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.
The start of the sequence as a triangular array read by rows: 0; 1, 2; 4, 3, 4; 9, 7, 7, 8; 16, 16, 14, 15, 16; 25, 32, 30, 29, 31, 32; 36, 57, 62, 59, 60, 63, 64;
T:= function(n,k)
if k=0 then return n^2;
elif k=n then return 2^n;
else return T(n-1,k-1) + T(n-1,k);
fi;
end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 12 2019
T:= proc(n, k) option remember;
if k=0 then n^2
elif k=n then 2^k
else T(n-1, k-1) + T(n-1, k)
fi
end:
seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Nov 12 2019
T[n_, k_]:= T[n, k] = If[k==0, n^2, If[k==n, 2^k, T[n-1, k-1] + T[n-1, k]]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 12 2019 *)
Flatten[Table[Sum[i^2 Binomial[n-1-i, n-k-i], {i,1,n-k}] + Sum[2^i Binomial[n-1-i, k-i], {i,1,k}], {n,0,10}, {k,0,n}]] (* Greg Dresden, Aug 06 2022 *)
T(n,k) = if(k==0, n^2, if(k==n, 2^k, T(n-1, k-1) + T(n-1, k) )); \\ G. C. Greubel, Nov 12 2019
def funcL(n):
q = n**2
return q
def funcR(n):
q = 2**n
return q
for n in range (1,9871):
t=int((math.sqrt(8*n-7) - 1)/ 2)
i=n-t*(t+1)/2-1
j=(t*t+3*t+4)/2-n-1
sum1=0
sum2=0
for m1 in range (1,i+1):
sum1=sum1+funcR(m1)*binomial(i+j-m1-1,i-m1)
for m2 in range (1,j+1):
sum2=sum2+funcL(m2)*binomial(i+j-m2-1,j-m2)
sum=sum1+sum2
@CachedFunction
def T(n, k):
if (k==0): return n^2
elif (k==n): return 2^n
else: return T(n-1, k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019
First few rows are: 1; 2, 3; 2, 5, 3; 2, 7, 8, 3; 2, 9, 15, 11, 3; ...
T:= function(n,k)
if n=0 and k=0 then return 1;
elif k=0 then return 2;
elif k=n then return 3;
else return T(n-1,k-1) + T(n-1,k);
fi;
end;
Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 12 2019
a029600 n k = a029600_tabl !! n !! k a029600_row n = a029600_tabl !! n a029600_tabl = [1] : iterate (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [2,3] -- Reinhard Zumkeller, Apr 08 2012
T:= proc(n, k) option remember;
if k=0 and n=0 then 1
elif k=0 then 2
elif k=n then 3
else T(n-1, k-1) + T(n-1, k)
fi
end:
seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Nov 12 2019
T[n_, k_]:= T[n, k]= If[n==0 && k==0, 1, If[k==0, 2, If[k==n, 3, T[n-1, k-1] + T[n-1, k] ]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 12 2019 *)
T(n,k) = if(n==0 && k==0, 1, if(k==0, 2, if(k==n, 3, T(n-1, k-1) + T(n-1, k) ))); \\ G. C. Greubel, Nov 12 2019
@CachedFunction
def T(n, k):
if (n==0 and k==0): return 1
elif (k==0): return 2
elif (k==n): return 3
else: return T(n-1,k-1) + T(n-1, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019
Triangle begins: 1; 4, 4; 4, 8, 4; 4, 12, 12, 4; 4, 16, 24, 16, 4; 4, 20, 40, 40, 20, 4;
[1] cat [4*Binomial(n,k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021
Table[4*Binomial[n,k] -3*Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 03 2021 *)
def A132200(n,k): return 4*binomial(n,k) - 3*bool(n==0) flatten([[A132200(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021
Given the square array 1 2 3 3 3 3 3 3 3 3 1 2 4 5 7 8 10 11 13 1 2 5 7 12 15 22 26 1 2 6 9 18 24 40 1 2 7 11 25 35 1 2 8 13 33 (Table A124458) 1 2 9 15 1 2 10 1 2 1 Omit these odd columns: 1 3 3 3 3 3 3 3 3 3 3 1 4 7 10 13 16 19 22 25 28 1 5 12 22 35 51 70 92 117 1 6 18 40 75 126 196 288 1 7 25 65 140 266 462 1 8 33 98 238 504 1 9 42 140 378 1 10 52 192 (Table A093560) 1 11 63 1 12 1 which yields the square array A124459
Reppasc := proc(n,k) binomial(n+floor(k/2),n) ; end: A124458 := proc(n,k) add(Reppasc(n,i), i=max(0,k-3)..k-1) ; end: A124459 := proc(n,k) A124458(n,2*k) ; end: for d from 1 to 19 do for k from d to 1 by -1 do n := d-k ; printf("%d,",A124459(n,k)) ; od: od: # R. J. Mathar, Jun 18 2008
Comments