A007576 -id:A007576 - cp's OEIS frontend

A086821 Duplicate of A007576.

Original entry on oeis.org

1, 1, 1, 3, 7, 15, 35, 87, 217, 547, 1417, 3735, 9911, 26513, 71581, 194681, 532481

Offset: 0

dead

A123762 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 2.

Original entry on oeis.org

1, 4, 37, 375, 4493, 56848, 753536, 10283622, 143607345, 2041497919, 29446248496, 429858432108

Offset: 1

Søren Eilers, Oct 29 2006

M. Abrahamsen and S. Eilers, On the asymptotic enumeration of LEGO structures, Exper Math. 20 (2) (2011) 145-152.
B. Durhuus and S. Eilers, On the entropy of LEGO, arXiv:math/0504039 [math.CO], 2005.
B. Durhuus and S. Eilers, On the entropy of LEGO, J. Appl. Math. Comput. 45 (1-2) (2014), 433-448.
S. Eilers, A LEGO Counting problem, 2005.
S. Eilers, The LEGO counting problem, Amer. Math. Monthly, 123 (May 2016), 415-426.
Index entry for sequences related to LEGO blocks

Cf. A007576, A082679, A112389, A112390.

A316706 Number of solutions to k_1 + 2k_2 + ... + nk_n = n, where k_i are from {-1,0,1}, i=1..n.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 27, 69, 178, 457, 1194, 3178, 8538, 23062, 62726, 171804, 473069, 1308397, 3634075, 10133154, 28352421, 79575702, 223981549, 632101856, 1788172541, 5069879063, 14403962756, 41001479103, 116921037003, 333971884899, 955443681814, 2737387314548, 7853533625522, 22560919253095, 64890249175438, 186854616134794

Offset: 0

Paul D. Hanna, Jul 10 2018

nonn

a(n) is the coefficient of both x^n and 1/x^n in Product_{k=1..n} (1/x^k + 1 + x^k), while A007576 gives the constant term in the symmetric product.

Alois P. Heinz, Table of n, a(n) for n = 0..700 (first 501 terms from Vaclav Kotesovec)

Cf. A007576, A059529.

nmax = 40; p = 1; Flatten[{1, Table[Coefficient[p = Expand[p*(1/x^n + 1 + x^n)], x^n], {n, 1, nmax}]}] (* Vaclav Kotesovec, Jul 11 2018 *)

{a(n) = polcoeff( prod(k=1,n, 1/x^k + 1 + x^k) + x*O(x^n),n)}
for(n=0,40, print1(a(n),", "))

from collections import Counter
def A316706(n):
    c = {0:1}
    for k in range(1,n+1):
        b = Counter(c)
        for j in c:
            a = c[j]
            b[j+k] += a
            b[j-k] += a
        c = b
    return c[n] # Chai Wah Wu, Feb 05 2024

a(n) = [x^n] Product_{k=1..n} (1/x^k + 1 + x^k).
a(n) = [x^(n*(n-1)/2)] Product_{k=1..n} (1 + x^k + x^(2*k)).
a(n) = [x^(n*(n+3)/2)] Product_{k=1..n} (1 + x^k + x^(2*k)).
a(n) ~ 3^(n + 1) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 11 2018

A225310 T(n,k)=Number of nXk -1,1 arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero and rows are nondecreasing (ways to put k thrusters pointing east or west at each of n positions 1..n distance from the hinge of a south-pointing gate without turning the gate).

Original entry on oeis.org

0, 1, 0, 0, 1, 2, 1, 0, 3, 2, 0, 3, 6, 7, 0, 1, 0, 9, 16, 15, 0, 0, 3, 12, 31, 0, 35, 8, 1, 0, 17, 52, 113, 0, 87, 14, 0, 5, 22, 83, 0, 443, 474, 217, 0, 1, 0, 27, 122, 427, 0, 1787, 1576, 547, 0, 0, 5, 34, 175, 0, 2341, 5304, 7445, 0, 1417, 70, 1, 0, 41, 238, 1165, 0, 13333, 26498

Offset: 1

R. H. Hardin May 05 2013

Table starts
..0....1....0......1.....0.......1......0........1......0.........1.......0
..0....1....0......3.....0.......3......0........5......0.........5.......0
..2....3....6......9....12......17.....22.......27.....34........41......48
..2....7...16.....31....52......83....122......175....238.......317.....410
..0...15....0....113.....0.....427......0.....1165......0......2591.......0
..0...35....0....443.....0....2341......0.....8221......0.....22351.......0
..8...87..474...1787..5304...13333..29638....60007.112790....199669..336342
.14..217.1576...7445.26498...77721.197440...449693.939130...1828785.3360554
..0..547....0..31593.....0..461973......0..3437315......0..17085339.......0
..0.1417....0.136351.....0.2791167......0.26700429......0.162204059.......0

			Some solutions for n=4 k=4
..1..1..1..1...-1.-1..1..1...-1.-1.-1..1....1..1..1..1...-1..1..1..1
..1..1..1..1...-1..1..1..1...-1.-1.-1..1...-1.-1.-1.-1...-1.-1..1..1
.-1.-1.-1.-1....1..1..1..1...-1..1..1..1....1..1..1..1...-1..1..1..1
.-1.-1..1..1...-1.-1.-1.-1...-1.-1..1..1...-1.-1.-1..1...-1.-1.-1..1

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A063865
Column 2 is A007576
Row 3 is A008810(n+1)

Empirical for row n:
n=1: a(n) = a(n-2)
n=2: a(n) = a(n-2) +a(n-4) -a(n-6)
n=3: a(n) = 2*a(n-1) -a(n-2) +a(n-3) -2*a(n-4) +a(n-5)
n=4: a(n) = a(n-1) +a(n-2) -2*a(n-5) +a(n-8) +a(n-9) -a(n-10)
n=5: [order 18]
n=6: [order 42]
n=7: [order 24]
n=8: [order 36]

A350249 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 7, 19, 43, 95, 189, 429, 1003, 2457, 6319, 16165, 41601, 107969, 280253, 737065, 1950865, 5201941, 13954313, 37593679, 101695957, 276296549, 753191093, 2061201397, 5658850121, 15583938539, 43040609115, 119182143639, 330841253283, 920550527585

Offset: 0

Ilya Gutkovskiy, Jan 28 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A007576, A158092, A350495, A350880.

b:= proc(n) option remember; `if`(n=0, 1,
      expand((x^(n^2)+1+1/x^(n^2))*b(n-1)))
    end:
a:= n-> coeff(b(n),x,0):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 28 2022

Table[Coefficient[Product[x^(k^2) + 1 + 1/x^(k^2), {k, 1, n}], x, 0], {n, 0, 30}] (* Vaclav Kotesovec, Feb 05 2022 *)

Conjecture: a(n) ~ sqrt(5) * 3^(n + 1/2) / (2*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Feb 04 2022

A156181 Number of solutions to e(1)1 + e(2)2 + ... + e(n)n = e(-1)1 + e(-2)2 + ... + e(-n)n, where e(j) are from {-1,0,1}, j=-n,...,n.

Original entry on oeis.org

1, 3, 13, 65, 403, 2669, 18759, 136477, 1020373, 7785741, 60395165, 474817833, 3775005799, 30298719855, 245167429681, 1997854542163, 16381233095985, 135050690760831, 1118800428892925, 9308791880014333, 77755512086256649

Offset: 0

Steven Finch, Feb 05 2009

nonn

a(n) = coefficient of x^(n*(n+1)) in the polynomial Product_{k=1..n} (1 + x^k + x^(2*k))^2, and is the maximal such coefficient as well.

Ray Chandler, Table of n, a(n) for n = 0..1052 (terms < 10^1000)
Steven R. Finch, Signum equations and extremal coefficients, February 7, 2009. [Cached copy, with permission of the author]

Cf. A007576, A047653, A063865.

Table[Coefficient[Expand[Product[(1 + x^k + x^(2*k))^2, {k, 1, n}]],x, n*(n + 1)], {n, 0, 20}]

a(n) is the constant term in expansion of Product_{k=1..n} (x^k + 1 + 1/x^k)^2. - Ilya Gutkovskiy, Jan 22 2024

A350880 a(n) is the constant term in expansion of Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).

Original entry on oeis.org

1, 1, 1, 3, 5, 7, 17, 39, 95, 233, 561, 1435, 3643, 9417, 24973, 66695, 177915, 475629, 1293017, 3517223, 9636365, 26676197, 73848517, 205382439, 571628347, 1588203787, 4435819313, 12474619295, 35194448271, 99782519701, 283514955585, 799783925547

Offset: 0

Ilya Gutkovskiy, Jan 20 2022

nonn

a(n) is the number of solutions to 0 = Sum_{i=1..n} c_i * prime(i) with c_i in {-1,0,1}. a(3) = 3: -2-3+5, +2+3-5, 0+0+0. - Alois P. Heinz, Dec 28 2023

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A000040, A007576, A022894.

s:= proc(n) s(n):= `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      b(n, i-1)+b(n+ithprime(i), i-1)+b(abs(n-ithprime(i)), i-1)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 28 2023

s[n_] := s[n] = If[n < 1, 0, Prime[n] + s[n-1]];
b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 0, 1, b[n, i-1] + b[n + Prime[i], i-1] + b[Abs[n - Prime[i]], i-1]]];
a[n_] := b[0, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 27 2025, after Alois P. Heinz *)

a(n) = polcoef (prod(k=1, n, x^prime(k) + 1 + 1/x^prime(k)), 0); \\ Michel Marcus, Jan 21 2022

A369628 Number of solutions to k_1 + 2k_2 + ... + nk_n = 1, where k_i are from {-1,0,1}, i=1..n.

Original entry on oeis.org

0, 1, 2, 3, 6, 15, 36, 85, 213, 549, 1423, 3723, 9882, 26508, 71579, 194533, 532120, 1463561, 4044075, 11221727, 31260192, 87386579, 245058185, 689209348, 1943530845, 5494106583, 15566303698, 44196212866, 125727934145, 358317169828, 1022916667066, 2924843243594

Offset: 0

Ilya Gutkovskiy, Jan 28 2024

nonn

Cf. A007576, A063866, A316706.

Table[Coefficient[Product[(x^k + 1 + 1/x^k), {k, 1, n}], x, 1], {n, 0, 31}]

from itertools import count, islice
from collections import Counter
def A369628_gen(): # generator of terms
    ccount = Counter({0:1})
    yield 0
    for i in count(1):
        bcount = Counter(ccount)
        for a in ccount:
            bcount[a+i] += ccount[a]
            bcount[a-i] += ccount[a]
        ccount = bcount
        yield(ccount[1])
A369628_list = list(islice(A369628_gen(),20)) # Chai Wah Wu, Jan 29 2024

a(n) = [x^1] Product_{k=1..n} (x^k + 1 + 1/x^k).

a(n) = [x^(n*(n+1)/2+1)] Product_{k=1..n} (1 + x^k + x^(2*k)).

A350305 a(n) is the constant term in the expansion of Product_{k=1..n} (x^k + 1 + 1/x^k)^n.

Original entry on oeis.org

1, 1, 13, 1437, 1884211, 24657701475, 3111336932350947, 3710920324904591897521, 41323213770479673319301068309, 4261037235228828189774620497534270303, 4045313784246510024420372971256850718016451185

Offset: 0

Seiichi Manyama, Dec 23 2021

nonn

a(n) is the coefficient of x^(n^2 * (n+1)/2) in Product_{k=0..n} (1 + x^k + x^(2*k))^n.

Robert Israel, Table of n, a(n) for n = 0..44

Cf. A007576, A156181, A225602, A350282, A350306, A350307.

f:= n -> coeff(mul(x^k+1+1/x^k,k=1..n)^n,x,0):
map(f, [$0..12]); # Robert Israel, Jan 15 2023

a[n_] := Coefficient[Series[Product[(x^k + 1 + 1/x^k)^n, {k, 1, n}], {x, 0, 0}], x, 0]; Array[a, 11, 0] (* Amiram Eldar, Dec 24 2021 *)

a(n) = polcoef(prod(k=1, n, x^k+1+1/x^k)^n, 0);

a(n) = polcoef(prod(k=1, n, 1+x^k+x^(2*k))^n, n^2*(n+1)/2);

A369344 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k(k+1)/2) + 1 + 1/x^(k(k+1)/2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 11, 27, 61, 133, 311, 761, 1839, 4575, 11573, 29641, 76487, 199617, 524067, 1384697, 3681069, 9841217, 26437741, 71369101, 193496241, 526685793, 1438816755, 3944034221, 10845006963, 29908325821, 82707648985, 229306378067, 637283978821

Offset: 0

Ilya Gutkovskiy, Jan 20 2024

nonn

All terms are odd.

a(n) is the number of solutions to 0 = Sum_{i=1..n} c_i * i*(i+1)/2 with c_i in {-1,0,1}.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000217, A007576, A158380, A350249.

b:= proc(n, i) option remember; `if`(n>i*(i+1)*(i+2)/6, 0, `if`(i=0, 1,
      b(n, i-1)+b(n+i*(i+1)/2, i-1)+b(abs(n-i*(i+1)/2), i-1)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 21 2024

Table[Coefficient[Product[x^(k (k + 1)/2) + 1 + 1/x^(k (k + 1)/2), {k, 1, n}], x, 0], {n, 0, 31}]

a(n) ~ sqrt(5) * 3^(n + 1/2) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Jan 22 2024

cp's OEIS Frontend

A086821 Duplicate of A007576.

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A123762 Number of ways, counted up to symmetry, to build a contiguous building with n LEGO blocks of size 1 X 2.

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A316706 Number of solutions to k_1 + 2*k_2 + ... + n*k_n = n, where k_i are from {-1,0,1}, i=1..n.

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A225310 T(n,k)=Number of nXk -1,1 arrays such that the sum over i=1..n,j=1..k of i*x(i,j) is zero and rows are nondecreasing (ways to put k thrusters pointing east or west at each of n positions 1..n distance from the hinge of a south-pointing gate without turning the gate).

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A350249 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

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A156181 Number of solutions to e(1)*1 + e(2)*2 + ... + e(n)*n = e(-1)*1 + e(-2)*2 + ... + e(-n)*n, where e(j) are from {-1,0,1}, j=-n,...,n.

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A350880 a(n) is the constant term in expansion of Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).

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Maple

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A369628 Number of solutions to k_1 + 2*k_2 + ... + n*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

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A350305 a(n) is the constant term in the expansion of Product_{k=1..n} (x^k + 1 + 1/x^k)^n.

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A316706 Number of solutions to k_1 + 2k_2 + ... + nk_n = n, where k_i are from {-1,0,1}, i=1..n.

A156181 Number of solutions to e(1)1 + e(2)2 + ... + e(n)n = e(-1)1 + e(-2)2 + ... + e(-n)n, where e(j) are from {-1,0,1}, j=-n,...,n.

A369628 Number of solutions to k_1 + 2k_2 + ... + nk_n = 1, where k_i are from {-1,0,1}, i=1..n.

A369344 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k(k+1)/2) + 1 + 1/x^(k(k+1)/2)).