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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A272122 a(n) is the number of positive divisors of A003266(n).

Original entry on oeis.org

1, 1, 2, 4, 8, 20, 40, 120, 288, 864, 1728, 4800, 9600, 28800, 84480, 304128, 608256, 2322432, 9289728, 40642560, 116121600, 348364800, 696729600, 3185049600, 8918138880, 26754416640, 149824733184, 624269721600, 1248539443200, 6522981580800, 26091926323200, 107629196083200
Offset: 1

Views

Author

Altug Alkan, Apr 28 2016

Keywords

Links

Crossrefs

Cf. A000005, A000045, A003266, A027423, A063375, A069744, A260622.

Programs

  • Mathematica
    a[n_] := DivisorSigma[0, Fibonorial[n]]; Array[a, 32] (* Amiram Eldar, Aug 09 2022 *)
  • PARI
    a(n) = numdiv(prod(k=1, n, fibonacci(k)));

Formula

a(n) = A000005(A003266(n)).
a(n+1) = 2*a(n) when n is in A069744.

A153757 a(n) = Sum_{k=1..n} A003266(k).

Original entry on oeis.org

1, 2, 4, 10, 40, 280, 3400, 68920, 2296600, 124819000, 11029312600, 1581276391000, 367448845658200, 138299522459392600, 84276864426837376600, 83129040425047907584600, 132705616446736897029760600, 342829213074356555028732544600
Offset: 1

Views

Author

Gary W. Adamson, Dec 31 2008

Keywords

Comments

Equals A000012 * A003266, where A000012 = the partial sum operator as an infinite lower triangular matrix.
a(n)+1 is divisible by 149 (a prime factor of Fibonacci(37)) for all n >= 36. The only values of n for which a(n)+1 is prime are: 1, 2, 3, 4, 5, 6, 10, 18. The corresponding primes are: 2, 3, 5, 11, 41, 281, 124819001, 342829213074356555028732544601. - Amiram Eldar, May 04 2017

Examples

			a(4) = 10 = (1 + 1 + 2 + 6), where A003266 = (1, 1, 2, 6, 30, 240, 3120,...).

Crossrefs

Cf. A003266, A153758 (partial sums).

Programs

  • Mathematica
    a[n_]:=Sum[Fibonorial[k], {k, n}]; Table[a[n],{n,1,10}]

Formula

Partial sums of A003266 terms.

Extensions

More terms from Amiram Eldar, Feb 26 2020

A270046 Integers n such that product of first n nonzero Fibonacci numbers (A003266) is the sum of 4 but no fewer nonzero squares.

Original entry on oeis.org

6, 8, 17, 34, 35, 60, 61, 62, 67, 72, 73, 74, 88, 114, 116, 126, 128, 144, 145, 146, 165, 171, 210, 212, 223, 231, 237, 247, 257, 269, 283, 288, 289, 290, 303, 317, 324, 325, 326, 330, 332, 339, 346, 347, 354, 356, 360, 361, 362, 376, 402, 404, 415, 423, 429, 438, 440
Offset: 1

Views

Author

Altug Alkan, Mar 09 2016

Keywords

Comments

How is the distribution of a(n), a(n+1), a(n+2) in this sequence where a(n+2) = a(n+1) + 1 = a(n) + 2?

Examples

			6 is a term because 1*1*2*3*5*8 = 240 and 240 = x^2 + y^2 + z^2 has no solution for integer values of x, y and z.

Links

Crossrefs

Cf. A003266, A004215.

Programs

  • PARI
    isA004215(n) = my(fouri, j) ; fouri=1 ; while( n >=7*fouri, if( n % fouri ==0, j= n/fouri-7 ; if( j % 8==0, return(1) ) ; ); fouri *= 4 ; ) ; return(0);
a003266(n) = prod(k=1, n, fibonacci(k));
for(n=1, 1e3, if(isA004215(a003266(n)), print1(n, ", ")));

A270475 Integers n such that A003266(n) is not divisible by n*(n+1)/2.

Original entry on oeis.org

2, 3, 4, 6, 7, 22, 23, 42, 43, 66, 67, 82, 83, 102, 103, 126, 127, 162, 163, 166, 167, 222, 223, 226, 227, 282, 283, 366, 367, 382, 383, 442, 443, 462, 463, 466, 467, 486, 487, 502, 503, 522, 523, 546, 547, 586, 587, 606, 607, 642, 643, 646, 647, 682, 683, 726, 727, 786, 787
Offset: 1

Views

Author

Altug Alkan, Mar 17 2016

Keywords

Comments

This sequence contains primes dividing all Fibonacci sequences.

Examples

			6 is a term because (1*1*2*3*5*8) is not divisible by (1+2+3+4+5+6).
5 is not a term because (1*1*2*3*5) is divisible by (1+2+3+4+5).

Crossrefs

Cf. A000040, A000057, A003266.

Programs

  • Mathematica
    nn = 800; Function[k, Select[Range@ nn, ! Divisible[k[[#]], # (# + 1)/2] &]]@ FoldList[Times, Array[Fibonacci@ # &, nn]] (* Michael De Vlieger, Mar 19 2016 *)
  • PARI
    a(n) = prod(k=1, n, fibonacci(k));
for(n=1, 1e3, if(a(n) % (n*(n+1)/2) != 0, print1(n, ", ")));

A270839 Integers k such that (A003266(k)/A000045(k-1)) is not divisible by k.

Original entry on oeis.org

2, 3, 4, 7, 9, 11, 19, 23, 31, 43, 59, 67, 71, 79, 83, 103, 127, 131, 163, 167, 179, 191, 223, 227, 239, 251, 271, 283, 311, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547, 571, 587, 599, 607, 631, 643, 647, 659, 683, 719, 727, 739, 751, 787, 823, 827, 839
Offset: 1

Views

Author

Altug Alkan, Mar 23 2016

Keywords

Comments

A270777 is a subsequence.
It appears that this sequence generates prime numbers except 4 and 9. [Verified for the first 500 terms. - Amiram Eldar, Apr 01 2021]

Examples

			3 is a term because 1*2 is not divisible by 3.
7 is a term because 1*1*2*3*5*13 is not divisible by 7.

Links

Crossrefs

Cf. A000045, A000057, A003266, A270777.

Programs

  • Mathematica
    Select[Range[2, 840], ! Divisible[Fibonorial@ #/Fibonacci[# - 1], #] &] (* Version 10, or *) Select[Range[2, 840], ! Divisible[Product[Fibonacci@ k, {k, #}]/Fibonacci[# - 1], #] &] (* Michael De Vlieger, Mar 24 2016 *)
  • PARI
    t(n) = fibonacci(n) * prod(k=1, n-2, Mod(fibonacci(k), n));
for(n=2, 1e3, if(lift(t(n)) != 0, print1(n, ", ")));

Extensions

Offset corrected by Amiram Eldar, Apr 01 2021

A382910 a(n) = A003266(n)^2.

Original entry on oeis.org

1, 1, 1, 4, 36, 900, 57600, 9734400, 4292870400, 4962558182400, 15011738501760000, 118907980672440960000, 2465675887223735746560000, 133859078241489389944995840000, 19025256931384645503492313743360000, 7079298104168226591849489943904256000000, 6896432754839457130755425769163265163264000000
Offset: 0

Views

Author

Edwin Hermann, Apr 08 2025

Keywords

Comments

For n>=3 number of valid symmetrical change ringing methods on n bells with the shortest number of rows per lead where the treble plain hunts out to the back. See Wikipedia and the Polster and Ross link for an explanation of bell ringing terminology.

Links

Crossrefs

Cf. A000045, A001622, A003266, A007598, A062073, A090281.

Programs

  • Maple
    a:= proc(n) a(n):= `if`(n=0, 1, a(n-1)*(<<0|1>, <1|1>>^n)[1, 2]^2) end:
seq(a(n), n=0..16);  # Alois P. Heinz, Apr 14 2025
  • Mathematica
    k = 1; {1, 1}~Join~Reap[Do[k *= Fibonacci[n]; Sow[k^2], {n, 16}] ][[-1, 1]] (* Michael De Vlieger, Apr 14 2025 *)

Formula

a(n) = Product_{j=1..n} Fibonacci(j)^2.
a(0) = 1; a(n) = a(n-1)*A007598(n). - Hugo Pfoertner, Apr 13 2025
a(n) ~ c^2 * phi^(n*(n+1)) / 5^n where phi is the golden ratio (A001622) and c = A062073. - Amiram Eldar, Aug 18 2025

A121284 Triangle T(n,k), 1<=k<=n, read by rows given by T(n,k) = A003266(n)/A000045(k).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 6, 3, 2, 30, 30, 15, 10, 6, 240, 240, 120, 80, 48, 30, 3120, 3120, 1560, 1040, 624, 390, 240, 65520, 65520, 32760, 21840, 13104, 8190, 5040, 3120, 2227680, 2227680, 1113840, 742560, 445536, 278460, 171360, 106080, 65520
Offset: 1

Views

Author

Philippe Deléham, Aug 24 2006

Keywords

A270491 a(n) = A256832(n) mod A003266(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 11138400, 2194264800, 970377408000, 194939999654400, 23386660116019200, 63018468582765696000, 81934202708323789824000, 118589068612624434080256000, 230237098382438262288036864000
Offset: 1

Views

Author

Altug Alkan, Mar 18 2016

Keywords

Examples

			a(5) = 0 because (1*2*5*12*29) mod (1*1*2*3*5) = 0.

Crossrefs

Cf. A000040, A000129, A003266, A256832.

Programs

  • Mathematica
    Table[Mod[Product[Expand[((1 + Sqrt@ 2)^j - (1 - Sqrt@ 2)^j)/(2 Sqrt@ 2)], {j, n}], Product[Fibonacci@ k, {k, n}]], {n, 18}] (* Michael De Vlieger, Mar 18 2016, after Vaclav Kotesovec at A256832 *)
  • PARI
    a000129(n) = ([2, 1; 1, 0]^n)[2, 1];
a256832(n) = prod(k=1, n, a000129(k));
a003266(n) = prod(k=1, n, fibonacci(k));
for(n=1, 20, print1(a256832(n) % a003266(n), ", "));

A271421 a(n) = fibonorial(3*n)/(fibonorial(2*n+1)*fibonorial(n+1)), where fibonorial(n) = A003266(n).

Original entry on oeis.org

1, 4, 119, 23496, 32149806, 300214157831, 19246160432331107, 8451529006578585976752, 25443734373070679510011112460, 524973397889459587964008354031908560, 74243674067972394056586805754940632245000310, 71965837912588688126721254257169744333502564695515911
Offset: 1

Views

Author

Vladimir Reshetnikov, May 21 2016

Keywords

References

  • Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.5.

Links

Crossrefs

Cf. A003266, A003267, A003268, A062073, A003150, A000045.

Programs

  • Mathematica
    Table[Fibonorial[3 n]/(Fibonorial[2 n + 1] Fibonorial[n + 1]), {n, 1, 30}] (* The sequence itself *)
QPochhammer[-1/GoldenRatio^2] (* The Fibonacci factorial constant C in the asymptotic expansion *)

Formula

a(n) ~ 5*phi^(2*n^2 - 3*n - 2)/C where phi = (1+sqrt(5))/2, and C = (-1/phi^2; -1/phi^2)_inf is the Fibonacci factorial constant whose decimal expansion is given in A062073.

A277363 Self-convolution of a(n)/4^n gives fibonorials (A003266).

Original entry on oeis.org

1, 2, 6, 52, 646, 13756, 458780, 24525352, 2094232006, 287618113900, 63647556127412, 22739228686869592, 13126310109506278556, 12250085882856201785816, 18488349380363585366790264, 45134497176992058331312333648, 178246891228174428563552421395782
Offset: 0

Views

Author

Vladimir Reshetnikov, Oct 10 2016

Keywords

Comments

Self-convolution of a(n) gives A003266(n)*4^n.

Crossrefs

Cf. A000045, A003266, A277362.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, (4^n
      *mul((<<0|1>, <1|1>>^i)[1, 2], i=1..n)-
      add(a(k)*a(n-k), k=1..n-1))/2)
    end:
seq(a(n), n=0...20);  # Alois P. Heinz, Oct 12 2016
  • Mathematica
    With[{n = 20}, Sqrt[Sum[Fibonorial[k] (4 x)^k, {k, 0, n - 1}] + O[x]^n][[3]]] (* before version 10.0 define Fibonorial[n_] := Product[Fibonacci[k], {k, 1, n}] *)

Formula

Sum_{k=0..n} a(k)/4^k * a(n-k)/4^(n-k) = A003266(n).
Previous Showing 11-20 of 89 results. Next

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