A077962 -id:A077962 - cp's OEIS frontend

A011656 A binary m-sequence: expansion of reciprocal of x^3 + x^2 + 1 (mod 2), shifted by 2 initial 0's.

0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1

Offset: 0

N. J. A. Sloane

Period 7.

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967.
H. D. Lueke, Korrelationssignale, Springer 1992, pp. 43-48.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier/North Holland, 1978, p. 408.

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

Cf. A077962.

Cf. A011655..A011751 for other binary m-sequences.

PadLeft[ Mod[ CoefficientList[ Series[1/(1 + x^2 + x^3), {x, 0, 102}], x], 2], 105] (* Robert G. Wilson v *)

A011656_vec(N)=concat([0,0],Vec(lift(O(x^(N-1))+Mod(1,2)/(1+x^2+x^3))))
A011656(n)=(n%7>3)||(n%7==2) \\ Faster than polcoeff(.../(1+x^2+x^3),n-2). - M. F. Hasler, Feb 17 2018

G.f.: (x^6 + x^5 + x^4 + x^2)/(1-x^7). a(n+7) = a(n). - Ralf Stephan, Aug 05 2013

G.f.: x^2/(1 + x^2 + x^3) in GF(2). - M. F. Hasler, Feb 16 2018

A219977 Expansion of 1/(1+x+x^2+x^3).

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

Offset: 0

Harvey P. Dale, Dec 02 2012

			G.f. = 1 - x + x^4 - x^5 + x^8 - x^9 + x^12 - x^13 + x^16 - x^17 + x^20 - x^21 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Elena Barcucci, Antonio Bernini, Stefano Bilotta, Renzo Pinzani, Non-overlapping matrices, arXiv:1601.07723 [cs.DM], 2016.
Stefano Bilotta, Variable-length Non-overlapping Codes, arXiv preprint arXiv:1605.03785, 2016
Kyu-Hwan Lee, Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1).

Cf. A036765, A038505, A049347, A077962, A133872.

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1+x+x^2+x^3))); // Vincenzo Librandi, Apr 22 2015

CoefficientList[Series[1/(1+x+x^2+x^3),{x,0,80}],x] (* or *) PadRight[{},120,{1,-1,0,0}]
LinearRecurrence[{-1,-1,-1},{1,-1,0},80] (* Harvey P. Dale, May 22 2021 *)

{a(n) = [1, -1, 0, 0][n%4 + 1]} /* Michael Somos, Dec 12 2012 */

Vec(1/(1+x+x^2+x^3) + O(x^100)) \\ Michel Marcus, Jan 28 2016

G.f.: 1/(1 +x +x^2 +x^3).
Euler transform of length 4 sequence [ -1, 0, 0, 1]. - Michael Somos, Dec 12 2012
a(n) = a(n+4) = -a(1-n). |a(n)| = A133872(n). REVERT transform is A036765. INVERT transform is A077962. - Michael Somos, Dec 12 2012
A038505(n+2) = p(-1)  where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n. - Michael Somos, Dec 12 2012
From Wesley Ivan Hurt, Apr 22 2015: (Start)
a(n) +a(n-1) +a(n-2) +a(n-3) = 0.
a(n) = (-1)^n/2 +(-1)^(n/2 +1/4 -(-1)^n/4)/2. (End)

A112455 a(n) = -a(n-2) - a(n-3).

Original entry on oeis.org

-3, 0, 2, 3, -2, -5, -1, 7, 6, -6, -13, 0, 19, 13, -19, -32, 6, 51, 26, -57, -77, 31, 134, 46, -165, -180, 119, 345, 61, -464, -406, 403, 870, 3, -1273, -873, 1270, 2146, -397, -3416, -1749, 3813, 5165, -2064, -8978, -3101, 11042, 12079, -7941

Offset: 0

Anthony C Robin, Dec 13 2005

This sequence resembles the Perrin sequence, A001608. Like many such sequences with a(1)=0, any prime p divides a(p). The first pseudoprime (composite n divides a(n)) is 121.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (0,-1,-1).

Cf. A001608, A112458.

a:=[-3,0,2];; for n in [4..60] do a[n]:=-a[n-2]-a[n-3]; od; a; # G. C. Greubel, May 19 2019

I:=[-3,0,2]; [n le 3 select I[n] else -Self(n-2) -Self(n-3): n in [1..60]]; // G. C. Greubel, May 19 2019

A112455 := proc(n)
    option remember ;
    if n <= 2 then
        op(n+1,[-3,0,2]) ;
    else
        -procname(n-2)-procname(n-3) ;
    end if;
end proc: # R. J. Mathar, Feb 18 2024

Table[ -Tr[MatrixPower[{{0, 0, -1}, {1, 0, -1}, {0, 1, 0}}, n]], {n, 1, 60}] (* Artur Jasinski, Jan 10 2007 *)
LinearRecurrence[{0,-1,-1}, {-3,0,2}, 60] (* G. C. Greubel, May 19 2019 *)

Vec(-(3+x^2)/(1+x^2+x^3)+O(x^60)) \\ Charles R Greathouse IV, May 15 2013

(-(3+x^2)/(1+x^2+x^3)).series(x, 60).coefficients(x, sparse=False) # G. C. Greubel, May 19 2019

a(n) = - trace({{0, 0, -1}, {1, 0, -1}, {0, 1, 0}})^n. - Artur Jasinski, Jan 10 2007
From R. J. Mathar, Oct 24 2009: (Start)
G.f.: -(3+x^2)/(1+x^2+x^3).
a(n) = -3*A077962(n) - A077962(n-2). (End)
a(n) = (-1)^(n+1)*(A001609(n)^2 - A001609(2*n))/2. - Greg Dresden, Apr 14 2023

Edited by Don Reble, Jan 25 2006

A077889 Expansion of 1/( (1-x)*(1 + x^2 + x^3) ).

Original entry on oeis.org

1, 1, 0, -1, 0, 2, 2, -1, -3, 0, 5, 4, -4, -8, 1, 13, 8, -13, -20, 6, 34, 15, -39, -48, 25, 88, 24, -112, -111, 89, 224, 23, -312, -246, 290, 559, -43, -848, -515, 892, 1364, -376, -2255, -987, 2632, 3243, -1644, -5874, -1598, 7519, 7473, -5920, -14991, -1552, 20912, 16544, -19359, -37455, 2816, 56815

Offset: 0

N. J. A. Sloane, Nov 17 2002

The Gi1 sums, see A180662 for the definition of these sums, of triangle A101950 equal the terms of this sequence. - Johannes W. Meijer, Aug 06 2011

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Eunmi Choi, Yuna Oh, Diagonal sums in negative trinomial table, Korean J. Math (2019) Vol. 27, No. 3, 723-734.
Index entries for linear recurrences with constant coefficients, signature (1,-1,0,1).

Cf. A077962, A101950.

a:=[1,1,0,-1];; for n in [5..60] do a[n]:=a[n-1]-a[n-2]+a[n-4]; od; a; # G. C. Greubel, Dec 30 2019

R:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)*(1+x^2+x^3)) )); // G. C. Greubel, Dec 30 2019

A101950 := proc(n,k) local j,k1: add((-1)^((n-j)/2)*binomial((n+j)/2,j)*(1+(-1)^(n+j))* binomial(j,k)/2, j=0..n) end: A077889 := proc(n): add(A101950(n-3*k,k), k=0..floor(n/4)) end: seq(A077889(n), n=0..60); # Johannes W. Meijer, Aug 06 2011

CoefficientList[Series[1/((1-x)*(1+x^2+x^3)),{x,0,60}],x] (* or *) LinearRecurrence[{1,-1,0,1},{1,1,0,-1},60] (* Harvey P. Dale, Jul 14 2017 *)

my(x='x+O('x^60)); Vec(1/((1-x)*(1+x^2+x^3))) \\ G. C. Greubel, Dec 30 2019

def A077952_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)*(1+x^2+x^3)) ).list()
A077952_list(60) # G. C. Greubel, Dec 30 2019

a(n) = Sum_{k=0..floor(n/4)} A101950(n-3*k, k).
a(n) = (1 + 2*A077962(n) + 2*A077962(n-1) + A077962(n-2))/3. - G. C. Greubel, Dec 30 2019
a(n)-a(n-1) = A077962(n). - R. J. Mathar, Mar 14 2021

A193736 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (n+1)-st Fibonacci polynomial and q(n,x) = (x+1)^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 2, 1, 4, 8, 8, 3, 1, 5, 13, 19, 15, 5, 1, 6, 19, 36, 42, 28, 8, 1, 7, 26, 60, 91, 89, 51, 13, 1, 8, 34, 92, 170, 216, 182, 92, 21, 1, 9, 43, 133, 288, 446, 489, 363, 164, 34, 1, 10, 53, 184, 455, 826, 1105, 1068, 709, 290, 55, 1, 11, 64, 246, 682, 1414, 2219, 2619, 2266, 1362, 509, 89

Offset: 0

Clark Kimberling, Aug 04 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

			First six rows:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  4,  2;
  1,  4,  8,  8,  3;
  1,  5, 13, 19, 15,  5;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000007, A005314 (diagonal sums), A052542 (row sums), A077962.

Cf. A029907, A193722, A193737, A324969.

function T(n,k) // T = A193736
  if k lt 0 or n lt 0 then return 0;
  elif n lt 3 then return Binomial(n,k);
  else return T(n-1, k) + T(n-1, k-1) + T(n-2, k-2);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 24 2023

(* First program *)
p[0, x_] := 1
p[n_, x_] := Fibonacci[n + 1, x] /; n > 0
q[n_, x_] := (x + 1)^n
t[n_, k_] := Coefficient[p[n, x], x^(n - k)];
t[n_, n_] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n - k + 1, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193736 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]          (* A193737 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[n<3, Binomial[n, k], T[n-1,k] +T[n-1,k-1] +T[n -2,k-2]];
Table[T[n, k], {n,0,12}, {k,0,n}]//TableForm (* G. C. Greubel, Oct 24 2023 *)

def T(n,k): # T = A193736
    if (n<3): return binomial(n,k)
    else: return T(n-1,k) +T(n-1,k-1) +T(n-2,k-2)
flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

T(0,0) = T(1,0) = T(1,1) = T(2,0) = T(2,2) = 1; T(n,k) = 0 if k<0 or k>n; T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2). - Philippe Deléham, Feb 13 2020
From G. C. Greubel, Oct 24 2023: (Start)
T(n, k) = A193737(n, n-k).
T(n, n) = Fibonacci(n) + [n=0] = A324969(n+1).
T(n, n-1) = A029907(n).
Sum_{k=0..n} T(n, k) = A052542(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A005314(n) + [n=0].
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = [n=0] + A077962(n-1). (End)

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A011656 A binary m-sequence: expansion of reciprocal of x^3 + x^2 + 1 (mod 2), shifted by 2 initial 0's.

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A219977 Expansion of 1/(1+x+x^2+x^3).

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A112455 a(n) = -a(n-2) - a(n-3).

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A077889 Expansion of 1/( (1-x)*(1 + x^2 + x^3) ).

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A193736 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (n+1)-st Fibonacci polynomial and q(n,x) = (x+1)^n.

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