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 1-6 of 6 results.

A119444 Triangle as described in A100461, except with t(1,n) = Fibonacci(n+1).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 5, 1, 2, 3, 4, 8, 3, 4, 6, 8, 10, 13, 7, 8, 9, 12, 15, 18, 21, 13, 14, 15, 16, 20, 24, 28, 34, 27, 28, 30, 32, 35, 36, 42, 48, 55, 63, 64, 66, 68, 70, 72, 77, 80, 81, 89, 109, 110, 111, 112, 115, 120, 126, 128, 135, 140, 144, 207, 208, 210, 212, 215, 216
Offset: 1

Views

Author

Joshua Zucker, May 20 2006

Keywords

Links

Crossrefs

Cf. A119445 (leading diagonal).
Cf. A100461 for powers of 2, A119446 for primes.

Programs

  • Magma
    function t(n,k)
  if k eq 1 then return Fibonacci(n+1);
  else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
  end if;
end function;
[t(n,n-k+1): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 07 2023
  • Mathematica
    t[n_, k_]:= t[n, k]= If[k==1, Fibonacci[n+1], (n-k+1)*Floor[(t[n, k-1] -1)/(n-k+1)]];
Table[t[n, n-k+1], {n,15}, {k,n}]//TableForm (* G. C. Greubel, Apr 07 2023 *)
  • SageMath
    def t(n, k):
    if (k==1): return fibonacci(n+1)
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
flatten([[t(n, n-k+1) for k in range(1,n+1)] for n in range(1, 16)]) # G. C. Greubel, Apr 07 2023

Formula

Form an array t(m,n) (n >= 1, 1 <= m <= n) by: t(1,n) = Fibonacci(n+1) for all n; t(m+1,n) = (n-m)*floor( (t(m,n) - 1)/(n-m) ) for 1 <= m <= n-1.

A119446 Triangle as described in A100461, except with t(1,n) = prime(n).

Original entry on oeis.org

2, 2, 3, 3, 4, 5, 3, 4, 6, 7, 3, 4, 6, 8, 11, 3, 4, 6, 8, 10, 13, 3, 4, 6, 8, 10, 12, 17, 3, 4, 6, 8, 10, 12, 14, 19, 3, 4, 6, 8, 10, 12, 14, 16, 23, 7, 8, 9, 12, 15, 18, 21, 24, 27, 29, 7, 8, 9, 12, 15, 18, 21, 24, 27, 30, 31, 7, 8, 9, 12, 15, 18, 21, 24, 27, 30, 33, 37
Offset: 1

Views

Author

Joshua Zucker, May 20 2006

Keywords

Examples

			Triangle begins as:
  2;
  2, 3;
  3, 4, 5;
  3, 4, 6,  7;
  3, 4, 6,  8, 11;
  3, 4, 6,  8, 10, 13;
  3, 4, 6,  8, 10, 12, 17;
  3, 4, 6,  8, 10, 12, 14, 19;
  3, 4, 6,  8, 10, 12, 14, 16, 23;
  7, 8, 9, 12, 15, 18, 21, 24, 27, 29;

Links

Crossrefs

Cf. A100461 for powers of 2, A119444 for Fibonacci and A119447 for leading diag. of this triangle.

Programs

  • Magma
    function t(n,k) // t = A119444
  if k eq 1 then return NthPrime(n);
  else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
  end if;
end function;
[t(n,n-n+1): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 07 2023
  • Mathematica
    t[n_, k_]:= t[n, k]= If[k==1, Prime[n], (n-k+1)*Floor[(t[n,k-1] -1)/(n -k+1)]];
Table[t[n, n-k+1], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 07 2023 *)
  • SageMath
    def t(n, k):
    if (k==1): return nth_prime(n)
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
flatten([[t(n, n-k+1) for k in range(1,n+1)] for n in range(1, 16)]) # G. C. Greubel, Apr 07 2023

Formula

Form an array t(m,n) (n >= 1, 1 <= m <= n) by: t(1,n) = prime(n) for all n; t(m+1,n) = (n-m)*floor( (t(m,n) - 1)/(n-m) ) for 1 <= m <= n-1.

A100462 Leading diagonal of array in A100461.

Original entry on oeis.org

1, 1, 1, 3, 7, 25, 49, 109, 229, 481, 1003, 2019, 4051, 8143, 16309, 32683, 65439, 131007, 262081, 524187, 1048423, 2097027, 4194147, 8388481, 16777039, 33554269, 67108699, 134217529, 268435227, 536870713, 1073741607, 2147483371, 4294967043, 8589934267, 17179868869
Offset: 1

Views

Author

N. J. A. Sloane, Nov 22 2004

Keywords

Links

Crossrefs

Cf. A100461, A100463.

Programs

  • Magma
    function t(n,k) // t = A100461
  if k eq 1 then return 2^(n-1);
  else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
  end if;
end function;
[t(n,n): n in [1..40]]; // G. C. Greubel, Apr 07 2023
  • Mathematica
    t[n_, k_]:= t[n, k]= If[k==1, 2^(n-1), (n-k+1)*Floor[(t[n,k-1] -1)/(n - k+1)]]; (* t = A100461 *)
Table[t[n,n], {n,40}] (* G. C. Greubel, Apr 07 2023 *)
  • SageMath
    def t(n,k): # t = A100461
    if (k==1): return 2^(n-1)
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
[t(n,n) for n in range(1,41)] # G. C. Greubel, Apr 07 2023

A100452 Triangle read by rows, based on array described below.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 13, 14, 15, 16, 19, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 39, 40, 42, 44, 45, 48, 49, 49, 50, 51, 52, 55, 60, 63, 64, 63, 64, 66, 68, 70, 72, 77, 80, 81, 79, 80, 81, 84, 85, 90, 91, 96, 99, 100, 91, 92, 93, 96, 100, 102, 105, 112, 117, 120, 121
Offset: 1

Views

Author

N. J. A. Sloane, Nov 22 2004

Keywords

Comments

The interesting property of this array is that the main diagonal gives A000960.

Examples

			Array begins:
  1 4 9 16 25 36 49 64 81 100 ...
    3 8 15 24 35 48 63 80  99 ...
      7 14 21 32 45 60 77  96 ...
        13 20 30 44 55 72  91 ...
           19 28 42 52 70  90 ...
and triangle begins:
   1
   3  4
   7  8  9
  13 14 15 16
  19 20 21 24 25
  27 28 30 32 35 36
  ...

Links

Crossrefs

Cf. A000960, A100453.
Column sums give A100454.
Row 1 = A000290, row 2 = A000290 - 1, row 3 = A100451.
See also A100461.

Programs

  • Magma
    function t(n,k) // t = A100452
  if k eq 1 then return n^2;
  else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
  end if;
end function;
[t(n,n-k+1): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 07 2023
  • Mathematica
    max=11; a[1, n_]:= n^2;
a[m_, n_]/; 1, ]=0;
t= Table[a[m, n], {m,max}, {n,m,max}];
Flatten[Table[t[[m-n+1, n]], {m,max}, {n,m}]] (* Jean-François Alcover, Feb 21 2012 *)
  • SageMath
    def t(n, k): # t = A100452
    if (k==1): return n^2
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
flatten([[t(n, n-k+1) for k in range(1,n+1)] for n in range(1, 16)]) # G. C. Greubel, Apr 07 2023

Formula

Form an array a(m,n) (n >= 1, 1 <= m <= n) by: a(1,n) = n^2 for all n; a(m+1,n) = (n-m)*floor( (a(m,n)-1)/(n-m) ) for 1 <= m <= n-1.

A100463 a(n) = 2^(n-1) - A100462(n).

Original entry on oeis.org

0, 1, 3, 5, 9, 7, 15, 19, 27, 31, 21, 29, 45, 49, 75, 85, 97, 65, 63, 101, 153, 125, 157, 127, 177, 163, 165, 199, 229, 199, 217, 277, 253, 325, 315, 365, 345, 379, 423, 449, 549, 529, 597, 409, 507, 473, 633, 569, 717, 523, 651, 655, 777, 793, 825, 835, 855, 833
Offset: 1

Views

Author

N. J. A. Sloane, Nov 23 2004

Keywords

Links

Crossrefs

Cf. A100461, A100462.

Programs

  • Magma
    function t(n,k) // t = A100461
  if k eq 1 then return 2^(n-1);
  else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
  end if;
end function;
[2^(n-1) - t(n,n): n in [1..60]]; // G. C. Greubel, Apr 07 2023
  • Maple
    A100461:= proc(m,n) option remember;
   if m=1 then 2^(n-1);
   else (n-m+1)*floor((A100461(m-1,n)-1)/(n-m+1));
fi; end:
A100462:= proc(n) A100461(n,n); end:
A100463:= proc(n) 2^(n-1) - A100462(n); end:
seq(A100463(n), n=1..100); # R. J. Mathar, Aug 06 2007
  • Mathematica
    t[n_, k_]:= t[n, k]= If[k==1, 2^(n-1), If[kA100461 *)
Table[2^(n-1) -t[n,n], {n,60}] (* G. C. Greubel, Apr 07 2023 *)
  • SageMath
    def t(n, k): # t = A100461
    if (k==1): return 2^(n-1)
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
[2^(n-1) - t(n, n) for n in range(1, 61)] # G. C. Greubel, Apr 07 2023

Extensions

More terms from R. J. Mathar, Aug 06 2007

A119445 Leading diagonal of triangle A119444.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 7, 13, 27, 63, 109, 207, 331, 553, 931, 1531, 2527, 4093, 6673, 10831, 17563, 28561, 46227, 74883, 121219, 196239, 317607, 514047, 831823, 1346041, 2178079, 3524323, 5702619, 9227161, 14930019, 24157471, 39087823, 63245551
Offset: 1

Views

Author

Joshua Zucker, May 20 2006

Keywords

Links

Crossrefs

Cf. A119444 for triangle corresponding to this sequence.
Cf. A100461 for powers of 2, A119446 for primes.

Programs

  • Magma
    function t(n,k) // t = A119444
  if k eq 1 then return Fibonacci(n+1);
  else return (n-k+1)*Floor((t(n,k-1) -1)/(n-k+1));
  end if;
end function;
[t(n,n): n in [1..60]]; // G. C. Greubel, Apr 07 2023
  • Mathematica
    t[1, n_]:= Fibonacci[n+1];  (* t = A119444 *)
t[m_, n_]/; 1, ]= 0;
A119445[n_]:= A119445[n]= t[n,n];
Table[A119445[n], {n,60}] (* G. C. Greubel, Apr 07 2023 *)
  • SageMath
    def t(n, k): # t = A119444
    if (k==1): return fibonacci(n+1)
    else: return (n-k+1)*((t(n, k-1) -1)//(n-k+1))
def A119445(n): return t(n,n)
[A119445(n) for n in range(1,61)] # G. C. Greubel, Apr 07 2023

Formula

a(n) = A119444(n, n).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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