A143211 -id:A143211 - cp's OEIS frontend

A259222 T(n,k) is the number of (n+1) X (k+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.

7, 13, 13, 24, 23, 24, 45, 40, 40, 45, 85, 71, 66, 71, 85, 162, 127, 112, 112, 127, 162, 311, 230, 192, 183, 192, 230, 311, 601, 421, 334, 303, 303, 334, 421, 601, 1168, 779, 588, 510, 487, 510, 588, 779, 1168, 2281, 1456, 1048, 869, 798, 798, 869, 1048, 1456, 2281

Offset: 1

R. H. Hardin, Jun 21 2015

Table starts
 13   24   45   85  162   311   601  1168  2281  4473   8802  17371
 23   40   71  127  230   421   779  1456  2747  5227  10022  19345
 40   66  112  192  334   588  1048  1890  3448  6360  11854  22308
 71  112  183  303  510   869  1499  2616  4619  8251  14910  27249
127  192  303  487  798  1325  2227  3784  6499 11283  19806  35161
230  334  510  798 1278  2078  3422  5694  9566 16222  27774  48030
421  588  869 1325 2078  3319  5377  8804 14545 24225  40670  68843
779 1048 1499 2227 3422  5377  8591 13888 22655 37231  61598 102589
1456 1890 2616 3784 5694  8804 13888 22210 35872 58368  95550 157276
2747 3448 4619 6499 9566 14545 22655 35872 57455 92767 150686 245965
Each row (and each column, by symmetry) has a rational generating function (and therefore a linear recurrence with constant coefficients) because the growth from an array to the next larger one is described by the transfer matrix method. - R. J. Mathar, Oct 09 2020

			Some solutions for n=4, k=4:
  0 0 1 0 1      1 1 1 0 1      0 0 0 0 1      0 0 0 1 0
  0 0 1 0 1      0 0 0 1 0      0 0 0 0 1      0 0 0 1 0
  1 1 0 1 0      0 0 0 1 0      1 1 1 1 0      0 0 0 1 0
  0 0 1 0 1      0 0 0 1 0      0 0 0 0 1      1 1 1 0 1
  1 1 0 1 0      0 0 0 1 0      0 0 0 0 1      0 0 0 1 0

R. H. Hardin, Table of n, a(n) for n = 1..480

Cf. A259215, A259216, A259217, A259218, A259219, A259220, A259221.

Empirical for diagonal and column k (k=3..7 recurrences work also for k=1,2):
diagonal: a(n) = 6*a(n-1) - 10*a(n-2) - 2*a(n-3) + 16*a(n-4) - 6*a(n-5) - 5*a(n-6) + 2*a(n-7).
k=1: a(n) = 3*a(n-1) -   a(n-2) - 2*a(n-3)
k=2: a(n) = 3*a(n-1) -   a(n-2) - 2*a(n-3)
k=3: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
k=4: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
k=5: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
k=6: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
k=7: a(n) = 4*a(n-1) - 4*a(n-2) -   a(n-3) + 2*a(n-4)
Empirical: T(n,k) = 2^(k+1) + 2^(n+1) + F(n+3)*F(k+3) - 2*F(n+3) - 2*F(k+3) + 2 = 2^(n+1) + A001911(k)*F(n+3) + A234933(k+1) = A234933(n+1) + A234933(k+1) + A143211(n+3,k+3) - 2, F=A000045. - Ehren Metcalfe, Dec 27 2018

A098356 Multiplication table of the Fibonacci numbers read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 2, 0, 0, 3, 2, 2, 3, 0, 0, 5, 3, 4, 3, 5, 0, 0, 8, 5, 6, 6, 5, 8, 0, 0, 13, 8, 10, 9, 10, 8, 13, 0, 0, 21, 13, 16, 15, 15, 16, 13, 21, 0, 0, 34, 21, 26, 24, 25, 24, 26, 21, 34, 0, 0, 55, 34, 42, 39, 40, 40, 39, 42, 34, 55, 0, 0, 89, 55, 68, 63, 65, 64, 65

Offset: 0

Douglas Stones (dssto1(AT)student.monash.edu.au), Sep 04 2004

Same as triangle T(n,k) = F(n)-F(k)*F(n-k+1), read by rows, F(i) = A000045(i). - Dale Gerdemann, Apr 24 2016

			Table begins:
   0   0   0   0   0   0   0   0   0 ...
   0   1   1   2   3   5   8  13  21...
   0   1   1   2   3   5   8  13  21...
   0   2   2   4   6  10  16  26  42...
   0   3   3   6   9  15  24  39  63...
   0   5   5  10  15  25  40  65 105...
   0   8   8  16  24  40  64 104 168...
   0  13  13  26  39  65 104 169 273...
   0  21  21  42  63 105 168 273 441...

Michael De Vlieger, Table of n, a(n) for n = 0..11324 (rows 0 <= n <= 149, flattened)

Cf. A003991, A058071, A001629 (antidiagonal sums).

Table[Fibonacci[n] - Fibonacci[k]*Fibonacci[n - k + 1], {n, 13}, {k, n}] // Flatten (* Michael De Vlieger, Dec 11 2020 *)

T(n,k) = T(k,n) = A000045(n)*A000045(k) = A143211(n,k). - R. J. Mathar, Dec 11 2020

A143212 a(n) = Fibonacci(n) * (Fibonacci(n+2) - 1).

Original entry on oeis.org

1, 2, 8, 21, 60, 160, 429, 1134, 2992, 7865, 20648, 54144, 141897, 371722, 973560, 2549421, 6675460, 17478176, 45761045, 119808150, 313668576, 821205937, 2149962768, 5628704256, 14736185425, 38579909330, 101003635304

Offset: 1

Gary W. Adamson, Jul 30 2008

nonn

Lim_{n -> oo} a(n)/a(n-1) tends to phi^2.

a(n) = Product of sum of first n Fibonacci numbers and Fibonacci number(n). - Vladimir Joseph Stephan Orlovsky, Oct 13 2009

			a(5) = 60 = F(5) * (F(7)-1) = 5*12.
a(5) = 60 = sum of row 5 terms of triangle A143211: (5 + 5 + 10 + 15 + 25).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Cf. A000045, A000071, A143211.

[Fibonacci(n)*(Fibonacci(n+2)-1): n in [1..40]]; // G. C. Greubel, Jul 21 2024

LinearRecurrence[{3,1,-5,-1,1}, {1,2,8,21,60}, 40] (* Vladimir Joseph Stephan Orlovsky, Oct 13 2009 *)
Table[Fibonacci[n](Fibonacci[n+2]-1),{n,30}] (* Harvey P. Dale, Dec 14 2012 *)

[fibonacci(n)*(fibonacci(n+2)-1) for n in range(1,41)] # G. C. Greubel, Jul 21 2024

a(n) = A000045(n) * A000071(n+2).
a(n) = Sum_{k=1..n} A143211(n, k) (row sums of A143211).
From R. J. Mathar, Sep 06 2008: (Start)
G.f.: (1-x+x^2)/((1+x)*(1-3*x+x^2)*(1-x-x^2)).
a(n) = (-5*A000045(n+1) + 3*(-1)^n + 7*A001906(n+1) -3*A001906(n))/5. (End)
a(n) = Fibonacci(n)*Sum_{k=0..n} Fibonacci(k). - Paul Barry, Jan 05 2009

cp's OEIS Frontend

A259222 T(n,k) is the number of (n+1) X (k+1) 0..1 arrays with each 2 X 2 subblock having clockwise pattern 0000 0011 or 0101.

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A098356 Multiplication table of the Fibonacci numbers read by antidiagonals.

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A143212 a(n) = Fibonacci(n) * (Fibonacci(n+2) - 1).

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