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 1-5 of 5 results.

A053210 Row sums of A051599.

Original entry on oeis.org

2, 6, 16, 36, 80, 164, 336, 676, 1360, 2732, 5468, 10948, 21904, 43812, 87632, 175276, 350564, 701132, 1402276, 2804560, 5609124, 11218260, 22436528, 44873068, 89746152, 179492312, 358984628, 717969264, 1435938532, 2871877072
Offset: 0

Views

Author

Asher Auel, Dec 14 1999

Keywords

Examples

			a(3) = 7 + 11 + 11 + 7 = 36.

Links

Crossrefs

Cf. A051599.

Programs

Extensions

Corrected and extended by James Sellers, Dec 15 1999

A381748 a(n) is the number of primes (counted with multiplicity) in row n of A051599.

Original entry on oeis.org

1, 2, 2, 4, 2, 4, 2, 6, 2, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2, 6, 2, 4, 2, 2, 2, 4, 4, 6, 2, 8, 6, 6, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 4, 2, 2, 4, 2, 6, 2, 4, 4, 8, 2, 4, 4, 2, 2, 4, 4, 2, 2, 2, 2, 10, 2, 2, 4, 4, 2, 4, 2, 2, 2, 2, 2, 2, 4, 2, 4
Offset: 0

Views

Author

Vladimir Igorevich Lukyanchikov, Mar 06 2025

Keywords

Examples

			2;                                                    1 prime
3,  3;                                                2 primes
5,  6,   5;                                           2 primes
7,  11,  11,  7;                                      4 primes
11, 18,  22,  18,  11;                                2 primes

Links

Crossrefs

Cf. A051599.

Programs

  • Python
    from sympy import *
pr = list(primerange(2, 200))
lst = []
a = []
lst.append(1)
for i in range(1, 31):
    c = []
    c.append(pr[i])
    for j in range(1, i):
        c.append(a[j-1] + a[j])
    c.append(pr[i])
    count_primes = sum(isprime(x) for x in c)
    lst.append(count_primes)
    a = c
print(*lst)

A227550 A triangle formed like Pascal's triangle, but with factorial(n) on the borders instead of 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 6, 4, 4, 6, 24, 10, 8, 10, 24, 120, 34, 18, 18, 34, 120, 720, 154, 52, 36, 52, 154, 720, 5040, 874, 206, 88, 88, 206, 874, 5040, 40320, 5914, 1080, 294, 176, 294, 1080, 5914, 40320, 362880, 46234, 6994, 1374, 470, 470, 1374, 6994, 46234, 362880, 3628800
Offset: 0

Views

Author

Vincenzo Librandi, Aug 04 2013

Keywords

Comments

A003422 gives the second column (after 0).

Examples

			Triangle begins:
       1;
       1,     1;
       2,     2,    2;
       6,     4,    4,    6;
      24,    10,    8,   10,  24;
     120,    34,   18,   18,  34, 120;
     720,   154,   52,   36,  52, 154,  720;
    5040,   874,  206,   88,  88, 206,  874, 5040;
   40320,  5914, 1080,  294, 176, 294, 1080, 5914, 40320;
  362880, 46234, 6994, 1374, 470, 470, 1374, 6994, 46234, 362880;

Links

Crossrefs

Cf. similar triangles with t on the borders: A007318 (t = 1), A028326 (t = 2), A051599 (t = prime(n)), A051601 (t = n), A051666 (t = n^2), A108617 (t = fibonacci(n)), A134636 (t = 2n+1), A137688 (t = 2^n), A227075 (t = 3^n).
Cf. A003422.
Cf. A227791 (central terms), A001563, A074911.

Programs

  • Haskell
    a227550 n k = a227550_tabl !! n !! k
a227550_row n = a227550_tabl !! n
a227550_tabl = map fst $ iterate
   (\(vs, w:ws) -> (zipWith (+) ([w] ++ vs) (vs ++ [w]), ws))
   ([1], a001563_list)
-- Reinhard Zumkeller, Aug 05 2013
  • Magma
    function T(n,k)
  if k eq 0 or k eq n then return Factorial(n);
  else return T(n-1,k-1) + T(n-1,k);
  end if; return T;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 02 2021
  • Mathematica
    t = {}; Do[r = {}; Do[If[k == 0||k == n, m = n!, m = t[[n, k]] + t[[n, k + 1]]]; r = AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t = Flatten[t]
  • Sage
    def T(n,k): return factorial(n) if (k==0 or k==n) else T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 02 2021

Formula

From G. C. Greubel, May 02 2021: (Start)
T(n, k) = T(n-1, k-1) + T(n-1, k) with T(n, 0) = T(n, n) = n!.
Sum_{k=0..n} T(n, k) = 2^n * (1 +Sum_{j=1..n-1} j*j!/2^j) = A140710(n). (End)

A051600 Rows of triangle formed using Pascal's rule except begin n-th row with (n+1)st prime and end it with (n+2)nd prime.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 13, 15, 19, 17, 19, 28, 34, 36, 23, 29, 47, 62, 70, 59, 31, 37, 76, 109, 132, 129, 90, 41, 43, 113, 185, 241, 261, 219, 131, 47, 53, 156, 298, 426, 502, 480, 350, 178, 59, 61, 209, 454, 724, 928, 982, 830, 528, 237, 67
Offset: 0

Views

Author

Asher Auel

Keywords

Examples

			2; 3,5; 7,8,11; 13,15,19,17; 19,28,34,36,23; ...

Crossrefs

Cf. A000040, A051599.

Extensions

a(31) onward corrected by Sean A. Irvine, Sep 20 2021

A317644 Triangle read by rows: multiplicative version of Pascal's triangle except n-th row begins and ends with (n+1)-st prime.

Original entry on oeis.org

2, 3, 3, 5, 9, 5, 7, 45, 45, 7, 11, 315, 2025, 315, 11, 13, 3465, 637875, 637875, 3465, 13, 17, 45045, 2210236875, 406884515625, 2210236875, 45045, 17, 19, 765765, 99560120034375, 899311160300888671875, 899311160300888671875, 99560120034375, 765765, 19, 23, 14549535, 76239655318123171875, 89535527067809533413858673095703125, 808760563041730681160065242862701416015625, 89535527067809533413858673095703125, 76239655318123171875, 14549535, 23
Offset: 0

Views

Author

Philipp O. Tsvetkov, Aug 02 2018

Keywords

Examples

			Triangle begins:
   2;
   3,      3;
   5,      9,      5;
   7,     45,     45,      7;
  11,    315,   2025,    315,     11;
  13,   3465, 637875, 637875,   3465,     13;
  ...
Formatted as a symmetric triangle:
.
                       2
.
                   3       3
.
               5       9       5
.
           7      45      45       7
.
      11      315    2025     315     11
.
  13     3465   637875  637875   3465     13
...

Crossrefs

Cf. A000040, A007318, A007949, A028326, A051599, A080046, A112765.

Programs

  • Mathematica
    t = {{2}};
Table[AppendTo[
    t, {Prime[i],
      Table[
       t[[i - 1]][[j]]*t[[i - 1]][[j + 1]], {j,
        1, (t[[i - 1]] // Length) - 1}], Prime[i]} // Flatten], {i, 2, 10}] //
   Last // Flatten
t={}; Do[r={}; Do[If[k==0||k==n, m=Prime[n + 1], m=t[[n, k]]t[[n, k + 1]]]; r=AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t (* Vincenzo Librandi, Sep 03 2018 *)

Formula

From Rémy Sigrist, Sep 02 2018: (Start)
A007949(T(n+1, k+1)) = A028326(n, k) for any n >= 0 and k = 0..n.
A112765(T(n+1, k+1)) = A007318(n, k) for any n > 0 and k = 0..n.
(End)
Showing 1-5 of 5 results.

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