A182890 -id:A182890 - cp's OEIS frontend

A108479 Antidiagonal sums of number triangle A086645.

1, 1, 2, 7, 17, 44, 117, 305, 798, 2091, 5473, 14328, 37513, 98209, 257114, 673135, 1762289, 4613732, 12078909, 31622993, 82790070, 216747219, 567451585, 1485607536, 3889371025, 10182505537, 26658145586, 69791931223, 182717648081

Offset: 0

Paul Barry, Jun 04 2005

Seiichi Manyama, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,1,2,-1).

Cf. A005252, A051286, A086645, A182890.
Cf. A381421, A382230, A382470, A382471, A382472, A382473, A382474.
Cf. A376729, A376730, A376731.

LinearRecurrence[{2,1,2,-1},{1,1,2,7},30] (* Harvey P. Dale, Jun 01 2021 *)

G.f.: (1 - x - x^2)/(1 - 2*x - x^2 - 2*x^3 + x^4).
a(n) = 2*(n-1) + a(n-2) + 2*a(n-3) - a(n-4).
a(n) = Sum_{k=0..floor(n/2)} C(2*(n-k), 2*k).
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(2*(n-2*k), j) * C(2*k, j).
a(n) = A005252(2*n). - Seiichi Manyama, Aug 11 2024

A375278 Expansion of 1/((1 - x - x^3)^2 - 4*x^4).

Original entry on oeis.org

1, 2, 3, 6, 15, 34, 70, 146, 317, 690, 1480, 3162, 6788, 14608, 31395, 67392, 144701, 310854, 667793, 1434310, 3080542, 6616676, 14212315, 30526804, 65567936, 140832740, 302495240, 649730544, 1395554885, 2997508382, 6438345511, 13828920758, 29703127299

Offset: 0

Seiichi Manyama, Aug 09 2024

nonn

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

Cf. A182890, A375283, A375285.

Cf. A246840, A375279.

my(N=40, x='x+O('x^N)); Vec(1/((1-x-x^3)^2-4*x^4))

a(n) = sum(k=0, n\3, binomial(2*n-4*k+2, 2*k+1))/2;

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) + 2*a(n-4) - a(n-6).

a(n) = (1/2) * Sum_{k=0..floor(n/3)} binomial(2*n-4*k+2,2*k+1).

A115730 a(n) = a(n-3) + A001654(n-1) with a(0)=0, a(1)=0 and a(2)=1.

Original entry on oeis.org

0, 0, 1, 2, 6, 16, 42, 110, 289, 756, 1980, 5184, 13572, 35532, 93025, 243542, 637602, 1669264, 4370190, 11441306, 29953729, 78419880, 205305912, 537497856, 1407187656, 3684065112, 9645007681, 25250957930, 66107866110

Offset: 0

Roger L. Bagula, Mar 13 2006

The a(n+1) represent the Ca2 and Ze4 sums of the Golden Triangle A180662. Furthermore the a(3*n) represent the Ze1 (terms doubled) and Ca3 sums of the Golden triangle. See A180662 for more information about these and other triangle sums.

			G.f. = x^2 + 2*x^3 + 6*x^4 + 16*x^5 + 42*x^6 + 110*x^7 + 289*x^8 + ... - _Michael Somos_, Sep 05 2023

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

Cf. A000045, A001654, A064831, A079962, A180662, A180664, A180665, A180666, A182890.

function A115730(n)
  if n lt 3 then return Floor(n/2);
  else return A115730(n-3) + Fibonacci(n-1)*Fibonacci(n);
  end if; return A115730;
end function;
[A115730(n): n in [0..40]]; // G. C. Greubel, Jan 20 2022

nmax:=31: with(combinat): for n from 0 to nmax do A001654(n):=fibonacci(n) * fibonacci(n+1) od: a(0):=0: a(1):=0: a(2):=1: for n from 3 to nmax do a(n):=a(n-3) + A001654(n-1) od: seq(a(n),n=0..nmax);

LinearRecurrence[{2,2,0,-2,-2,1}, {0,0,1,2,6,16}, 40] (* modified by G. C. Greubel, Jan 20 2022 *)
a[ n_] := Floor[(2*Fibonacci[2*n+1] + Fibonacci[2*n+2] + 2)/20]; (* Michael Somos, Sep 05 2023 *)

{a(n) = (2*fibonacci(2*n+1) + fibonacci(2*n+2) + 2)\20}; /* Michael Somos, Sep 05 2023 */

U=chebyshev_U
def A115730(n): return (1/60)*((-1)^n*(6 - 5*U(n, 1/2) + 10*U(n-1, 1/2)) - (10 - 9*U(n, 3/2) + 6*U(n-1, 3/2)))
[A115730(n) for n in (0..40)] # G. C. Greubel, Jan 20 2022

a(n) = -floor(g(Fibonacci(n+1))) where g(x) = (1-x^2)^2/(-4*x^2).
G.f.: x^2/( (1-x)*(1+x)*(1+x+x^2)*(1-3*x+x^2) ). - R. J. Mathar, Jun 20 2015
a(n) - a(n-2) = A182890(n-1). - R. J. Mathar, Jun 20 2015
a(n) = (1/60)*((-1)^n*(6 - 5*ChebyshevU(n, 1/2) + 10*ChebyshevU(n-1, 1/2)) - (10 - 9*ChebyshevU(n, 3/2) + 6*ChebyshevU(n-1, 3/2))). - G. C. Greubel, Jan 20 2022
a(n) = floor((2*Fibonacci(2*n+1) + Fibonacci(2*n+2) + 2)/20). - Michael Somos, Sep 05 2023

Corrected and information added by Johannes W. Meijer, Sep 22 2010

Edited by Editors-in-Chief. - N. J. A. Sloane, Jun 20 2015

A376723 Expansion of 1/((1 - x^2 - x^3)^2 - 4*x^5).

Original entry on oeis.org

1, 0, 2, 2, 3, 10, 7, 28, 33, 64, 132, 170, 408, 578, 1119, 2002, 3194, 6310, 10021, 18666, 32353, 55450, 101443, 170672, 308744, 534820, 935936, 1663892, 2872669, 5111652, 8898082, 15641802, 27538647, 48049562, 84813451, 148219128, 260572901, 457451088

Offset: 0

Seiichi Manyama, Oct 02 2024

nonn

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

Cf. A182890, A376724, A376725.
Cf. A376726, A376729.
Cf. A375278.

my(N=40, x='x+O('x^N)); Vec(1/((1-x^2-x^3)^2-4*x^5))

a(n) = sum(k=0, n\2, binomial(2*k+2, 2*n-4*k+1))/2;

a(n) = 2*a(n-2) + 2*a(n-3) - a(n-4) + 2*a(n-5) - a(n-6).

a(n) = (1/2) * Sum_{k=0..floor(n/2)} binomial(2*k+2,2*n-4*k+1).

A376724 Expansion of 1/((1 - x^3 - x^4)^2 - 4*x^7).

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 3, 10, 3, 4, 28, 28, 9, 60, 126, 66, 115, 396, 403, 292, 1007, 1724, 1281, 2366, 5736, 6128, 6468, 16202, 24888, 23664, 43055, 85158, 97156, 124044, 257474, 374538, 421785, 740324, 1294129, 1577756, 2217676, 4085272, 5813587, 7319572, 12370630

Offset: 0

Seiichi Manyama, Oct 02 2024

nonn

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

Cf. A182890, A376723, A376725.

Cf. A376727, A376730.

my(N=50, x='x+O('x^N)); Vec(1/((1-x^3-x^4)^2-4*x^7))

a(n) = sum(k=0, n\3, binomial(2*k+2, 2*n-6*k+1))/2;

a(n) = 2*a(n-3) + 2*a(n-4) - a(n-6) + 2*a(n-7) - a(n-8).

a(n) = (1/2) * Sum_{k=0..floor(n/3)} binomial(2*k+2,2*n-6*k+1).

A375255 Expansion of 1/(1 - 2x + 3x^2 + 2*x^3 + x^4).

Original entry on oeis.org

1, 2, 1, -6, -20, -26, 19, 162, 339, 180, -1000, -3380, -4459, 3042, 27221, 57614, 31940, -166446, -571161, -764478, 485479, 4573160, 9790000, 5654040, -27693719, -96502718, -131022359, 77196834, 768159900, 1663276734, 998702459, -4605941918, -16302704581

Offset: 0

Seiichi Manyama, Aug 09 2024

sign

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

Cf. A182890, A375273.

Cf. A375021, A375275.

my(N=40, x='x+O('x^N)); Vec(1/(1-2*x+3*x^2+2*x^3+x^4))

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

a(n) = 2*a(n-1) - 3*a(n-2) - 2*a(n-3) - a(n-4).

a(n) = (1/2) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-2*k+2,2*k+1).

A375283 Expansion of 1/((1 - x - x^4)^2 - 4*x^5).

Original entry on oeis.org

1, 2, 3, 4, 7, 16, 35, 68, 122, 220, 417, 816, 1588, 3028, 5707, 10784, 20547, 39322, 75150, 143144, 272212, 517990, 987005, 1881824, 3586808, 6832874, 13013780, 24789200, 47229672, 89991518, 171459667, 326651952, 622295173, 1185547900, 2258689217, 4303264572

Offset: 0

Seiichi Manyama, Aug 09 2024

nonn

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

Cf. A182890, A375278, A375285.

Cf. A246883, A375282.

my(N=40, x='x+O('x^N)); Vec(1/((1-x-x^4)^2-4*x^5))

a(n) = sum(k=0, n\4, binomial(2*n-6*k+2, 2*k+1))/2;

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-4) + 2*a(n-5) - a(n-8).

a(n) = (1/2) * Sum_{k=0..floor(n/4)} binomial(2*n-6*k+2,2*k+1).

A376716 Expansion of (1 - x + x^2)/((1 - x + x^2)^2 - 4*x^2).

Original entry on oeis.org

1, 1, 4, 11, 27, 72, 189, 493, 1292, 3383, 8855, 23184, 60697, 158905, 416020, 1089155, 2851443, 7465176, 19544085, 51167077, 133957148, 350704367, 918155951, 2403763488, 6293134513, 16475640049, 43133785636, 112925716859, 295643364939, 774004377960

Offset: 0

Seiichi Manyama, Oct 02 2024

nonn

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

Cf. A000302, A376717, A376718.

Cf. A182890.

my(N=30, x='x+O('x^N)); Vec((1-x+x^2)/((1-x+x^2)^2-4*x^2))

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

a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4).

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-2*k+1,2*k).

A376725 Expansion of 1/((1 - x^4 - x^5)^2 - 4*x^9).

Original entry on oeis.org

1, 0, 0, 0, 2, 2, 0, 0, 3, 10, 3, 0, 4, 28, 28, 4, 5, 60, 126, 60, 11, 110, 396, 396, 117, 188, 1001, 1716, 1009, 462, 2191, 5720, 5729, 2592, 4564, 15920, 24320, 16482, 12036, 39168, 84000, 84750, 51927, 93024, 249292, 353738, 269962, 258324, 666932, 1250142

Offset: 0

Seiichi Manyama, Oct 02 2024

nonn

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

Cf. A182890, A376723, A376724.

Cf. A376728, A376731.

my(N=50, x='x+O('x^N)); Vec(1/((1-x^4-x^5)^2-4*x^9))

a(n) = sum(k=0, n\4, binomial(2*k+2, 2*n-8*k+1))/2;

a(n) = 2*a(n-4) + 2*a(n-5) - a(n-8) + 2*a(n-9) - a(n-10).

a(n) = (1/2) * Sum_{k=0..floor(n/4)} binomial(2*k+2,2*n-8*k+1).

A182888 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps at level 0. These are paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 8, 7, 6, 4, 0, 1, 17, 20, 12, 8, 5, 0, 1, 38, 44, 36, 18, 10, 6, 0, 1, 89, 104, 82, 56, 25, 12, 7, 0, 1, 206, 253, 204, 132, 80, 33, 14, 8, 0, 1, 485, 604, 513, 344, 195, 108, 42, 16, 9, 0, 1, 1152, 1466, 1262, 891, 530, 272, 140, 52, 18, 10, 0, 1

Offset: 0

Emeric Deutsch, Dec 11 2010

			T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, namely hH and Hh, have exactly two (1,0)-steps at level 0.
Triangle starts:
   1;
   0,   1;
   1,   0,  1;
   2,   2,  0,  1;
   3,   4,  3,  0,  1;
   8,   7,  6,  4,  0,  1;
  17,  20, 12,  8,  5,  0, 1;
  38,  44, 36, 18, 10,  6, 0, 1;
  89, 104, 82, 56, 25, 12, 7, 0, 1;
  ...

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini and N. Zagaglia Salvi, On the Rank Polynomial of the Lattice of Order Ideals of Fences and Crowns, Discrete Mathematics 259 (2002), 163-177.

Row sums give A051286.
Column k=0 gives A182889.
Cf. A182890.

G:=1/(z-t*z+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,k),k=0..n) od; # yields sequence in triangular form

G.f.: G(t,z) = 1/( z-tz+sqrt((1+z+z^2)(1-3z+z^2)) ).

Sum_{k=0..n} k*T(n,k) = A182890(n).

cp's OEIS Frontend

A108479 Antidiagonal sums of number triangle A086645.

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A375278 Expansion of 1/((1 - x - x^3)^2 - 4*x^4).

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A115730 a(n) = a(n-3) + A001654(n-1) with a(0)=0, a(1)=0 and a(2)=1.

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A376723 Expansion of 1/((1 - x^2 - x^3)^2 - 4*x^5).

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A376724 Expansion of 1/((1 - x^3 - x^4)^2 - 4*x^7).

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A375255 Expansion of 1/(1 - 2*x + 3*x^2 + 2*x^3 + x^4).

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A375283 Expansion of 1/((1 - x - x^4)^2 - 4*x^5).

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A376716 Expansion of (1 - x + x^2)/((1 - x + x^2)^2 - 4*x^2).

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A375255 Expansion of 1/(1 - 2x + 3x^2 + 2*x^3 + x^4).