cp's OEIS Frontend

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.

Showing 1-4 of 4 results.

A126257 Number of distinct new terms in row n of Pascal's triangle.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 7, 5, 7, 8, 9, 9, 9, 8, 11, 11, 12, 12, 13, 13, 13, 14, 15, 15, 16, 16, 17, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 21, 23, 23, 24, 24, 25, 25, 26, 26, 27, 26, 26, 28, 29, 29, 30, 30, 31, 31, 32, 32, 32, 33, 34, 34, 34, 35, 36, 36, 37, 37, 38
Offset: 0

Views

Author

Nick Hobson, Dec 24 2006

Keywords

Comments

Partial sums are in A126256.
n occurs a(n) times in A265912. - Reinhard Zumkeller, Dec 18 2015

Examples

			Row 6 of Pascal's triangle is: 1, 6, 15, 20, 15, 6, 1. Of these terms, only 15 and 20 do not appear in rows 0-5. Hence a(6)=2.

Links

Crossrefs

Cf. A007318, A062854, A126254, A126255, A126256.
Cf. A034868, A265912.

Programs

  • Haskell
    import Data.List.Ordered (minus, union)
a126257 n = a126257_list !! n
a126257_list = f [] a034868_tabf where
   f zs (xs:xss) = (length ys) : f (ys `union` zs) xss
                   where ys = xs `minus` zs
-- Reinhard Zumkeller, Dec 18 2015
  • PARI
    lim=77; z=listcreate(1+lim^2\4); print1(1, ", "); r=1; for(a=1, lim, for(b=1, a\2, s=Str(binomial(a, b)); f=setsearch(z, s, 1); if(f, listinsert(z, s, f))); print1(1+#z-r, ", "); r=1+#z)
  • Python
    def A126257(n):
    if n:
        s, c = (1,), {1}
        for i in range(n-1):
            c.update(set(s:=(1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1))+(1,)))
        return len(set((1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1))+(1,))-c)
    return 1 # Chai Wah Wu, Oct 17 2023

A126256 Number of distinct terms in rows 0 through n of Pascal's triangle.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 9, 12, 16, 20, 24, 29, 35, 41, 48, 53, 60, 68, 77, 86, 95, 103, 114, 125, 137, 149, 162, 175, 188, 202, 217, 232, 248, 264, 281, 297, 314, 332, 351, 370, 390, 410, 431, 452, 474, 495, 518, 541, 565, 589, 614, 639, 665, 691, 718, 744, 770, 798
Offset: 0

Views

Author

Nick Hobson, Dec 24 2006

Keywords

Comments

An easy upper bound is 1 + floor(n^2/4) = A033638(n).
First differences are in A126257.

Examples

			There are 9 distinct terms in rows 0 through 6 of Pascal's triangle (1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1); hence a(6)=9.

Links

Crossrefs

Cf. A007318, A027424, A061786, A126254, A126255, A126257.
Cf. A199425.
Cf. A258318.

Programs

  • Haskell
    -- import Data.List.Ordered (insertSet)
a126256 n = a126256_list !! n
a126256_list = f a007318_tabl [] where
   f (xs:xss) zs = g xs zs where
     g []     ys = length ys : f xss ys
     g (x:xs) ys = g xs (insertSet x ys)
-- Reinhard Zumkeller, May 26 2015, Nov 09 2011
  • Maple
    seq(nops(`union`(seq({seq(binomial(n,k),k=0..n)},n=0..m))),m=0..57); # Emeric Deutsch, Aug 26 2007
  • Mathematica
    Table[Length[Union[Flatten[Table[Binomial[n,k],{n,0,x},{k,0,n}]]]],{x,0,60}] (* Harvey P. Dale, Sep 10 2022 *)
  • PARI
    lim=57; z=listcreate(1+lim^2\4); for(n = 0, lim, for(r=1, n\2, s=Str(binomial(n, r)); f=setsearch(z, s, 1); if(f, listinsert(z, s, f))); print1(1+#z, ", "))
  • Python
    def A126256(n):
    s, c = (1,), {1}
    for i in range(n):
        s = (1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1)) + (1,)
        c.update(set(s))
    return len(c) # Chai Wah Wu, Oct 17 2023

A126255 Number of distinct terms i^j for 2 <= i,j <= n.

Original entry on oeis.org

1, 4, 8, 15, 23, 34, 44, 54, 69, 88, 106, 129, 152, 177, 195, 226, 256, 291, 324, 361, 399, 442, 483, 519, 564, 600, 648, 703, 755, 814, 856, 915, 976, 1039, 1085, 1156, 1224, 1295, 1365, 1444, 1519, 1602, 1681, 1762, 1846, 1937, 2023, 2095, 2184, 2279
Offset: 2

Views

Author

Nick Hobson, Dec 24 2006

Keywords

Comments

An easy upper bound is (n-1)^2 = A000290(n-1).

Examples

			a(4) = 8 as there are 8 distinct terms in 2^2=4, 2^3=8, 2^4=16, 3^2=9, 3^3=27, 3^4=81, 4^2=16, 4^3=64, 4^4=256.

Links

Crossrefs

Cf. A027424, A061786, A126254, A126256, A126257.

Programs

  • Mathematica
    SetAttributes[a, {Listable, NumericFunction}]
a[n_ /; n < 2] := "error"
a[2] := 1
a[n_Integer?IntegerQ /; n > 2] :=
 Length[DeleteDuplicates[
   Distribute[f[Range[2, n], Range[2, n]], List,
     f] /. {f ->
      Power}]](*By using Distribute instead of Outer I avoid having to use Flatten on Outer*)
a[Range[2, 100]]
(* Peter Cullen Burbery, Aug 15 2023 *)
  • PARI
    lim=51; z=listcreate((lim-1)^2); for(m=2, lim, for(i=2, m, x=factor(i); x[, 2]*=m; s=Str(x); f=setsearch(z, s, 1); if(f, listinsert(z, s, f))); t=factor(m); for(j=2, m, x=t; x[, 2]=j*t[, 2]; s=Str(x); f=setsearch(z, s, 1); if(f, listinsert(z, s, f))); print1(#z, ", "))
  • Python
    def A126255(n): return len({i**j for i in range(2,n+1) for j in range(2,n+1)}) # Chai Wah Wu, Oct 17 2023

A303748 a(n) is the number of distinct terms of the form i^j where 0 <= i,j <= n.

Original entry on oeis.org

1, 2, 4, 8, 12, 20, 29, 41, 51, 61, 77, 97, 116, 140, 164, 190, 208, 240, 271, 307, 341, 379, 418, 462, 504, 540, 586, 622, 671, 727, 780, 840, 882, 942, 1004, 1068, 1114, 1186, 1255, 1327, 1398, 1478, 1554, 1638, 1718, 1800, 1885, 1977, 2064, 2136, 2226, 2322
Offset: 0

Views

Author

Ralph-Joseph Tatt, Apr 30 2018

Keywords

Examples

			For n=3 the distinct terms are 0,1,2,3,4,8,9,27 so a(3) = 8.

Links

Crossrefs

Cf. A126254.

Programs

  • Mathematica
    {1}~Join~Array[Length@ Union@ Map[#1^#2 & @@ # &, Rest@ Tuples[Range[0, #], {2}]] &, 51] (* Michael De Vlieger, Jan 31 2019 *)
  • Python
    def distinct(limit):
    unique = set()
    for i in range(limit+1):
        for j in range(limit+1):
            if i**j not in unique:
                unique.add(i**j)
    return len(unique)
print([distinct(i) for i in range(40)])

Formula

a(n) = A126254(n) + 1.
Showing 1-4 of 4 results.

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