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.
a016754 n = a016754_list !! n a016754_list = scanl (+) 1 $ tail a008590_list -- Reinhard Zumkeller, Apr 02 2012
[n^2: n in [1..100 by 2]]; // Vincenzo Librandi, Jan 03 2017
Range[1, 100, 2]^2 (* Paolo Xausa, Feb 13 2025 *)
A016754(n):=(n+n+1)^2$ makelist(A016754(n),n,0,20); /* Martin Ettl, Nov 12 2012 */
A016754(n)=(n<<1+1)^2 \\ Charles R Greathouse IV, Jun 16 2011, corrected and edited by M. F. Hasler, Apr 11 2023
def A016754(n): return ((n<<1)|1)**2 # Chai Wah Wu, Jul 06 2023
a(50) = #{49+1} = 1;
a(130) = #{121+9, 81+49} = 2.
a167700 = p a016754_list where p _ 0 = 1 p (q:qs) m = if m < q then 0 else p qs (m - q) + p qs m -- Reinhard Zumkeller, Mar 15 2014
nmax = 100; CoefficientList[Series[Product[1 + x^((2*k-1)^2), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 18 2017 *)
a(12) = 5 because we have [9, 1, 1, 1], [1, 9, 1, 1], [1, 1, 9, 1], [1, 1, 1, 9] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].
nmax = 63; CoefficientList[Series[1/(1 - Sum[x^(2 k + 1)^2, {k, 0, nmax}]), {x, 0, nmax}], x]
a(27) = 2 because we have [27] and [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].
nmax = 110; CoefficientList[Series[Product[1/(1 - x^((2*k-1)^3)), {k, 1, Floor[nmax^(1/3)/2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 18 2017 *)
E.g. a(10)=6 because we can write it as 91,91*,9*1,9*1*,82,811*.
series(product((1+x^((2*k-1)^2))/(1-x^(2*k-1)^2)),k=1..100),x=0,100);
nmax = 100; CoefficientList[Series[Product[(1 + x^((2*k-1)^2)) / (1 - x^((2*k-1)^2)), {k, 1, Floor[Sqrt[nmax]/2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 19 2017 *)
a(35) = 6 because we have [25, 9, 1], [25, 1, 9], [9, 25, 1], [9, 1, 25], [1, 25, 9] and [1, 9, 25].
N:= 200: # for a(0)..a(N) G:= mul(1+t*x^(i^2),i=1..floor(sqrt(N)),2): 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
a(28) = 2 because we have [4,4,4,4,4,4,4] and [4,4,4,16]. a(32) = 3 because we have [4,4,4,4,4,4,4,4], [4,4,4,4,16], and [16,16].
g := 1/mul(1-x^(4*i^2), i = 1 .. 150): gser := series(g, x = 0, 105): seq(coeff(gser, x, n), n = 0 .. 100);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+ `if`(i^2>n, 0, b(n-i^2, i))))
end:
a:= n-> `if`(irem(n, 4, 'm')=0, b(m, isqrt(m)), 0):
seq(a(n), n=0..120); # Alois P. Heinz, Jan 27 2016
a[n_] := If[n==0, 1, If[Divisible[n, 4], PowersRepresentations[n/4, n/4, 2] // Length, 0]]; Array[a, 100, 0] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
For n=0, 72*0+42 = 42 = 25+9+1+1+1+1+1+1+1+1 = 9+9+9+9+1+1+1+1+1+1, so a(0)=2. For n=1, 72*1+42 = 114 = 81+25+1+1+1+1+1+1+1+1 = 81+9+9+9+1+1+1+1+1+1 = 49+49+9+1+1+1+1+1+1+1 = 49+25+25+9+1+1+1+1+1+1 = 49+25+9+9+9+9+1+1+1+1 = 49+9+9+9+9+9+9+9+1+1 = 25+25+25+25+9+1+1+1+1+1 = 25+25+25+9+9+9+9+1+1+1 = 25+25+9+9+9+9+9+9+9+1, so a(1)=9.
[#RestrictedPartitions(72*n+42, 10, {(2*d+1)^2:d in [0..100]}): n in [0..100]];
S:= proc(n, k, m)
option remember;
local p,j;
if k = 0 then if n = 0 then return 1 else return 0 fi
elif m < 1 then return 0
elif n < k then return 0
elif n > k*m^2 then return 0
fi;
if m^2 > n then
p:= floor(sqrt(n));
if p::even then p:= p-1 fi;
return procname(n, k, p)
fi;
add(procname(n-j*m^2,k-j,m-2), j=0..n/m^2)
end proc:
seq(S(72*n+42, 10, 72*n+42), n=0..100); # Robert Israel, Feb 24 2019
a[n_] := IntegerPartitions[72n+42, {10}, Select[ Range[1, 72n+42, 2], IntegerQ@Sqrt@#&]] // Length;
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 19 2022 *)
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