This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
b:= proc(n) option remember; local i; if n=0 then 1 else 0;
for i while i*(i+1)/2<=n do %+b(n-i*(i+1)/2) od; % fi
end:
a:= n-> b(n^2):
seq(a(n), n=0..20); # Alois P. Heinz, Feb 05 2018
b[n_] := b[n] = Module[{i, j = If[n == 0, 1, 0]}, For[i = 1, i(i+1)/2 <= n, i++, j += b[n-i(i+1)/2]]; j];
a[n_] := b[n^2];
a /@ Range[0, 20] (* Jean-François Alcover, Nov 04 2020, after Alois P. Heinz *)
{a(n)=polcoeff(1/(1-sum(r=1,n+1, x^(r*(r+1)/2)+x*O(x^(n^2)))), n^2)}
for(n=0, 20, print1(a(n), ", "))
a(7) = 5 because we have [4, 1, 1, 1], [1, 4, 1, 1], [1, 1, 4, 1], [1, 1, 1, 4] and [1, 1, 1, 1, 1, 1, 1].
nmax = 50; CoefficientList[Series[1/(1 - Sum[x^(3 k (k + 1)/2 + 1), {k, 0, nmax}]), {x, 0, nmax}], x]
b:= proc(n) option remember; `if`(n=0, 1,
add(`if`(issqr(8*j+1), b(n-j), 0), j=1..n))
end:
a:= proc(n) option remember;
`if`(n<0, 0, b(n)+a(n-1))
end:
seq(a(n), n=0..50); # Alois P. Heinz, Apr 28 2018
nmax = 41; CoefficientList[Series[1/((1 - x) (2 - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)))), {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[1/((1 - x) (1 - Sum[x^(k (k + 1)/2), {k, 1, nmax}])), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, 8 k + 1] a[n - k], {k, 1, n}]/2; Accumulate[Table[a[n], {n, 0, 41}]]
1 = x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 + ... x = 0.6452227032360209791342516639440263322547274436405712210742201839013654671
x /. FindRoot[ EllipticTheta[2, 0, Sqrt[x]] == 4*x^(1/8), {x, 1/2}, WorkingPrecision -> 110] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Feb 13 2013 *)
solve(x=0.6,0.7,1-sum(n=1,60,x^(n*(n+1)/2)))
The a(10) = 4 such compositions are: : : 1: [ 1 2 1 2 1 2 1 ] (composition) : : o o o : ooooooo (rendering as composition) : : O O O : O O O O O O O (rendering as fountain of coins) : : : 2: [ 1 2 1 2 3 1 ] : : o : o oo : oooooo : : O : O O O : O O O O O O : : : 3: [ 1 2 3 1 2 1 ] : : o : oo o : oooooo : : O : O O O : O O O O O O : : : 4: [ 1 2 3 4 ] : : o : oo : ooo : oooo : : O : O O : O O O : O O O O :
b:= proc(n, i) option remember; `if`(n=0, 1, add(
`if`(i=j, 0, b(n-j, j)), j=1..min(n, i+1)))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..60); # Alois P. Heinz, Mar 11 2014
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[i == j, 0, b[n-j, j]], {j, 1, Min[n, i+1]}]];
a[n_] := b[n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, Nov 07 2020, after Alois P. Heinz *)
# translation of the Maple program by Alois P. Heinz @CachedFunction def F(n, i): if n == 0: return 1 return sum( (i!=j) * F(n-j, j) for j in [1..min(n,i+1)] ) # A238870 # return sum( F(n-j, j) for j in [1, min(n,i+1)] ) # A005169 def a(n): return F(n, 0) print([a(n) for n in [0..50]]) # Joerg Arndt, Mar 20 2014
a(15) = 6 because we have [10, 4, 1], [10, 1, 4], [4, 10, 1], [4, 1, 10], [1, 10, 4] and [1, 4, 10].
N:= 200: # for a(0)..a(N) G:= mul(1+t*x^(i*(i+1)*(i+2)/6), i=1..floor((6*N)^(1/3))): F:= proc(n) local R, k, v; R:= coeff(G, x, n); add(k!*coeff(R, t, k), k=1..degree(R, t)) end proc: F(0):= 1: map(F, [$0..N]); # Robert Israel, Feb 03 2020
M = 100;
G = Product[1 + t x^(i(i+1)(i+2)/6), {i, 1, Floor[(6M)^(1/3)]}];
F[n_] := Module[{R, k, v}, R = Coefficient[G, x, n]; Sum[k! Coefficient[R, t, k], {k, 1, Exponent[R, t]}]];
F[0] = 1;
F /@ Range[0, M] (* Jean-François Alcover, Jun 20 2020, after Robert Israel *)
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