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.
Benedict W. J. Irwin's wiki page.
Benedict W. J. Irwin has authored 20 sequences. Here are the ten most recent ones:
a(4) = 2, this is from the partitions (2,4,1,3) and (3,4,1,2).
a(4) = 8; this is from the partitions (2,1,4,3), (2,4,3,1), (3,2,4,1), (3,4,2,1), (4,1,3,2), (4,2,1,3), (4,2,3,1), (4,3,1,2).
a(2) = 1, this is from the partition (2,1). a(3) = 4, this is from the partitions (1,3,2),(2,1,3),(2,3,1),(3,1,2).
1.15690515375402895450134581557232146535255402894879536470039938959...
Prime7[n_] := If[n < 16, Prime[n], If[n == 16, 7^2, Prime[n - 1]]];
RealDigits[N[Pi^4*Sum[Sum[2^(4-n-8*k)*15^(-n-8*k)/Product[Gamma[ Prime7[2+m+n]/30+k], {m,1,8}],{n,1,8}],{k,0,Infinity}], 100]][[1]]
Left-align the binary representation of the Catalan numbers:
1
1
10
101
1110
101010
10000100
110101101
10110010110
The first column is all 1's, by definition.
The next column appears to have blocks of 1's and 0's.
The sizes of the blocks make this sequence.
The values of this column may be generated with the MMA code:
Table[If[2^Floor[Log2[CatalanNumber[n]]] + 2^(Floor[Log2[CatalanNumber[n]]] - 1) < CatalanNumber[n], 1, 0], {n, 1, 1000}].
Num = 10000;
T = Table[If[2^Floor[Log2[CatalanNumber[n]]] +2^(Floor[Log2[CatalanNumber[n]]] - 1) < CatalanNumber[n], 1, 0], {n, 1, Num}]; S = {};
For[i = 1, i < Num, i++, j = 0; If[T[[i]] == 1, While[T[[i]] == 1, i++; j++]; AppendTo[S, j];];]; S
1.2824271291006... = 1/1 + 1/(1*4) + 1/(1*4*8) + 1/(1*4*8*27) + ...
EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]} &, {Ceiling[1/(A - Floor[A])], A - Floor[A]}, n - 1]];
EngelExp[Glaisher, 22]
NN=200;MM=PrimePi[NN]+1; Table[Boole[n>2]+Sum[Boole[(n>Prime[k])&&(Mod[n-Prime[k]+k-1,k] == 0)], {k, 2, MM}], {n, 1, NN}]
CoefficientList[Series[Sum[x^Prime[k]/(1 - x)^k, {k, 1, 40}], {x, 0, 40}], x]
The sequence of numerals starts 0, 1, 10, 11, 12, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123. To get the numeral following 12, we first increment the final digit: 13. But the digits before the 3 are 0 (implied) and 1, and 3 is greater than either of those by more than 1. So we set the last digit to 0, and increment the previous one: 20. Again, 2 is too large for the two implicit zeros in front of it, so we set it to 0 and increment the preceding digit, an implicit zero; so we get 100, which presents no problems. The length 3 numerals come from the numerals less than 100: 0, 1, 10, 11, 12. Inserting leading zeros to length 3 gives 000, 001, 010, 011, 012. The values of 1, 10, 100, 1000, etc. make up the sequence. a(5) = 51 = 52 - 1 = A000110(5) - 1 counts all set partitions of [5] except: 134|2|5. - _Alois P. Heinz_, May 27 2017
b:= proc(n, i, j) option remember; `if`(n=0, 1,
add(b(n-1, max(j, k), k), k=1..i+1))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30); # Alois P. Heinz, May 26 2017
SIZ = 30;MAX = 100000;
M = Table[0, {n, 1, SIZ + 2}];
For[i = 0, i <= MAX, i++,sum = 0;For[j = 1, j <= SIZ, j++,sum += M[[j]];]
If[sum == 1, Print[i]]M[[1]]++;
For[j = 1, j <= SIZ, j++,If[M[[j]] > M[[j + 1]] + 1 && M[[j]] > M[[j + 2]] + 1, M[[j]] = 0; M[[j + 1]]++]]] (* Benedict W. J. Irwin, Nov 14 2016 *)
b[n_, i_, j_] := b[n, i, j] = If[n==0, 1, Sum[b[n-1, Max[j, k], k], {k, 1, i+1} ] ];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 23 2017, after Alois P. Heinz *)
S:= series(RootOf(x-x^2-x^6-t,x),t,41): seq(coeff(S,t,i),i=1..40); # Robert Israel, Nov 04 2024
Table[Sum[Binomial[n-1-4*j,j]Binomial[2*n-4*j-2,n-1],{j,0,(n-1)/2}]/n, {n, 1, 25}]
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