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.

A202148 Sum of rows of the triangle in A080381.

Original entry on oeis.org

1, 2, 4, 8, 12, 32, 36, 100, 132, 392, 384, 1168, 1500, 5332, 5284, 15740, 16804, 60896, 62872, 222948, 243780, 927176, 876004, 2999432, 3284180, 12706832, 12636656, 45043700, 46679920, 176783132, 177597976, 652158968, 700632804, 2696835032, 2735898216
Offset: 0

Views

Author

Jacques ALARDET, Dec 12 2011

Keywords

Examples

			a(0)= 1.
a(4)= 1 + 2 + 6 + 2 + 1 = 12.

Crossrefs

Cf. A080381.

Programs

  • Mathematica
    Table[Total[Table[GCD[Binomial[n, j], Binomial[n, Floor[n/2]]], {j, 0, n}]], {n, 0, 50}]

A204087 Reduced Pascal triangle: C_R(n,m) = A003418(n) / max(A003418(m), A003418(n-m)), m=0,...,n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 2, 6, 2, 1, 1, 5, 10, 10, 5, 1, 1, 1, 5, 10, 5, 1, 1, 1, 7, 7, 35, 35, 7, 7, 1, 1, 2, 14, 14, 70, 14, 14, 2, 1, 1, 3, 6, 42, 42, 42, 42, 6, 3, 1, 1, 1, 3, 6, 42, 42, 42, 6, 3, 1, 1, 1, 11, 11, 33, 66, 462, 462, 66, 33, 11, 11, 1
Offset: 0

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 10 2012

Keywords

Comments

The sixth row is the first one which differs from triangles A080381, A080396.

Examples

			Triangle begins:
n/m.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....1
.2..|..1.....2.....1
.3..|..1.....3.....3.....1
.4..|..1.....2.....6.....2.....1
.5..|..1.....5....10....10.....5.....1
.6..|..1.....1.....5....10.....5.....1.....1
.7..|..1.....7.....7....35....35.....7.....7.....1

Links

Crossrefs

Cf. A007318, A080381, A080396, A186430, A202917, A202941.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, ilcm(g(n-1), n)) end:
CR:= proc(n, m) option remember; g(n)/max(g(m), g(n-m)) end:
seq (seq (CR(n,m), m=0..n), n=0..11); # Alois P. Heinz, Jan 11 2012
  • Mathematica
    g[n_] := g[n] = If[n == 0, 1, LCM[g[n-1], n]]; CR[n_, m_] := CR[n, m] = g[n]/Max[ g[m], g[n-m]]; Table[Table[CR[n, m], {m, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Mar 12 2015, after Alois P. Heinz *)

A080379 Least n such that n consecutive values in A080378 equals 2; i.e., exactly n differences between consecutive primes give residues 2 when divided by 4.

Original entry on oeis.org

5, 2, 9, 15, 39, 32, 305, 51, 2631, 3685, 170, 1156, 8775, 98, 5295, 41914, 106469, 167115, 186917, 1098776, 187784, 976193, 1166047, 423098, 77442332, 2643158, 11004239, 36330320, 259652255, 307899596, 2573725031, 411764049, 4080634008, 14841740642, 6022532018, 17035372732, 35045523209
Offset: 1

Views

Author

Labos Elemer, Mar 04 2003

Keywords

Comments

a(43) = 147618899630. - Donovan Johnson

Examples

			n=4: a(4)=15,differences between {47,53,59,61,67} are {6,6,2,6} corresponds to exactly four differences congruent to 2 mod 4,since before and after 47-43=4 or 71-67=4 are congruent to 0 mod 4.

Crossrefs

Cf. A001223, A080378, A080381.

Programs

  • Mathematica
    dp[x_] := Mod[Prime[x+1]-Prime[x], 4] pat[x_, h_] := Table[dp[x+j], {j, 0, h-1}] up[x_, h_] := Union[pat[x, h]] Table[fa=1; k=0; Do[s=up[n, h]; s1=Length[s]; s2=Part[u=pat[n+1, h], Length[u]]; s3=Part[w=pat[n-1, h], 1]; If[Equal[s1, 1]&&Equal[fa, 1]&&Equal[s2, 0]&&Equal[s3, 0], k=k+1; Print[{k, h, n, Prime[n], s, s1}]; fa=0], {n, 2, 200000}], {h, 1, 19}]
With[{c=Mod[Differences[Prime[Range[12*10^5]]],4]},Join[{5,2},Drop[ Flatten[ Table[ SequencePosition[ c,Join[ {0},PadRight[ {},n,2],{0}],1][[All,1]],{n,0,25}]]+1,3]]] (* The program generates the first 24 terms of the sequence. *) (* Harvey P. Dale, Dec 01 2022 *)

Formula

a(n)=Min{x; Union[{Mod[A001223(x), 4], ..., Mod[A001223(x+n-1), 4]}]=2}

Extensions

a(20)-a(37) from Donovan Johnson, Nov 16 2010

A080382 Triangle read by rows: T(n,k) = C(n,floor(n/2))/gcd(C(n,floor(n/2)),C(n,k)), k=0..n; central binomial coefficient is divided by greatest common divisor of binomial coefficients and corresponding central binomial coefficient.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 1, 3, 6, 3, 1, 3, 6, 10, 2, 1, 1, 2, 10, 20, 10, 4, 1, 4, 10, 20, 35, 5, 5, 1, 1, 5, 5, 35, 70, 35, 5, 5, 1, 5, 5, 35, 70, 126, 14, 7, 3, 1, 1, 3, 7, 14, 126, 252, 126, 28, 21, 6, 1, 6, 21, 28, 126, 252, 462, 42, 42, 14, 7, 1, 1, 7, 14, 42, 42, 462, 924, 77, 14, 21, 28
Offset: 1

Views

Author

Labos Elemer, Mar 12 2003

Keywords

Examples

			Triangle begins:
   1;
   1,  1;
   2,  1,  2;
   3,  1,  1,  3;
   6,  3,  1,  3,  6;
  10,  2,  1,  1,  2, 10;
  20, 10,  4,  1,  4, 10, 20;
  35,  5,  5,  1,  1,  5,  5, 35;

Crossrefs

Cf. A007318, A001405, A080381.

Programs

  • Mathematica
    Table[Table[Binomial[n, Floor[n/2]]/GCD[Binomial[n, j], Binomial[n, Floor[n/2]]], {j, 0, n}], {n, 1, 10}]
Showing 1-4 of 4 results.

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