).

CoefficientList[Series[(1 + x)^2/(1 + x + x^2)^2, {x, 0, 100}], x] (* Vincenzo Librandi, Apr 13 2014 *)
Print[Table[(-1)^(n+1)*Sum[Binomial[n-i, i]*(n-2*i-1)*(-1)^i, {i, 0, Floor[n/2]}], {n, 0, 100}]] ;  (* John M. Campbell, Jan 08 2016 *)

Vec((1+x)^2/(1+x+x^2)^2 + O(x^100)) \\ Altug Alkan, Jan 08 2015

A122918(n)=(-1)^(n+1)*sum(i=0,n\2,(-1)^i*binomial(n-i,i)*(n-2*i-1)) \\ M. F. Hasler, Jan 12 2016

a(n) = 4 * sqrt(3) * cos(2*Pi*n/3 + Pi/6)/9 + 2(n+1) * sin(2*Pi*n/3 + Pi/6)/3. a(n) = sum{k=0..n} A057078(k) * A057078(n-k).

a(n) = (-1)^(n+1)*sum((-1)^i*binomial(n-i,i)*(n-2*i-1), i=0..[n/2]). - John M. Campbell, Jan 08 2016

A138187 Hankel transform of binomial(2*n+3, n).

Original entry on oeis.org

1, -4, 3, 3, -8, 5, 5, -12, 7, 7, -16, 9, 9, -20, 11, 11, -24, 13, 13, -28, 15, 15, -32, 17, 17, -36, 19, 19, -40, 21, 21, -44, 23, 23, -48, 25, 25, -52, 27, 27, -56, 29, 29, -60, 31, 31, -64, 33, 33, -68, 35, 35, -72, 37, 37, -76, 39, 39, -80, 41

Offset: 0

Paul Barry, Mar 04 2008

Hankel transform of A002054(n+1).
Hankel transform of A002054(n) is A057078(n+1).
Partial sums are A138188.

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

Cf. A002054, A057078, A138188.

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( (1-2*x-2*x^2-x^3)/(1+x+x^2)^2 )); // G. C. Greubel, Jun 16 2021

a[n_]:= a[n]= Sum[(-1)^(n-k+1)*(n+k+2)*Binomial[n+k+1, 2*k], {k, 0, n+1}];
Table[a[n], {n, 0, 65}] (* G. C. Greubel, Jun 16 2021 *)

@CachedFunction
def A138187(n):
    if (n%3==0): return 2*(n//3) +1
    elif (n%3==1): return -4*((n//3) +1)
    else: return 2*(n//3) +3
[A138187(n) for n in (0..65)] # G. C. Greubel, Jun 16 2021

G.f.: (1 -2*x -2*x^2 -x^3)/(1 +x +x^2)^2.
a(n) = Sum_{k=0..n} (-1)^(n-k+1)*(n+k+2)*binomial(n+k+1, 2*k). - Paul Barry, Apr 19 2010
a(n) = 2*floor(n/3) + 1 if (n mod 3) = 0, -4*(floor(n/3) + 1) if (n mod 3) = 1 and 2*floor(n/3) + 3 if (n mod 3) = 2. - G. C. Greubel, Jun 16 2021

A360705 Expansion of Sum_{k>=0} (x * (1 + (-1)^k * x))^k.

Original entry on oeis.org

1, 1, 0, 3, -1, 8, 1, 21, 0, 55, -1, 144, 1, 377, 0, 987, -1, 2584, 1, 6765, 0, 17711, -1, 46368, 1, 121393, 0, 317811, -1, 832040, 1, 2178309, 0, 5702887, -1, 14930352, 1, 39088169, 0, 102334155, -1, 267914296, 1, 701408733, 0, 1836311903, -1, 4807526976, 1

Offset: 0

Seiichi Manyama, Feb 17 2023

Winston de Greef, Table of n, a(n) for n = 0..4762
Index entries for linear recurrences with constant coefficients, signature (0,2,0,1,0,2,0,-1)

Cf. A360592, A360696, A360704.

Cf. A000045, A001906, A057078.

my(N=50, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+(-1)^k*x))^k))

a(n) = sum(k=0, n\2, (-1)^(k*(n-k))*binomial(n-k, k));

a(n) = if(n%2, fibonacci(n+1), [1, 0, -1][n/2%3+1]);

a(n) = Sum_{k=0..floor(n/2)} (-1)^(k*(n-k)) * binomial(n-k,k).
a(2*n) = A057078(n), a(2*n+1) = A000045(2*n+2).
G.f.: ( 1+x+x^3-2*x^4+x^5+x^6-2*x^2 ) / ( (x^2-x-1)*(x^2+x-1)*(1+x+x^2)*(x^2-x+1) ). - R. J. Mathar, Mar 12 2023

A375372 Expansion of 1/( (1 + x) * (1 - x^2*(1 + x)^2) ).

Original entry on oeis.org

1, -1, 2, 0, 2, 2, 5, 5, 12, 16, 28, 44, 73, 115, 190, 304, 494, 798, 1293, 2089, 3384, 5472, 8856, 14328, 23185, 37511, 60698, 98208, 158906, 257114, 416021, 673133, 1089156, 1762288, 2851444, 4613732, 7465177, 12078907, 19544086, 31622992, 51167078, 82790070

Offset: 0

Seiichi Manyama, Aug 13 2024

Index entries for linear recurrences with constant coefficients, signature (-1,1,3,3,1).

Cf. A093040, A375373.

my(N=50, x='x+O('x^N)); Vec(1/((1+x)*(1-x^2*(1+x)^2)))

a(n) = sum(k=0, n\2, binomial(2*k-1, n-2*k));

a(n) = -a(n-1) + a(n-2) + 3*a(n-3) + 3*a(n-4) + a(n-5).
a(n) = Sum_{k=0..floor(n/2)} binomial(2*k-1,n-2*k).
a(n) = A375373(n) + A375373(n-1).
2*a(n) = 2*(-1)^n + A000045(n) + A057078(n+1). - R. J. Mathar, Aug 14 2024

A005289 Number of graphs on n nodes with 3 cliques.

Original entry on oeis.org

0, 0, 1, 4, 12, 31, 67, 132, 239, 407, 657, 1019, 1523, 2211, 3126, 4323, 5859, 7806, 10236, 13239, 16906, 21346, 26670, 33010, 40498, 49290, 59543, 71438, 85158, 100913, 118913, 139398, 162609, 188817, 218295, 251349, 288285

Offset: 1

N. J. A. Sloane

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letter to N. J. A. Sloane, Apr 1988
R. K. Guy, Monthly research problems, 1969-73, Amer. Math. Monthly, 80 (1973), 1120-1128.
R. K. Guy, Monthly research problems, 1969-75, Amer. Math. Monthly, 82 (1975), 995-1004.
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
K. B. Reid, The number of graphs on N vertices with 3 cliques, J. London Math. Soc. (2) 8 (1974), 94-98.
Eric Weisstein's World of Mathematics, Clique.
Index entries for linear recurrences with constant coefficients, signature (3,-1,-4,2,2,2,-4,-1,3,-1)

A005289p := proc(n)
    n*(2*n^2+3*n-6)/72 ;
    round(%) ;
end proc:
A005289 := proc(n)
    if type(n,'even') then
        n*(n^2-4)*(n^2-6)/240+A005289p(n) ;
    else
        n*(n^2-1)*(n^2-9)/240+A005289p(n) ;
    end if;
end proc:
seq(A005289(n),n=1..40) ; # R. J. Mathar, Aug 23 2015

s = x^2*(3*x^3+x^2+x+1) / ((x-1)^6*(x+1)^2*(x^2+x+1)) + O[x]^40; CoefficientList[s, x] (* Jean-François Alcover, Nov 27 2015 *)

G.f.: x^3*(1+x+3*x^3+x^2) / ( (1+x+x^2)*(1+x)^2*(x-1)^6 ). - Simon Plouffe in his 1992 dissertation

288*a(n) = -4*n^3+12*n^2-21*n/5-14+6*n^5/5+(-1)^n*9*(n-2) +32*A057078(n). - R. J. Mathar, Jul 30 2024

A131640 First differences are periodic: 50, 50, 75, 50, 50, 75, ...

Original entry on oeis.org

985, 1035, 1085, 1160, 1210, 1260, 1335, 1385, 1435, 1510, 1560, 1610, 1685, 1735, 1785, 1860, 1910, 1960, 2035, 2085, 2135, 2210, 2260, 2310, 2385, 2435, 2485, 2560, 2610, 2660, 2735, 2785, 2835, 2910, 2960, 3010, 3085, 3135, 3185, 3260, 3310, 3360

Offset: 0

Eric M. Adler (eadler(AT)simi.k12.ca.us), Sep 05 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A057078, A102283.

A131640:= func< n | (5/3)*(35*n + 591 - 5*(n mod 3) ) >;
[A131640(n): n in [0..50]]; // G. C. Greubel, Sep 08 2025

A131640 := proc(n) option remember ; if n =0 then 985 ; elif n mod 3 = 0 then A131640(n-1)+75 ; else A131640(n-1)+50 ; fi ; end: seq(A131640(n),n=0..80) ; # R. J. Mathar, Oct 24 2007

LinearRecurrence[{1,0,1,-1},{985,1035,1085,1160},50] (* Ray Chandler, Aug 25 2015 *)
Table[5*(35*n +591 -5*Mod[n,3])/3, {n,0,50}] (* G. C. Greubel, Sep 08 2025 *)

Vec(5*(197+10*x+10*x^2-182*x^3)/((1-x)^2*(1+x+x^2)) + O(x^40)) \\ Andrew Howroyd, Feb 20 2018

def A131640(n): return 5*(35*n + 591 - 5*(n%3))//3
print([A131640(n) for n in range(51)]) # G. C. Greubel, Sep 08 2025

G.f.: 5*(197 + 10*x + 10*x^2 - 182*x^3)/((1-x)^2*(1+x+x^2)). - R. J. Mathar, Nov 14 2007
From G. C. Greubel, Sep 08 2025: (Start)
a(n) = (5/3)*(35*(n+1) + 551 + 5*(A102283(n+1) + A102283(n))).
a(n) = (5/3)*(35*n + 586 + 5*A057078(n)).
E.g.f.: (5/3)*( 5*exp(-x/2)*( cos((sqrt(3)*x)/2) + (1/sqrt(3))*sin((sqrt(3)*x)/2)) + (586 + 35*x)*exp(x) ). (End)

Definition supplied by N. J. A. Sloane, Sep 14 2007

More terms from R. J. Mathar, Oct 24 2007

A146994 a(n) = (n+1)^2/4 + (floor((n+5)/6) - 1/4) * ((n+1) mod 2).

Original entry on oeis.org

1, 3, 4, 7, 9, 13, 16, 22, 25, 32, 36, 44, 49, 59, 64, 75, 81, 93, 100, 114, 121, 136, 144, 160, 169, 187, 196, 215, 225, 245, 256, 278, 289, 312, 324, 348, 361, 387, 400, 427, 441, 469, 484, 514, 529, 560, 576, 608, 625, 659, 676, 711, 729, 765, 784, 822, 841

Offset: 1

Mitch Phillipson, Manda Riehl, and Tristan Williams, Nov 04 2008

This sum appears when calculating the number of elements of S_3 wreath C_k which avoid 12. We use a nonstandard ordering, where we consider an element of S_3 wreath C_k to be a permutation sigma of S_3 with each sigma_i colored one of k colors. We then create a string au with au_i being defined as sigma_i times its color (where, e.g., the 3rd color has value 3). We then consider the reduced string of au identically to reducing permutations as done in standard pattern avoidance. When calculating the number of these reduced strings which avoid 12, we encounter this sequence as one of our subcases.

G. C. Greubel, Table of n, a(n) for n = 1..1000
T. Mansour, Pattern Avoidance in Coloured Permutations, Séminaire Lotharingien de Combinatoire, 46, 2001.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,0,1,-1,-1,1).

Cf. A000290, A010892, A057078.

[(n mod 2) eq 0 select n*(n+2)/4 + Floor((n+5)/6) else (n+1)^2/4: n in [1..60]]; // G. C. Greubel, Jan 09 2020

a := n -> `if`(irem(n,2)=1,(n+1)^2/4, ((n+1)^2-1)/4 + floor((n+5)/6)): seq(a(n), n=1..57); # Peter Luschny, Feb 01 2015

LinearRecurrence[{1,1,-1,0,0,1,-1,-1,1},{1,3,4,7,9,13,16,22,25},60] (* Harvey P. Dale, Dec 17 2012 *)
Table[If[EvenQ[n], n*(n+2)/4 + Floor[(n+5)/6], (n+1)^2/4], {n, 60}] (* G. C. Greubel, Jan 09 2020 *)

a(n) = if(n%2==0, n*(n+2)/4 + (n+5)\6, (n+1)^2/4);
vector(60, n, a(n)) \\ G. C. Greubel, Jan 09 2020

def a(n):
    if (mod(n,2)==0): return n*(n+2)/4 + floor((n+5)/6)
    else: return (n+1)^2/4
[a(n) for n in (1..60)] # G. C. Greubel, Jan 09 2020

a(2*n-1) = n^2 for n >= 1.
a(2*n) = n*(n+1) + floor((2*n+5)/6) for n >= 0.
From R. J. Mathar, Nov 21 2008: (Start)
a(n) = (-4*A057078(n) - 4*A010892(n+1) + 6*n^2 + 14*n + 7 + (-1)^n*(2n+1))/24.
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-6) - a(n-7) - a(n-8) + a(n-9).
G.f.: x*(1 + 2*x + x^3 + x^4 + x^5)/((1 + x + x^2)*(1 - x + x^2)*(1+x)^2*(1-x)^3). (End)

More terms from R. J. Mathar, Nov 21 2008

Name corrected and partial edit by Peter Luschny, Feb 01 2015

A164360 Period 3: repeat [5, 4, 3].

Original entry on oeis.org

5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3

Offset: 0

Stephen Crowley, Aug 14 2009

From Klaus Brockhaus, May 29 2010: (Start)
Continued fraction expansion of (32+sqrt(1297))/13.
Decimal expansion of 181/333. (End)

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

Cf. A007877 (repeat 0,1,2,1), A068073 (repeat 1,2,3,2), A028356 (repeat 1,2,3,4,3,2), A130784 (repeat 1,3,2), A158289 (repeat 0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1).

Cf. A178566 (decimal expansion of (32+sqrt(1297))/13). [Klaus Brockhaus, May 29 2010]

[ n le 3 select 6-n else Self(n-3):n in [1..105] ]; // Klaus Brockhaus, Sep 17 2009

&cat [[5, 4, 3]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([5, 4, 3]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016

PadRight[{}, 100, {5, 4, 3}] (* Wesley Ivan Hurt, Jul 01 2016 *)

a(n) = 4+(-1)^n*((1/2+I*sqrt(3)/6)*((1+I*sqrt(3))/2)^n+(1/2-I*sqrt(3)/6)*((1-I*sqrt(3))/2)^n). [Corrected by Klaus Brockhaus, Sep 17 2009]
a(n) = 4+(1/3)*sqrt(3)*sin(2*n*Pi/3)+cos(2*n*Pi/3). [Corrected by Klaus Brockhaus, Sep 17 2009]
a(n) = a(n-3) for n > 2, with a(0) = 5, a(1) = 4, a(2) = 3.
G.f.: (5+4*x+3*x^2)/((1-x)*(1+x+x^2)). [Klaus Brockhaus, Sep 17 2009]
E.g.f.: 4*exp(x)+(1/3)*sqrt(3)*exp(-(1/2)*x)*sin((1/2)*x*sqrt(3))+exp(-(1/2)*x)*cos((1/2)*x*sqrt(3)).
a(n) = 4 + A057078(n). - Wesley Ivan Hurt, Jul 01 2016

Edited by Klaus Brockhaus, Sep 17 2009

Offset changed to 0 and formulas adjusted by Klaus Brockhaus, May 18 2010

A191272 Expansion of x(4+5x)/( (1-4x)(1 + x + x^2) ).

Original entry on oeis.org

0, 4, 17, 63, 256, 1025, 4095, 16384, 65537, 262143, 1048576, 4194305, 16777215, 67108864, 268435457, 1073741823, 4294967296, 17179869185, 68719476735, 274877906944, 1099511627777, 4398046511103, 17592186044416, 70368744177665

Offset: 0

Paul Curtz, May 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..175
Index entries for linear recurrences with constant coefficients, signature (3,3,4).

m:=24; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(4+5*x)/((1-4*x)*(1+x+x^2))));  // Bruno Berselli, Jul 04 2011

LinearRecurrence[{3,3,4},{0,4,17},30] (* or *) CoefficientList[ Series[ x (4+5x)/((1-4x)(1+x+x^2)),{x,0,30}],x] (* Harvey P. Dale, Jun 19 2011 *)

makelist(coeff(taylor(x*(4+5*x)/((1-4*x)*(1+x+x^2)), x, 0, n), x, n), n, 0, 23);  /* Bruno Berselli, Jun 06 2011 */

a(n)=4^n-[1,0,-1][n%3+1] \\ Charles R Greathouse IV, Jun 06 2011

G.f.:  x*(4+5*x)/(1 - 3*x - 3*x^2 - 4*x^3).
a(n) = 4^n-A057078(n) = 4^n - (n-th element of periodic length 3 repeat 1, 0, -1)
a(n) = A024495(2*n) + A024495(1+2*n).
a(n+1) = 4*a(n) + (n-th element of periodic length 3 repeat 4, 1, -5).
a(n) = A052539(n) - A080425(n+1). - Bruno Berselli, Jun 06 2011
a(0)=0, a(1)=4, a(2)=17, a(n) = 3*a(n-1) + 3*a(n-2) + 4*a(n-3). - Harvey P. Dale, Jun 19 2011
a(n) = 4*a(n-1) + a(n-3) - 4*(n-4) (n>3). - Bruno Berselli, Jul 04 2011

cp's OEIS Frontend

A062000 a(n) = a(n-1)^2 - a(n-2)^2 with a(0) = 0, a(1) = 2.

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A122918 Expansion of (1+x)^2/(1+x+x^2)^2.

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A138187 Hankel transform of binomial(2*n+3, n).

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A360705 Expansion of Sum_{k>=0} (x * (1 + (-1)^k * x))^k.

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A375372 Expansion of 1/( (1 + x) * (1 - x^2*(1 + x)^2) ).

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A005289 Number of graphs on n nodes with 3 cliques.

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A131640 First differences are periodic: 50, 50, 75, 50, 50, 75, ...

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A191272 Expansion of x(4+5x)/( (1-4x)(1 + x + x^2) ).