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.

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A027932 T(n, 2n-9), T given by A027926.

Original entry on oeis.org

1, 3, 8, 21, 55, 143, 364, 894, 2098, 4685, 9955, 20175, 39130, 72905, 130965, 227612, 383911, 630191, 1009242, 1580345, 2424289, 3649547, 5399802, 7863034, 11282400, 15969161, 22317933, 30824563, 42106956, 56929205
Offset: 5

Views

Author

Clark Kimberling

Keywords

Links

Crossrefs

Cf. A228074.

Programs

  • GAP
    List([5..40], n-> Sum([0..4], k-> Binomial(n-k, 9-2*k)) ); # G. C. Greubel, Sep 27 2019
  • Magma
    [&+[Binomial(n-k, 9-2*k): k in [0..4]] : n in [5..40]]; // G. C. Greubel, Sep 27 2019
  • Maple
    A027932 := proc(n)
1/362880 *(n-4) *(n^8 -32*n^7 +490*n^6 -4592*n^5 +30289*n^4 -147728*n^3 +543780*n^2 -1359648*n +1905120)
end proc:
seq(A027932(n),n=5..30) ; # R. J. Mathar, Jun 29 2012
  • Mathematica
    Sum[Binomial[Range[5, 40] -k, 9-2*k], {k,0,4}] (* G. C. Greubel, Sep 27 2019 *)
  • PARI
    vector(40, n, sum(k=0,4, binomial(n+4-k, 9-2*k)) ) \\ G. C. Greubel, Sep 27 2019
  • Sage
    [sum(binomial(n-k, 9-2*k) for k in (0..4)) for n in (5..40)] # G. C. Greubel, Sep 27 2019

Formula

a(n) = Sum_{k=0..4} binomial(n-k, 9-2*k). - Len Smiley, Oct 20 2001
a(n) = C(n,n-1) + C(n+1,n-2) + C(n+2,n-3) + C(n+3,n-4) + C(n+4,n-5), n>=1 . - Zerinvary Lajos, May 29 2007
G.f.: x^5*(1 -7*x +23*x^2 -44*x^3 +55*x^4 -44*x^5 +23*x^6 -7*x^7 +x^8) / (1-x)^10 . - R. J. Mathar, Oct 31 2015

A228078 a(n) = 2^n - Fibonacci(n) - 1.

Original entry on oeis.org

0, 0, 2, 5, 12, 26, 55, 114, 234, 477, 968, 1958, 3951, 7958, 16006, 32157, 64548, 129474, 259559, 520106, 1041810, 2086205, 4176592, 8359950, 16730847, 33479406, 66987470, 134021309, 268117644, 536356682, 1072909783, 2146137378, 4292788986, 8586410013
Offset: 0

Views

Author

Reinhard Zumkeller, Aug 15 2013

Keywords

Comments

a(n+1) = sum of n-th row of the triangle in A228074.

Links

Programs

  • Haskell
    a228078 = subtract 1 . a099036
  • Magma
    [2^n - Fibonacci(n) - 1: n in [0..40]]; // Vincenzo Librandi, Aug 16 2013
  • Mathematica
    Table[(2^n - Fibonacci[n] - 1), {n, 0, 40}] (* Vincenzo Librandi, Aug 16 2013 *)
  • PARI
    concat([0,0], Vec(x^2*(3*x-2)/((x-1)*(2*x-1)*(x^2+x-1)) + O(x^100))) \\ Colin Barker, Mar 20 2015

Formula

a(n) = A000079(n) - A000045(n) - 1 = A000225(n) - A000045(n) = A000079(n) - A001611(n) = A099036(n) - 1.
From Colin Barker, Mar 20 2015: (Start)
a(n) = 4*a(n-1)-4*a(n-2)-a(n-3)+2*a(n-4) for n>3.
G.f.: x^2*(3*x-2) / ((x-1)*(2*x-1)*(x^2+x-1)). (End)
a(n) = (-1+2^n+(((1-sqrt(5))/2)^n-((1+sqrt(5))/2)^n)/sqrt(5)). - Colin Barker, Nov 02 2016

A261507 Fibonacci-numbered rows of Pascal's triangle. Triangle read by rows: T(n,k)= binomial(Fibonacci(n), k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 10, 10, 5, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1, 1, 21, 210, 1330, 5985, 20349, 54264, 116280, 203490, 293930, 352716, 352716, 293930, 203490, 116280, 54264, 20349, 5985, 1330, 210, 21, 1
Offset: 0

Views

Author

Maghraoui Abdelkader, Aug 22 2015

Keywords

Comments

Subsequence of A007318.

Examples

			1,
1,  1,
1,  1,
1,  2,  1,
1,  3,  3,   1,
1,  5, 10,  10,   5,    1,
1,  8, 28,  56,  70,   56,   28,    8,    1,
1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1

Links

Crossrefs

Cf. A000045, A007318.
Cf. A036355, A037027, A228074, A162741, A034871, A034870, A191797.

Programs

  • Mathematica
    Table[Binomial[Fibonacci[n], k], {n, 0, 8}, {k, 0, Fibonacci[n]}]//Flatten (* Jean-François Alcover, Nov 12 2015*)
  • PARI
    v = vector(101,j,fibonacci(j)); i=0; n=0; while(n<100, for(k=0, n, print1(binomial(n, k), ", ","")); print(); i=i+1; n=v[i] ;)

Formula

T(n, k) = binomial(fibonacci(n), k).
T(n, 1) = fibonacci(n) = A000045(n).
T(n, 2) = A191797(n) for n>3.

A354267 A Fibonacci-Pascal triangle read by rows: T(n, n) = 1, T(n, n-1) = n - 1, T(n, 0) = T(n-1, 1) and T(n, k) = T(n-1, k-1) + T(n-1, k) for 0 < k < n-1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 4, 3, 1, 3, 5, 7, 7, 4, 1, 5, 8, 12, 14, 11, 5, 1, 8, 13, 20, 26, 25, 16, 6, 1, 13, 21, 33, 46, 51, 41, 22, 7, 1, 21, 34, 54, 79, 97, 92, 63, 29, 8, 1, 34, 55, 88, 133, 176, 189, 155, 92, 37, 9, 1, 55, 89, 143, 221, 309, 365, 344, 247, 129, 46, 10, 1
Offset: 0

Views

Author

Peter Luschny, May 31 2022

Keywords

Examples

			[0]  1;
[1]  0,  1;
[2]  1,  1,  1;
[3]  1,  2,  2,  1;
[4]  2,  3,  4,  3,  1;
[5]  3,  5,  7,  7,  4,  1;
[6]  5,  8, 12, 14, 11,  5,  1;
[7]  8, 13, 20, 26, 25, 16,  6,  1;
[8] 13, 21, 33, 46, 51, 41, 22,  7, 1;
[9] 21, 34, 54, 79, 97, 92, 63, 29, 8, 1;

Links

Crossrefs

Cf. A212804 (first column, which is also row 0 of A352744), A099036 (row sums), A228074 (subtriangle), A000045 (Fibonacci), A371870 (central terms).

Programs

  • Maple
    T := proc(n, k) option remember;
if n = k then 1 elif k = n-1 then n-1 elif k = 0 then T(n-1, 1) else
T(n-1, k) + T(n-1, k-1) fi end: seq(seq(T(n, k), k = 0..n), n = 0..11);
  • Mathematica
    T[n_, k_] := Which[n == k, 1, k == n-1, n-1, k == 0, T[n-1, 1], True, T[n-1, k] + T[n-1, k-1]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023 *)
  • Python
    from functools import cache
@cache
def A354267row(n):
    if n == 0: return [1]
    if n == 1: return [0, 1]
    row = A354267row(n - 1) + [1]
    s = row[1]
    for k in range(n-1, 0, -1):
        row[k] += row[k - 1]
    row[0] = s
    return row
for n in range(10): print(A354267row(n))

Formula

T(n, 0) = Fibonacci(n - 1).
Previous Showing 31-34 of 34 results.

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