A007307 -id:A007307 - cp's OEIS frontend

A113214 Riordan array (1+2x,x(1+x)).

1, 2, 1, 0, 3, 1, 0, 2, 4, 1, 0, 0, 5, 5, 1, 0, 0, 2, 9, 6, 1, 0, 0, 0, 7, 14, 7, 1, 0, 0, 0, 2, 16, 20, 8, 1, 0, 0, 0, 0, 9, 30, 27, 9, 1, 0, 0, 0, 0, 2, 25, 50, 35, 10, 1, 0, 0, 0, 0, 0, 11, 55, 77, 44, 11, 1, 0, 0, 0, 0, 0, 2, 36, 105, 112, 54, 12, 1, 0, 0, 0, 0, 0, 0, 13, 91, 182, 156, 65, 13, 1

Offset: 0

Paul Barry, Oct 18 2005

Row sums are Lucas numbers A000204. Diagonal sums are A007307(n+1). Inverse is (-1)^(n-k)A092392(n,k). Product with Pascal triangle (1/(1-x),x/(1-x)) is A111125.

			Triangle begins
  1;
  2,  1;
  0,  3,  1;
  0,  2,  4,  1;
  0,  0,  5,  5,  1;
  0,  0,  2,  9,  6,  1;
  0,  0,  0,  7, 14,  7,  1;
  0,  0,  0,  2, 16, 20,  8,  1;
Row 4: (1 + x*c(-x))^5 = 1 + 5*x + 5*x^2 + O(x^5). - _Peter Bala_, Sep 10 2021

R. Tauraso, Edge cover time for regular graphs, JIS 11 (2008) 08.4.4.

Cf. A000108, A000204, A007307, A092392, A111125.

T(n, k) = C(k, n-k) + 2*C(k, n-k-1).
T(n, k) = Sum_{j = 0..n} (-1)^(n-j)*C(n, j)*C(j+k, 2*k)*(2*j+1)/(2*k+1).
From Peter Bala, Sep 10 2021: (Start)
T(n,k) = T(n-1,k-1) + T(n-2,k-1) with boundary conditions T(n,n) = 1, T(1,0) = 2 and T(n,k) = 0 for k < 0 or k > n.
The entries in row n, read in reverse order, are the coefficients in the n-th degree Taylor polynomial of (1 + x*c(-x))^(n+1) at x = 0, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)

A117363 Expansion of (1-2x)/(1-x^2+x^3).

Original entry on oeis.org

1, -2, 1, -3, 3, -4, 6, -7, 10, -13, 17, -23, 30, -40, 53, -70, 93, -123, 163, -216, 286, -379, 502, -665, 881, -1167, 1546, -2048, 2713, -3594, 4761, -6307, 8355, -11068, 14662, -19423, 25730, -34085, 45153, -59815, 79238

Offset: 0

Paul Barry, Mar 10 2006

Diagonal sums of number triangle A117362.

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

Cf. A109248, A007307.

CoefficientList[Series[(1-2x)/(1-x^2+x^3),{x,0,60}],x] (* or *) LinearRecurrence[ {0,1,-1},{1,-2,1},60] (* Harvey P. Dale, Aug 04 2021 *)

a(n)=a(n-2)-a(n-3); a(n)=sum{k=0..floor(n/2), (-1)^n*(C(k,n-2k)+2*C(k,n-2k-1))}.

A168637 a(n) = a(n-1) + a(n-2) - a(n-4) starting a(0)=0, a(1)=1, a(2)=a(3)=3.

Original entry on oeis.org

0, 1, 3, 3, 6, 8, 11, 16, 21, 29, 39, 52, 70, 93, 124, 165, 219, 291, 386, 512, 679, 900, 1193, 1581, 2095, 2776, 3678, 4873, 6456, 8553, 11331, 15011, 19886, 26344, 34899, 46232, 61245, 81133, 107479, 142380, 188614, 249861, 330996, 438477, 580859, 769475

Offset: 0

Roger L. Bagula and Gary W. Adamson, Dec 01 2009

R. Pallu de la Barriere, Optimal Control Theory, Dover Publications, New York, 1967, pages 339-344

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

Cf. A007307 (for a different starting vector of the Mma program).

Cf. A000931, A060006.

R:=PowerSeriesRing(Integers(), 60);
[0] cat Coefficients(R!( x*(1+2*x-x^2)/((1-x)*(1-x^2-x^3)) )); // G. C. Greubel, Apr 20 2025

LinearRecurrence[{1,1,0,-1},{0,1,3,3},50] (* or *) CoefficientList[ Series[ x*(-1-2x+x^2)/((1-x)(x^3+x^2-1)),{x,0,50}],x] (* Harvey P. Dale, Jun 22 2011 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -1,0,1,1]^n*[0;1;3;3])[1,1] \\ Charles R Greathouse IV, Jul 29 2016

def A168637_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(1+2*x-x^2)/((1-x)*(1-x^2-x^3)) ).list()
print(A168637_list(60)) # G. C. Greubel, Apr 20 2025

Limit_{n -> oo} a(n+1)/a(n) = A060006 (also a limiting value of A000931).
G.f.: x*(1 + 2*x - x^2)/((1-x)*(1 - x^2 - x^3)). [Dec 03 2009]
a(n) = 3*A000931(n+4) + 2*A000931(n+3) - 2. [Dec 03 2009]
a(n) = a(n-2) + a(n-3) + 2. - Greg Dresden, May 18 2020

Precise definition and more formulas supplied by the Assoc. Editors of the OEIS, Dec 03 2009

A276275 Padovan like sequence: a(n) = a(n-2) + a(n-3) for n>3, a(1)=2, a(2)=2, a(3)=0.

Original entry on oeis.org

2, 2, 0, 4, 2, 4, 6, 6, 10, 12, 16, 22, 28, 38, 50, 66, 88, 116, 154, 204, 270, 358, 474, 628, 832, 1102, 1460, 1934, 2562, 3394, 4496, 5956, 7890, 10452, 13846, 18342, 24298, 32188, 42640, 56486, 74828, 99126, 131314, 173954, 230440, 305268, 404394, 535708

Offset: 1

Nicolas Bègue, Aug 26 2016

Obtained from Padovan Spiral number (A134816) modulo 3 reduction periodic sequence 1112201210010, 111 112 122 220 ... fourth initialization values 220, it satisfies the same recurrence a(n) = a(n-2) + a(n-3).

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

Cf. A134816, A084338.

RecurrenceTable[{a[n] == a[n - 2] + a[n - 3], a[1] == 2, a[2] == 2, a[3] == 0}, a, {n, 1, 48}] (* or *) CoefficientList[Series[2 x (1 + x - x^2)/(1 - x^2 - x^3), {x, 0, 47}], x] (* Michael De Vlieger, Sep 02 2016 *)
LinearRecurrence[{0,1,1},{2,2,0},60] (* Harvey P. Dale, Jan 27 2023 *)

G.f.: 2*x*(1 + x - x^2)/(1 - x^2 - x^3).
a(n) = A134816(n) + A007307(n-3) for n>=4.
a(n) = 2*A084338(n-3) for n>=4.

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A113214 Riordan array (1+2x,x(1+x)).

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A117363 Expansion of (1-2x)/(1-x^2+x^3).

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A168637 a(n) = a(n-1) + a(n-2) - a(n-4) starting a(0)=0, a(1)=1, a(2)=a(3)=3.

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A276275 Padovan like sequence: a(n) = a(n-2) + a(n-3) for n>3, a(1)=2, a(2)=2, a(3)=0.

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