A127546 -id:A127546 - cp's OEIS frontend

A220436 a(n) = A127546(n)^2.

4, 36, 196, 1444, 9604, 66564, 454276, 3118756, 21362884, 146458404, 1003749124, 6880038916, 47155859716, 323212716324, 2215328606404, 15184099435684, 104073336269956, 713329336067076, 4889231802533764, 33511293841053604, 229689823620354244, 1574317475335503396, 10790532493690424836

Offset: 0

N. J. A. Sloane, Dec 16 2012

nonn

Giacomo Candido, A Relationship Between the Fourth Powers of the Terms of the Fibonacci Series, Scripta Mathematica, Vol. 17, No. 3-4 (1951), p. 230.
Shalosh B. Ekhad and Doron Zeilberger, Automatic Counting of Tilings of Skinny Plane Regions, in: Simon R. Blackburn, Stefanie Gerke and Mark Wildon, eds., Surveys in Combinatorics 2013, Cambridge University Press, 2013, pp. 363-378.

Claudi Alsina and Roger B. Nelsen, On Candido's Identity, Mathematics Magazine, Vol. 80, No. 3 (2007), pp. 226-228; alternative link.
Alexander Bogomolny, Candido's Identity, Cut the Knot.
Brother Alfred Brousseau, Ye Olde Fibonacci Curiosity Shoppe, The Fibonacci Quarterly, Vol. 10, No. 4 (1972), pp. 441-443.
Shalosh B. Ekhad and Doron Zeilberger, Automatic Counting of Tilings of Skinny Plane Regions, arXiv preprint, arXiv:1206.4864 [math.CO], 2012.
R. S. Melham, Ye Olde Fibonacci Curiosity Shoppe Revisited, The Fibonacci Quarterly, Vol. 42, No. 2 (2004), pp. 155-160.
Roger B. Nelsen, Proof without Words: Candido's Identity, Mathematics Magazine, Vol. 78, No. 2 (2005), p. 131.
Darko Veljan, A note on Candido's identity and Heron's formula, Proceedings of the 1st Croatian Combinatorial Days, Zagreb, September 29-30, 2016 (2016).
Wikipedia, Candido's identity.

Cf. A000045, A127546.

Table[Total[Fibonacci[Range[n, n + 2]]^2]^2, {n, 0, 22}] (* or *)
Table[4 (4 ((-1)^(n + 1) LucasL[2 (n + 1)] + LucasL[4 (n + 1)]) + 9)/25, {n, 0, 22}] (* Michael De Vlieger, Feb 18 2017 *)

Empirical g.f.: -4*(x^4-4*x^3-11*x^2+4*x+1) / ((x-1)*(x^2-7*x+1)*(x^2+3*x+1)). - Colin Barker, Jul 22 2013
a(n) = 4*(4*((-1)^(n + 1)*Lucas(2*(n + 1)) + Lucas(4*(n + 1))) + 9)/25. - Ehren Metcalfe, Feb 18 2017
a(n) = 2 * (F(n)^4 + F(n+1)^4 + F(n+2)^4), where F(n) is the n-th Fibonacci number (A000045) (Candido, 1951). - Amiram Eldar, Jan 11 2022

A059779 A Lucas triangle: T(m,n), m >= n >= 0.

Original entry on oeis.org

2, 1, 1, 3, 2, 3, 4, 3, 3, 4, 7, 5, 6, 5, 7, 11, 8, 9, 9, 8, 11, 18, 13, 15, 14, 15, 13, 18, 29, 21, 24, 23, 23, 24, 21, 29, 47, 34, 39, 37, 38, 37, 39, 34, 47, 76, 55, 63, 60, 61, 61, 60, 63, 55, 76, 123, 89, 102, 97, 99, 98, 99, 97, 102, 89, 123, 199, 144, 165, 157, 160, 159

Offset: 0

N. J. A. Sloane, Feb 22 2001

From Amiram Eldar, May 15 2023: (Start)
Named "Lucas triangle" by Josef (1983), and "Josef's triangle" by Koshy (2007).
The rows of the triangle are the antidiagonals of the array in which the 0th row is T(0, k) = Lucas(k) = A000032(k), the 1st row is T(1, k) = Fibonacci(k+2) = A000045(k+2), and each subsequent row is the sum of the previous 2 rows.
The central elements in the even rows are in A127546, starting from the 2nd row, i.e., the central element of the k-th row, for even k >= 2, is A127546(k/2-1). (End)

			Triangle starts:
  2;
  1,1;
  3,2,3;
  4,3,3,4;
  ...

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 28.
Alexander Engstrom, Graph colouring and the total Betti number, arXiv preprint, arXiv:1412.8460 [math.CO], 2014.
Šána Josef, Lucas Triangle, The Fibonacci Quarterly, Vol. 21, No. 3 (1983), pp. 192-195.
Thomas Koshy, 91.01 The central elements in Josef's triangle, The Mathematical Gazette, Vol. 91, No. 520 (2007), pp. 63-68.

Cf. A000045, A000032, A127546.

T := proc(m, n) option remember: if m=0 and n=0 then RETURN(2) fi: if m=1 and n=0 then RETURN(1) fi: if m=1 and n=1 then RETURN(1) fi: if m=2 and n=1 then RETURN(2) fi: if m<=n+1 then RETURN(T(m, m-n)) fi: if mJames Sellers, Feb 22 2001

T[0, k_] := T[0, k] = LucasL[k]; T[1, k_] := T[1, k] = Fibonacci[k + 2]; T[n_, k_] := T[n, k] = T[n - 1, k] + T[n - 2, k]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, May 15 2023 *)

T(m, n) = T(m-1, n) + T(m-2, n); T(0, 0)=2, T(1, 0)=1, T(1, 1)=1, T(2, 1)=2.

More terms from James Sellers, Feb 22 2001

A256661 Rectangular array by antidiagonals: row n shows the numbers k such that R(k) consists of n terms, where R(k) is the minimal alternating Fibonacci representation of k.

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 5, 7, 14, 25, 8, 10, 15, 38, 64, 13, 11, 17, 40, 98, 169, 21, 12, 22, 41, 103, 258, 441, 34, 16, 23, 46, 104, 271, 674, 1156, 55, 18, 24, 59, 106, 273, 708, 1766, 3025, 89, 19, 27, 61, 119, 274, 713, 1855, 4622, 7921, 144, 20, 28, 62, 153

Offset: 1

Clark Kimberling, Apr 08 2015

See A256655 for definitions.  Every positive integer occurs exactly once.
(row 1):  A000045 (Fibonacci numbers)
(col 1):  A007598 (squared Fibonacci numbers)
(col 2):  A127546 (conjectured)

			Northwest corner:
1     2     3     5     8     13    21
4     6     7     10    11    12    62
9     14    15    17    22    23    24
25    38    40    41    46    59    61
64    98    103   104   106   119   153
169   258   271   273   274   279   313
R(1) = 1, in row 1
R(2) = 2, in row 1
R(3) = 3, in row 1
R(4) = 5 - 1, in row 2
R(9) = 13 - 5 + 1, in row 3
R(25) = 34 - 13 + 5 - 1, in row 4
R(64) = 89 - 34 + 13 - 5 + 1, in row 5

Cf. A000045, A256655, A007598, A127546.

b[n_] = Fibonacci[n]; bb = Table[b[n], {n, 1, 70}];
h[0] = {1}; h[n_] := Join[h[n - 1], Table[b[n + 2], {k, 1, b[n]}]];
g = h[23];  r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]];
u = Table[Length[r[n]], {n, 1, 6000}];
TableForm[Table[Flatten[Position[u, k]], {k, 1, 9}]]

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A220436 a(n) = A127546(n)^2.

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A059779 A Lucas triangle: T(m,n), m >= n >= 0.

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A256661 Rectangular array by antidiagonals: row n shows the numbers k such that R(k) consists of n terms, where R(k) is the minimal alternating Fibonacci representation of k.

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