A001607 -id:A001607 - cp's OEIS frontend

A110523 Expansion of (1 + x)/(1 + x + 3*x^2).

1, 0, -3, 3, 6, -15, -3, 48, -39, -105, 222, 93, -759, 480, 1797, -3237, -2154, 11865, -5403, -30192, 46401, 44175, -183378, 50853, 499281, -651840, -846003, 2801523, -263514, -8141055, 8931597, 15491568, -42286359, -4188345, 131047422, -118482387, -274659879, 630107040, 193872597

Offset: 0

Paul Barry, Jul 24 2005

Row sums of number triangle A110522.

The sequence a(n) is conjugate with A214733 since the following alternative relations: either ((-1 + i*sqrt(11))/2)^n = a(n) + A214733(n)*(-1 + i*sqrt(11))/2 or ((-1 - i*sqrt(11))/2)^n = a(n) + A214733(n)*(-1 - i*sqrt(11))/2. We have a(n+1) = -3*A214733(n), A214733(n+1) = a(n) - A214733(n). We note that sequences A110512 and A001607 are conjugated in a similar way. The above relations are connected with the Gauss sums, for example if e := exp(i*2Pi/11) then e + e^3 + e^4 + e^5 + e^9 = (-1 + i*sqrt(11))/2, and e^2 + e^6 + e^7 + e^8 + e^10 = (-1 - i*sqrt(11))/2, which is equivalent to the system of sums: Sum_{k=1..5} cos(2Pi*k/11) = -1/2 and Sum_{k=1..5} sin(2Pi*k/11) = sqrt(11)/2, and which is equivalent to the system of products: P_{k=1..5} cos(2Pi*k/11) = -1/32 and P_{k=1..5} sin(2Pi*k/11) = sqrt(11)/32 - for details see Witula's book. Lastly we note that ((-1 + i*sqrt(11))/2)^n + ((-1 - i*sqrt(11))/2)^n = 2*a(n) - A214733(n). - Roman Witula, Jul 27 2012

Roman Witula, On Some Applications of Formulae for Unimodular Complex Numbers, Jacek Skalmierski's Press, Gliwice 2011.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg Matrices with Generalized Leonardo Number Entries, Ann. Math. Silesianae, Vol. 38, No. 2 (2024), pp. 284-313. See p. 298.
Index entries for linear recurrences with constant coefficients, signature (-1,-3).

Cf. A001607, A106852, A110512, A110522, A214733.

[n le 2 select 2-n else -(Self(n-1) +3*Self(n-2)): n in [1..50]]; // G. C. Greubel, Dec 28 2023

CoefficientList[Series[(1+x)/(1+x+3*x^2), {x,0,50}], x] (* G. C. Greubel, Aug 30 2017 *)
LinearRecurrence[{-1,-3},{1,0},40] (* Harvey P. Dale, Jul 02 2022 *)

my(x='x+O('x^50)); Vec((1+x)/(1+x+3*x^2)) \\ G. C. Greubel, Aug 30 2017

@CachedFunction # a = A110523
def a(n): return 1-n if n<2 else -a(n-1) -3*a(n-2)
[a(n) for n in range(41)] # G. C. Greubel, Dec 28 2023

a(n) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-3)^(j-k)*C(k, j-k).
From Roman Witula, Jul 27 2012: (Start)
a(n+2) + a(n+1) + 3*a(n) = 0.
a(n+1) = (-1)^n*(3*i*sqrt(11)/11)*(((1 + i*sqrt(11))/2)^(n-1) - ((1 - i*sqrt(11))/2)^(n-1)). (End)
From G. C. Greubel, Dec 28 2023: (Start)
a(n) = (-1)^n*3^((n-1)/2)*( sqrt(3)*ChebyshevU(n, 1/(2*sqrt(3))) - ChebyshevU(n-1, 1/(2*sqrt(3))) ).
a(n) = A106852(n) - A106852(n-1).
a(n) = (-1)^n*( A214733(n+1) + A214733(n) ). (End)
E.g.f.: exp(-x/2)*(sqrt(11)*cos(sqrt(11)*x/2) + sin(sqrt(11)*x/2))/sqrt(11). - Stefano Spezia, Jul 27 2025

A167433 Row sums of the Riordan array (1-4x+4x^2, x(1-2x)) (A167431).

Original entry on oeis.org

1, -3, -1, 5, 7, -3, -17, -11, 23, 45, -1, -91, -89, 93, 271, 85, -457, -627, 287, 1541, 967, -2115, -4049, 181, 8279, 7917, -8641, -24475, -7193, 41757, 56143, -27371, -139657, -84915, 194399, 364229, -24569, -753027, -703889, 802165, 2209943

Offset: 0

Paul Barry, Nov 03 2009

The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.

Variants are A107920 and A001607.

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

a[n_] := Sin[n*ArcTan[Sqrt[7]]]; FullSimplify[Join[{1}, Table[- (2^(n/2 + 1)/Sqrt[7])*(2*a[n] + Sqrt[2]*a[n + 1]), {n, 1, 100}]]] (* or *) Join[{1}, LinearRecurrence[{1,-2},{-3,-1},100]] (* G. C. Greubel, Jun 13 2016 *)

G.f.: (1-4x+4x^2)/(1-x+2x^2).
From G. C. Greubel, Jun 13 2016: (Start)
a(n) = a(n-1) - 2*a(n-2).
a(n) = -(2^((n+2)/2)/sqrt(7))*( 2*sin(n*arctan(sqrt(7))) + sqrt(2)*sin((n+1)*arctan(sqrt(7))) ), n>=1, and a(0)=1. (End)

A105578 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 1, a(2) = 0.

Original entry on oeis.org

1, 1, 0, -1, 0, 3, 4, -1, -8, -5, 12, 23, 0, -45, -44, 47, 136, 43, -228, -313, 144, 771, 484, -1057, -2024, 91, 4140, 3959, -4320, -12237, -3596, 20879, 28072, -13685, -69828, -42457, 97200, 182115, -12284, -376513, -351944, 401083, 1104972, 302807, -1907136, -2512749, 1301524, 6327023, 3723976

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: ibaseiseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]

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

Cf. A002249, A014551, A078020, A105577, A105579.

Equals (A107920(n) + 1)/2.

-Join[{-1,-1,a=0,b=1},Table[c=1*b-2*a-1;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2011 *)
LinearRecurrence[{2,-3,2},{1,1,0},50] (* Harvey P. Dale, Mar 28 2019 *)

Vec(-(x^2-x+1)/((x-1)*(2*x^2-x+1)) + O(x^100)) \\ Colin Barker, Feb 08 2015

a(n) - a(n+1) = A001607(n); a(n+2) - 2a(n+1) + a(n) = - A078020(n).

G.f.: -(x^2-x+1) / ((x-1)*(2*x^2-x+1)). - Colin Barker, Feb 08 2015

A169998 a(0)=1, a(1)=1; thereafter a(n) = -a(n-1) - 2*a(n-2).

Original entry on oeis.org

1, 1, -3, 1, 5, -7, -3, 17, -11, -23, 45, 1, -91, 89, 93, -271, 85, 457, -627, -287, 1541, -967, -2115, 4049, 181, -8279, 7917, 8641, -24475, 7193, 41757, -56143, -27371, 139657, -84915, -194399, 364229, 24569, -753027, 703889, 802165, -2209943, 605613, 3814273, -5025499, -2603047

Offset: 0

N. J. A. Sloane, Aug 29 2010

Cassels, following Nagell, shows that a(n) = +- 1 only for n = 1, 2, 3, 5, 13.

The sequences A001607, A077020, A107920, A167433, A169998 are all essentially the same except for signs.

J. W. S. Cassels, Local Fields, Cambridge, 1986, see p. 67.

F. Beukers, The multiplicity of binary recurrences, Compositio Mathematica, Tome 40 (1980) no. 2 , p. 251-267. See Theorem 2 p. 259.
M. Mignotte, Propriétés arithmétiques des suites récurrentes, Besançon, 1988-1989, see p. 14. In French.
Index entries for linear recurrences with constant coefficients, signature (-1,-2).

f:=proc(n) option remember; if n <= 1 then 1 else -f(n-1)-2*f(n-2); fi; end;

LinearRecurrence[{-1, -2}, {1, 1}, 46] (* Jean-François Alcover, Feb 23 2024 *)

a(n)=([0,1;-2,-1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Jun 11 2015

G.f.: ( 1+2*x ) / ( 1+x+2*x^2 ). - R. J. Mathar, Jul 14 2011

A113726 A Jacobsthal convolution.

Original entry on oeis.org

1, 0, 1, 4, 5, 8, 25, 44, 77, 176, 353, 660, 1365, 2776, 5417, 10876, 21981, 43648, 87153, 175076, 349669, 698280, 1398585, 2797260, 5590381, 11184720, 22373761, 44735284, 89474165, 178969208, 357910345, 715807004, 1431683837, 2863325216

Offset: 0

Paul Barry, Nov 08 2005

Convolution of A001045(n+1) and A001607(n+1).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1,4,4).

Cf. A094686, A089977, A001045, A001607.

LinearRecurrence[{0,1,4,4},{1,0,1,4},40] (* Harvey P. Dale, Apr 30 2025 *)

G.f.: 1/((1-x-2*x^2)*(1+x+2*x^2)).
a(n) = a(n-2) + 4*a(n-3) + 4*a(n-4).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*2^k*(1+(-1)^(n-k))/2.
a(n) = 2^n/3 + (-1)^n/6 + A001607(n+1)/2. - R. J. Mathar, Aug 23 2011
a(n) = sum(A128099(n, n-2*k), k=0..floor(n/2)). - Johannes W. Meijer, Aug 28 2013

A107443 Expansion of g.f.: (1+3x^2)/((1-x)(1+x+2x^2)(1-x+2*x^2)).

Original entry on oeis.org

1, 1, 1, 1, -3, -3, 9, 9, -11, -11, 1, 1, 45, 45, -135, -135, 229, 229, -143, -143, -483, -483, 2025, 2025, -4139, -4139, 4321, 4321, 3597, 3597, -28071, -28071, 69829, 69829, -97199, -97199, 12285, 12285, 351945, 351945, -1104971, -1104971, 1907137, 1907137, -1301523, -1301523, -3723975, -3723975

Offset: 0

Creighton Dement, May 26 2005

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,-3,3,-4,4).

Cf. A001607, A174565.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1+3*x^2)/((1-x)*(1+3*x^2+4*x^4)) )); // G. C. Greubel, Mar 24 2024

with(gfun): seriestolist(series((3*x^2+1)/((1-x)*(2*x^2+x+1)*(2*x^2-x+1)), x=0,50));

CoefficientList[Series[(1+3*x^2)/((1-x)*(1+3*x^2+4*x^4)), {x,0,50}], x] (* G. C. Greubel, Mar 24 2024 *)

def A107443_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+3*x^2)/((1-x)*(1+3*x^2+4*x^4)) ).list()
A107443_list(50) # G. C. Greubel, Mar 24 2024

a(2n) = a(2n+1) = A174565(n).

a(n) = (1 - 2*(-1)^n*A001607(n) + A001607(n+1))/2. - G. C. Greubel, Mar 24 2024

A172250 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,...] DELTA [1,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 0, 2, -1, 0, 0, 1, 1, -1, 0, 0, 0, 3, -2, 0, 0, 0, 0, 1, 3, -4, 1, 0, 0, 0, 0, 4, -2, -2, 1, 0, 0, 0, 0, 1, 6, -9, 3, 0, 0, 0, 0, 0, 0, 5, 0, -9, 6, -1, 0, 0, 0, 0, 0, 1, 10, -15, 3, 3, -1, 0, 0, 0, 0, 0, 0, 6, 5, -24, 18, -4, 0, 0, 0, 0, 0, 0, 0, 1, 15, -20, -6, 18, -8, 1

Offset: 0

Philippe Deléham, Jan 29 2010

			Triangle begins:
  1;
  0,  1;
  0,  1,  0;
  0,  0,  2, -1;
  0,  0,  1,  1, -1;
  0,  0,  0,  3, -2,  0;
  0,  0,  0,  1,  3, -4,  1;
  0,  0,  0,  0,  4, -2, -2,  1; ...

Cf. A101950.

T(n,k) = T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0)=1, T(n,k) = 0 if k > n or if k < 0.
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A088139(n+1), A001607(n+1), A000007(n), A000012(n), A099087(n), A190960(n+1) for x = -2, -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Feb 15 2012
G.f.: 1/(1-y*x+y*(y-1)*x^2). - Philippe Deléham, Feb 15 2012

A078033 Expansion of (1-x) / (1+x^2+2*x^3).

Original entry on oeis.org

1, -1, -1, -1, 3, 3, -1, -9, -5, 11, 23, -1, -45, -45, 47, 135, 43, -229, -313, 143, 771, 483, -1057, -2025, 91, 4139, 3959, -4321, -12237, -3597, 20879, 28071, -13685, -69829, -42457, 97199, 182115, -12285, -376513, -351945, 401083, 1104971, 302807, -1907137, -2512749, 1301523, 6327023

Offset: 0

N. J. A. Sloane, Nov 17 2002

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

a:=[1,-1,-1];; for n in [4..40] do a[n]:=-a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Nov 22 2019

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x)/((1+x)*(1-x+2*x^2)) )); // G. C. Greubel, Nov 22 2019

seq(coeff(series((1-x)/((1+x)*(1-x+2*x^2)), x, n+1), x, n), n = 0..40); # G. C. Greubel, Nov 22 2019

LinearRecurrence[{0,-1,-2}, {1,-1,-1}, 40] (* G. C. Greubel, Nov 22 2019 *)
CoefficientList[Series[(1-x)/(1+x^2+2x^3),{x,0,60}],x] (* Harvey P. Dale, Mar 31 2023 *)

Vec((1-x)/((1+x)*(1-x+2*x^2)) + O(x^40)) \\ Colin Barker, May 18 2019

def A078033_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/((1+x)*(1-x+2*x^2)) ).list()
A078033_list(40) # G. C. Greubel, Nov 22 2019

2*((-1)^n)*a(n) + A001607(n+2) = 1 - Creighton Dement, Oct 30 2004

a(n) = -a(n-2) - 2*a(n-3) for n>2. - Colin Barker, May 18 2019

A105576 a(n) = 2a(n-1) - 3a(n-2) + 2*a(n-3) with a(0) = 3, a(1) = 4, a(2) = 0.

Original entry on oeis.org

3, 4, 0, -6, -4, 10, 20, 2, -36, -38, 36, 114, 44, -182, -268, 98, 636, 442, -828, -1710, -52, 3370, 3476, -3262, -10212, -3686, 16740, 24114, -9364, -57590, -38860, 76322, 154044, 1402, -306684, -309486, 303884, 922858, 315092, -1530622, -2160804, 900442, 5222052, 3421170

Offset: 0

Creighton Dement, Apr 14 2005

Floretion Algebra Multiplication Program, FAMP Code: 1vesseq[.5'j + .5'k + .5j' + .5k' + .5'ii' + .5e]

Index entries for linear recurrences with constant coefficients, signature (2,-3,2).

Cf. A105577, A105578, A105579, A105580.

Equals 2*A107920(n) + A107920(n-1) + 1.

LinearRecurrence[{2,-3,2},{3,4,0},50] (* Harvey P. Dale, Jul 05 2022 *)

2*a(n) = A105225(n) + A105577(n) + 4*((-1)^n)*A001607(n+1)

G.f.: (3-2x+x^2)/((1-x)(1-x+2x^2)). a(n)=1+A107920(n)+2*A107920(n+1). [From R. J. Mathar, Feb 04 2009]

A386489 Expansion of (1-x)/((1+x+2x^2)(1-4*x+x^2)).

Original entry on oeis.org

1, 2, 7, 30, 109, 402, 1511, 5638, 21021, 78474, 292887, 1093006, 4079181, 15223810, 56815879, 212039702, 791343293, 2953333114, 11021988791, 41134623134, 153516503405, 572931388658, 2138209053735, 7979904827430, 29781410249821, 111145736175722

Offset: 0

Greg Dresden and Madison Lingchen Zhou, Aug 20 2025

a(n) is the number of ways to tile a 2 X n board with squares, dominoes, and L-shaped quadrominoes. Here is one of the a(4)=109 possible tilings of a 2 X 4 board:
    _____
   | | |||
   ||____|
Compare to A030186 which counts the tilings with just squares and dominos.

Index entries for linear recurrences with constant coefficients, signature (3,1,7,-2).

Cf. A030186, A033505.

LinearRecurrence[{3, 1, 7, -2}, {1, 2, 7, 30}, 30]

a(n) = 3*a(n-1) + a(n-2) + 7*a(n-3) - 2*a(n-4).
a(n) = A030186(n) + 2*sum_{i=0..n-2}(A033505(n-i-3)*a(i) + A030186(n-i-3)*(a(i)+2*sum_{j=0..i} a(j)).
a(n) ~ (2 + sqrt(3))^(n+2) / (18 + 4*sqrt(3)). - Vaclav Kotesovec, Aug 21 2025
23*a(n) = -4*A001353(n)+13*A001353(n+1) +10*A001607(n+1)+8*A001607(n) . - R. J. Mathar, Aug 26 2025

cp's OEIS Frontend

A110523 Expansion of (1 + x)/(1 + x + 3*x^2).

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A167433 Row sums of the Riordan array (1-4x+4x^2, x(1-2x)) (A167431).

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A105578 a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 1, a(2) = 0.

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A169998 a(0)=1, a(1)=1; thereafter a(n) = -a(n-1) - 2*a(n-2).

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A113726 A Jacobsthal convolution.

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A107443 Expansion of g.f.: (1+3*x^2)/((1-x)*(1+x+2*x^2)*(1-x+2*x^2)).

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A172250 Triangle, read by rows, given by [0,1,-1,0,0,0,0,0,0,0,...] DELTA [1,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A078033 Expansion of (1-x) / (1+x^2+2*x^3).

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A107443 Expansion of g.f.: (1+3x^2)/((1-x)(1+x+2x^2)(1-x+2*x^2)).

A105576 a(n) = 2a(n-1) - 3a(n-2) + 2*a(n-3) with a(0) = 3, a(1) = 4, a(2) = 0.

A386489 Expansion of (1-x)/((1+x+2x^2)(1-4*x+x^2)).