A006054 -id:A006054 - cp's OEIS frontend

A234713 Triangle, read by rows, based on the Fibonacci numbers.

0, 1, 1, 1, 3, 2, 2, 6, 7, 3, 3, 13, 20, 14, 4, 5, 25, 51, 51, 25, 5, 8, 48, 118, 154, 111, 41, 6, 13, 89, 260, 416, 393, 217, 63, 7, 21, 163, 548, 1042, 1218, 890, 392, 92, 8, 34, 294, 1121, 2465, 3435, 3127, 1842, 666, 129, 9, 55, 525, 2236, 5586, 9035, 9845

Offset: 0

Philippe Deléham, Dec 29 2013

First column is the Fibonacci sequence.

Sum_{k=0..n} T(n,k)*2^k = -A106732(n).

			Triangle begins:
0
1, 1
1, 3, 2
2, 6, 7, 3
3, 13, 20, 14, 4
5, 25, 51, 51, 25, 5
8, 48, 118, 154, 111, 41, 6
13, 89, 260, 416, 393, 217, 63, 7
21, 163, 548, 1042, 1218, 890, 392, 92, 8

Cf. Diagonals: A001477, A004006.
Cf. Columns: A000045 (Fibonacci), A131913, A261054.
Cf. A025192 (row sums for n>0), A006054 (diagonal sums)

G.f.: (y+1)*x/(1-(2y+1)*x+(y^2-1)*x^2).

T(n,k)=T(n-1,k)+2*T(n-1,k-1)+T(n-2,k)-T(n-2,k-2), T(0,0)=0, T(1,0)=1, T(1,1)=1, T(n,k)=0 if k<0 or if k>n.

A370377 a(n) is the number of symmetrical linear hydrocarbon chains with n C-C bonds.

Original entry on oeis.org

1, 3, 2, 6, 5, 14, 11, 31, 25, 70, 56, 157, 126, 353, 283, 793, 636, 1782, 1429, 4004, 3211, 8997, 7215, 20216, 16212, 45425, 36428, 102069, 81853, 229347, 183922, 515338, 413269, 1157954, 928607, 2601899, 2086561, 5846414, 4688460, 13136773, 10534874

Offset: 0

Tomasz Dziekanski, Feb 18 2024

			For n = 1: a(1) = A006356(1) = 3
 CH3-CH3, CH2=CH2, CH≡CH
For n = 3: a(3) = A006356(2) = 6
 CH3-CH2-CH2-CH3, CH3-CH=CH-CH3, CH3-C≡C-CH3, CH2=CH-CH=CH2, CH≡C-C≡CH, CH2=C=C=CH2
For n = 4: a(4) = A006356(2) - A006356(0) = 6 - 1 = 5
 CH3-CH2-CH2-CH2-CH3, CH3-CH=C=CH-CH3, CH2=CH-CH2-CH=CH2, CH≡C-CH2-C≡CH, CH2=C=C=C=CH2

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

Cf. A006356, A006054, A306334.

LinearRecurrence[{0, 2, 0, 1, 0, -1}, {1, 3, 2, 6, 5, 14}, 50] (* Paolo Xausa, Feb 22 2024 *)

Vec(O(x^55)+(1+3*x-x^5)/(1-2*x^2-x^4+x^6)) \\ Joerg Arndt, Feb 18 2024

a = [1, 3, 2, 6, 5, 14]
for i in range(30):
    a.append(2*a[-2]+a[-4]-a[-6])
print(a)

a(n) = 2*A306334(n) - A006356(n).
Also:
a(0) = 1;
a(2) = 2;
a(n) = A006356((n+1)/2) if n is odd;
a(n) = A006356(n/2) - A006356((n-4)/2) if n is even.
G.f.: (1+3*x-x^5)/(1-2*x^2-x^4+x^6). - Joerg Arndt, Feb 18 2024

A120771 Expansion of ( 1-x^3+x^4+x^5-x^8 ) / ( 1-2*x^3-x^6+x^9 ).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 2, 1, 6, 5, 3, 14, 11, 6, 31, 25, 14, 70, 56, 31, 157, 126, 70, 353, 283, 157, 793, 636, 353, 1782, 1429, 793, 4004, 3211, 1782, 8997, 7215, 4004, 20216, 16212, 8997, 45425, 36428, 20216, 102069, 81853, 45425, 229347, 183922, 102069, 515338, 413269, 229347, 1157954, 928607, 515338

Offset: 0

Gary W. Adamson, Jul 03 2006

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

Cf. A077998 (trisection), A006054 (trisection), A006356 (trisection), A038196.

CoefficientList[Series[(1-x^3+x^4+x^5-x^8)/(1-2*x^3-x^6+x^9),{x,0,60}],x] (* or *) LinearRecurrence[{0,0,2,0,0,1,0,0,-1},{1,0,0,1,1,1,3,2,1},60] (* Harvey P. Dale, Feb 19 2016 *)

Three consecutive coefficients are generated from the left row of the n-th power of the matrix [1,1,1; 1,1,0; 1,0,0].

A122499 Semiprimes in A006053.

Original entry on oeis.org

4, 9, 14, 155, 2993, 9707, 184183, 331981, 1942071, 1263047761, 140390505643, 455845099957, 296452328830865, 32951342156444219, 2381929669709247097441, 9063289616192276216577361, 34485996673867704851362967681, 426068342298911680872493712146539, 117190394374593808526426397401539675762247

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 15 2006

nonn

Intersection of A001358 and A006053.

Cf. A006054.

SemiprimeQ[n_Integer] := Plus @@ (Last /@ FactorInteger[n]) == 2;
a = Table[ SeriesCoefficient[ Series[x/(x^3 - 2*x^2 - x + 1), {x, 0, 50}], n], {n, 0, 50}];
f[n_] = If[SemiprimeQ[a[[n]]] == True, a[[n]], {}];
Flatten[Table[f[n], {n, 1, Length[a]}]]

Edited by N. J. A. Sloane, Sep 17 2006

More terms from Amiram Eldar, Jun 06 2025

A122504 a(n) = -a(n-6) + 3a(n-5) + a(n-4) - 7a(n-3) + a(n-2) + 3*a(n-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, -3, -13, -39, -107, -273, -675, -1624, -3847, -8995, -20851, -47995, -109915, -250695, -570024, -1292915, -2926953, -6616051, -14936895, -33690357, -75931283, -171029936, -385046687, -866536007, -1949510615, -4384874471, -9860587191, -22170707871, -49842661456

Offset: 1

Roger L. Bagula and Gary W. Adamson, Sep 15 2006

sign

Original name started "Bi_Steinbach heptagon recursion".

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

Cf. A006054, A006053.

a[0] = 1; a[1] = 1; a[2] = 1; a[3] = 1; a[4] = 1; a[5] = 1; a[n_] := a[n] = -a[n - 6] + 3 a[n - 5] + a[n - 4] - 7 a[n - 3] + a[n - 2] + 3 a[n - 1] Table[a[n], {n, 0, 30}]
M = {{0, 1, 0, 0, 0, 0}, {0, 0, 1, 0, 0, 0}, {0, 0, 0, 1, 0, 0}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, 0, 0, 1}, {-1, 3, 1, -7, 1, 3}} v[1] = {1, 1, 1, 1, 1, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Floor[v[n][[1]]], {n, 1, 50}]
LinearRecurrence[{3,1,-7,1,3,-1},{1,1,1,1,1,1},40] (* or *) Rest[ CoefficientList[ Series[x(1-x-3x^2)(1-x-x^2)/((1-2x-x^2+x^3)(1-x-2x^2+x^3)),{x,0,40}],x]] (* Harvey P. Dale, Jun 24 2011 *)

O.g.f.: x*(1-x-3*x^2)*(1-x-x^2)/((1-2*x-x^2+x^3)*(1-x-2*x^2+x^3)). - R. J. Mathar, Aug 22 2008

A122517 G.f.: 1/(1 - x^3 - 2 x^4 + x^5).

Original entry on oeis.org

1, 0, 0, 1, 2, -1, 1, 4, 2, -3, 7, 9, -3, -1, 26, 8, -16, 27, 61, -26, -13, 131, 69, -126, 131, 344, -119, -190, 732, 438, -772, 471, 2092, -628, -1511, 3806, 3085, -4859, 1412, 12208, -2495, -11391, 19891, 20509, -28589, -396, 71682, -7462, -78083, 99479

Offset: 0

Roger L. Bagula, Sep 16 2006

sign

Based on characteristic polynomial x^5 - x^2 - 2x + 1.

Cf. A006054, A006053.

p[x_] := x^5 - x^2 - 2x + 1 q[x_] := ExpandAll[x^5*p[1/x]] Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 50}], n], {n, 0, 50}]

Edited by N. J. A. Sloane, Oct 01 2006

A122518 G.f.: 1/(1 -2 x^3 - x^4 + x^5).

Original entry on oeis.org

1, 0, 0, 2, 1, -1, 4, 4, -3, 6, 13, -6, 5, 35, -5, -9, 81, 20, -58, 158, 130, -177, 238, 476, -382, 169, 1367, -526, -520, 3285, 146, -2933, 6576, 4097, -9005, 10073, 17703, -20489, 7044, 54484, -33348, -24104, 136501, -19256, -136040, 282246, 122093, -427837, 447708, 662472

Offset: 0

Roger L. Bagula, Sep 16 2006

Based on characteristic polynomial x^5 - 2x^2 - x + 1.

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

Cf. A006054, A006053.

p[x_] := x^5 - 2x^2 - x + 1 q[x_] := ExpandAll[x^5*p[1/x]] Table[ SeriesCoefficient[ Series[x/q[x], {x, 0, 50}], n], {n, 0, 50}]

Edited by N. J. A. Sloane, Oct 01 2006

A319106 Expansion of Product_{k>=1} (1 + x^k)^ceiling(k/2).

Original entry on oeis.org

1, 1, 1, 3, 4, 7, 11, 17, 26, 40, 60, 88, 131, 190, 276, 398, 568, 806, 1142, 1603, 2242, 3124, 4328, 5973, 8214, 11249, 15349, 20879, 28297, 38235, 51513, 69190, 92674, 123811, 164961, 219248, 290705, 384537, 507515, 668376, 878339, 1151899, 1507679, 1969503, 2567976, 3342227

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Weigh transform of 1, 1, 2, 2, 3, 3, 4, 4, ... (A110654).

N. J. A. Sloane, Transforms

Cf. A003293, A006054, A026007, A110654, A263140.

a:=series(mul((1+x^k)^ceil(k/2),k=1..100),x=0,46): seq(coeff(a,x,n),n=0..45); # Paolo P. Lava, Apr 02 2019

nmax = 45; CoefficientList[Series[Product[(1 + x^k)^Ceiling[k/2], {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Product[((1 + x^(2 k - 1))(1 + x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d Ceiling[d/2], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]

G.f.: Product_{k>=1} (1 + x^k)^A110654(k).
G.f.: Product_{k>=1} ((1 + x^(2*k-1))*(1 + x^(2*k)))^k.
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*ceiling(d/2) ) * x^k/k).
a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(5/3) + Pi^2 * n^(1/3) / (2^(10/3) * 3^(4/3) * Zeta(3)^(1/3)) - Pi^4 / (2^7 * 3^4 * Zeta(3))) * Zeta(3)^(1/6) / (2^(7/8) * 3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Sep 11 2018

cp's OEIS Frontend

A234713 Triangle, read by rows, based on the Fibonacci numbers.

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A370377 a(n) is the number of symmetrical linear hydrocarbon chains with n C-C bonds.

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A120771 Expansion of ( 1-x^3+x^4+x^5-x^8 ) / ( 1-2*x^3-x^6+x^9 ).

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A122499 Semiprimes in A006053.

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A122504 a(n) = -a(n-6) + 3*a(n-5) + a(n-4) - 7*a(n-3) + a(n-2) + 3*a(n-1).

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A122517 G.f.: 1/(1 - x^3 - 2 x^4 + x^5).

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A122518 G.f.: 1/(1 -2 x^3 - x^4 + x^5).

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A319106 Expansion of Product_{k>=1} (1 + x^k)^ceiling(k/2).

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Maple

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A122504 a(n) = -a(n-6) + 3a(n-5) + a(n-4) - 7a(n-3) + a(n-2) + 3*a(n-1).