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 1-3 of 3 results.

A094567 Associated with alternating row sums of array in A094566.

Original entry on oeis.org

1, 4, 30, 203, 1394, 9552, 65473, 448756, 3075822, 21081995, 144498146, 990405024, 6788337025, 46527954148, 318907342014, 2185823439947, 14981856737618, 102687173723376, 703828359326017, 4824111341558740, 33064951031585166, 226630545879537419
Offset: 0

Views

Author

Clark Kimberling, May 12 2004

Keywords

Examples

			Obtain 4,30,203 from a(0)=1 and Fibonacci numbers 1,5,34,233: 4=5-1, 30=34-4, 203=233-30.

Links

Crossrefs

Cf. A000045, A094566.

Programs

  • Mathematica
    RecurrenceTable[{a[0]==1,a[n]==Fibonacci[4n+1]-a[n-1]},a[n],{n,30}] (* or *) LinearRecurrence[{6,6,-1},{1,4,30},31] (* Harvey P. Dale, Jul 13 2011 *)
  • PARI
    Vec(-(2*x-1)/((x+1)*(x^2-7*x+1)) + O(x^100)) \\ Colin Barker, Nov 19 2014
  • PARI
    vector(30, n, n--; (fibonacci(4*n+3) + (-1)^n)/3) \\ Michel Marcus, Nov 19 2014

Formula

a(n) = F(4n+1) - a(n-1) for n >= 1, with a(0) = 1.
a(n) = (Fib(4n+3) + (-1)^n)/3. - Ralf Stephan, Dec 04 2004
a(n) = 6*a(n-1)+6*a(n-2)-a(n-3), with a(0)=1, a(1)=4, a(2)=30. - Harvey P. Dale, Jul 13 2011
G.f.: (1-2*x)/(1-6*x-6*x^2+x^3). - Harvey P. Dale, Jul 13 2011
a(n) = (-1)^n*sum((-1)^k*Fibonacci(4*k+1), k=0..n). - Gary Detlefs, Jan 22 2013
a(n) = (2^(-n)*(5*(-2)^n+(7-3*sqrt(5))^n*(5-2*sqrt(5))+(5+2*sqrt(5))*(7+3*sqrt(5))^n))/15. - Colin Barker, Mar 05 2016

Extensions

More terms from Harvey P. Dale, Jul 13 2011

A094565 Triangle read by rows: binary products of Fibonacci numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 13, 15, 16, 21, 34, 39, 40, 42, 55, 89, 102, 104, 105, 110, 144, 233, 267, 272, 273, 275, 288, 377, 610, 699, 712, 714, 715, 720, 754, 987, 1597, 1830, 1864, 1869, 1870, 1872, 1885, 1974, 2584, 4181, 4791, 4880, 4893, 4895, 4896, 4901, 4935, 5168, 6765
Offset: 1

Views

Author

Clark Kimberling, May 12 2004

Keywords

Comments

Row n consists of n numbers, first F(2n-1) and last F(2n).
Central numbers: (1,6,40,273,...) = A081016.
Row sums: A001870.
Alternating row sums: 1,1,7,7,48,48,329,329; the sequence b=(1,7,48,329,...) is A004187, given by b(n)=F(4n+2)-b(n-1) for n>=2, with b(1)=1.
In each row, the difference between neighboring terms is a Fibonacci number.

Examples

			Triangle begins:
   1;
   2,   3;
   5,   6    8;
  13,  15,  16,  21;
  34,  39,  40,  42,  55;
  89, 102, 104, 105, 110, 144; ...

Links

Crossrefs

Cf. A000045, A094566, A094568.

Programs

  • GAP
    Flat(List([1..12], n-> List([1..n], k-> Fibonacci(2*k)*Fibonacci(2*n-2*k+1) ))); # G. C. Greubel, Jul 15 2019
  • Magma
    [Fibonacci(2*k)*Fibonacci(2*n-2*k+1): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 15 2019
  • Mathematica
    Table[Fibonacci[2*k]*Fibonacci[2*n-2*k+1], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Jul 15 2019 *)
  • PARI
    row(n) = vector(n, k, fibonacci(2*k)*fibonacci(2*n-2*k+1));
tabl(nn) = for(n=1, nn, print(row(n))); \\ Michel Marcus, May 03 2016
  • Sage
    [[fibonacci(2*k)*fibonacci(2*n-2*k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 15 2019

Formula

Row n: F(2)F(2n-1), F(4)F(2n-3), ..., F(2n)F(1).

A094568 Triangle of binary products of Fibonacci numbers.

Original entry on oeis.org

2, 3, 5, 8, 10, 13, 21, 24, 26, 34, 55, 63, 65, 68, 89, 144, 165, 168, 170, 178, 233, 377, 432, 440, 442, 445, 466, 610, 987, 1131, 1152, 1155, 1157, 1165, 1220, 1597, 2584, 2961, 3016, 3024, 3026, 3029, 3050, 3194, 4181, 6765, 7752, 7896, 7917, 7920, 7922
Offset: 1

Views

Author

Clark Kimberling, May 12 2004

Keywords

Comments

Start with the triangle in A094566: starting with row 2, expel from each row the term that is a square of a Fibonacci number (A007598). The remaining triangle is this sequence.
In each row, the difference between neighboring terms is a Fibonacci number. For n>1, row n consists of n numbers, first F(2n) and last F(2n+1).
Central numbers: (2,10,65,442,...), essentially A064170.
Alternating row sums: 2,2,11,11,78,78,...; the sequence b=(2,11,78,...) is A094569.

Examples

			First four rows:
2
3 5
8 10 13
21 24 26 34

Links

Crossrefs

Cf. A000045, A007598, A094565, A094566, A094569.

Programs

  • PARI
    pef(k, n) = fibonacci(2*k)*fibonacci(2*n-2*k);
pof(k, n) = fibonacci(2*n-2*k+1)*fibonacci(2*k-1);
isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8)); \\ from A010056
isfib2(x) = issquare(x) && isfib(sqrtint(x));
tabl(nn) = {for (n=2, nn, if (n % 2 == 0, for (k=1, n/2, if (! isfib2(x = pef(k,n)), print1(x, ", "));); forstep (k=n/2, 1, -1, if (! isfib2(x = pof(k,n)), print1(x, ", "));), for (k=1, n\2, if (! isfib2(x = pef(k,n)), print1(x, ", "));); forstep (k=n\2+1, 1, -1, if (! isfib2(x = pof(k,n)), print1(x, ", ")););); print(););} \\ Michel Marcus, May 04 2016
Showing 1-3 of 3 results.

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