A180662 The Golden Triangle: T(n,k) = A001654(k) for n>=0 and 0<=k<=n.

0, 0, 1, 0, 1, 2, 0, 1, 2, 6, 0, 1, 2, 6, 15, 0, 1, 2, 6, 15, 40, 0, 1, 2, 6, 15, 40, 104, 0, 1, 2, 6, 15, 40, 104, 273, 0, 1, 2, 6, 15, 40, 104, 273, 714, 0, 1, 2, 6, 15, 40, 104, 273, 714, 1870, 0, 1, 2, 6, 15, 40, 104, 273, 714, 1870, 4895

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010

Keywords

• easy
• nonn
• tabl

Comments

The terms in the n-th row of the Golden Triangle are the first (n+1) golden rectangle numbers. The golden rectangle numbers are A001654(n)=F(n)*F(n+1), with F(n) the Fibonacci numbers. The mirror image of the Golden Triangle is A180663.

We define below 24 mostly new triangle sums. The Row1 and Row2 sums are the ordinary and alternating row sums respectively and the Kn11 and Kn12 sums are commonly known as antidiagonal sums. Each of the names of these sums, except for the row sums, comes from a (fairy) chess piece that moves in its own peculiar way over a chessboard, see Hooper and Whyld. All pieces are leapers: knight (sqrt(5) or 1,2), fil (sqrt(8) or 2,2), camel (sqrt(10) or 3,1), giraffe (sqrt(17) or 4,1) and zebra (sqrt(13) or 3,2). Information about the origin of these chess sums can be found in "Famous numbers on a chessboard", see Meijer.

Each triangle or chess sum formula adds up numbers on a chessboard using the moves of its namesake. Converting a number triangle to a square array of numbers shows this most clearly (use the table button!). The formulas given below are for number triangles.

The chess sums of the Golden Triangle lead to six different sequences, see the crossrefs. As could be expected all these sums are related to the golden rectangle numbers.

Some triangles with complete sets of triangle sums are: A002260 (Natural Numbers), A007318 (Pascal), A008288 (Delannoy) A013609 (Pell-Jacobsthal), A036561 (Nicomachus), A104763 (Fibonacci(n)), A158405 (Odd Numbers) and of course A180662 (Golden Triangle).

#..Name....Type..Code....Definition of triangle sums.

1. Row......1....Row1.. a(n) = Sum_{k=0..n} T(n, k).

2. Row Alt..2....Row2.. a(n) = Sum_{k=0..n} (-1)^(n+k)*T(n, k).

3. Knight...1....Kn11.. a(n) = Sum_{k=0..floor(n/2)} T(n-k, k).

4. Knight...1....Kn12.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, k+1).

5. Knight...1....Kn13.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+2, k+2).

6. Knight...2....Kn21.. a(n) = Sum_{k=0..floor(n/2)} T(n-k, n-2*k).

7. Knight...2....Kn22.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, n-2*k).

8. Knight...2....Kn23.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+2, n-2*k).

9. Knight...3....Kn3... a(n) = Sum_{k=0..n} T(n+k, 2*k).

10. Knight...4....Kn4... a(n) = Sum_{k=0..n} T(n+k, n-k).

11. Fil......1....Fi1... a(n) = Sum_{k=0..floor(n/2)} T(n, 2*k).

12. Fil......2....Fi2... a(n) = Sum_{k=0..floor(n/2)} T(n, n-2*k).

13. Camel....1....Ca1... a(n) = Sum_{k=0..floor(n/3)} T(n-2*k, k).

14. Camel....2....Ca2... a(n) = Sum_{k=0..floor(n/3)} T(n-2*k, n-3*k).

15. Camel....3....Ca3... a(n) = Sum_{k=0..n} T(n+2*k, 3*k).

16. Camel....4....Ca4... a(n) = Sum_{k=0..n} T(n+2*k, n-k).

17. Giraffe..1....Gi1... a(n) = Sum_{k=0..floor(n/4)} T(n-3*k, k).

18. Giraffe..2....Gi2... a(n) = Sum_{k=0..floor(n/4)} T(n-3*k, n-4*k).

19. Giraffe..3....Gi3... a(n) = Sum_{k=0..n} T(n+3*k, 4*k).

20. Giraffe..4....Gi4... a(n) = Sum_{k=0..n} T(n+3*k, n-k).

21. Zebra....1....Ze1... a(n) = Sum_{k=0..floor(n/2)} T(n+k, 3*k).

22. Zebra....2....Ze2... a(n) = Sum_{k=0..floor(n/2)} T(n+k, n-2*k).

23. Zebra....3....Ze3... a(n) = Sum_{k=0..floor(n/3)} T(n-k, 2*k).

24. Zebra....4....Ze4... a(n) = Sum_{k=0..floor(n/3)} T(n-k, n-3*k).

Examples

			The first few rows of the Golden Triangle are:
  0;
  0, 1;
  0, 1, 2;
  0, 1, 2, 6;
  0, 1, 2, 6, 15;
  0, 1, 2, 6, 15, 40;
		

References

• David Hooper and Kenneth Whyld, The Oxford Companion to Chess, p. 221, 1992.

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
• Verner E. Hoggatt, Jr., A new angle on Pascal’s triangle, The Fibonacci Quarterly, Vol. 6, Number 4, pp. 228-230, Oct. 1968.
• Edouard Lucas, Recherches sur plusieurs ouvrages de Léonard de Pise, Ch. 1, pp. 12-14, 1877.
• Johannes W. Meijer, Famous numbers on a chessboard, Acta Nova, Volume 4, No.4, December 2010; pp. 589-598.
• Johannes W. Meijer, Illustrations of the triangle sums, Mar 07 2013.
• S. Northshield, Sums across Pascal's triangle modulo 2, Congressus Numerantium, 200, pp. 35-52, 2010.

Crossrefs

Cf. A180663 (Mirror), A001654 (Golden Rectangle), A000045 (F(n)).

Triangle sums: A064831 (Row1, Kn11, Kn12, Kn4, Ca1, Ca4, Gi1, Gi4), A077916 (Row2), A180664 (Kn13), A180665 (Kn21, Kn22, Kn23, Fi2, Ze2), A180665(2*n) (Kn3, Fi1, Ze3), A115730(n+1) (Ca2, Ze4), A115730(3*n+1) (Ca3, Ze1), A180666 (Gi2), A180666(4*n) (Gi3).

Programs

Haskell

import Data.List (inits)
a180662 n k = a180662_tabl !! n !! k
a180662_row n = a180662_tabl !! n
a180662_tabl = tail $ inits a001654_list
-- Reinhard Zumkeller, Jun 08 2013

Magma

[Fibonacci(k)*Fibonacci(k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, May 25 2021

Maple

F:= combinat[fibonacci]:
T:= (n, k)-> F(k)*F(k+1):
seq(seq(T(n, k), k=0..n), n=0..10); # revised Johannes W. Meijer, Sep 13 2012

Mathematica

Table[Times @@ Fibonacci@ {k, k + 1}, {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 18 2016 *)
Module[{nn=20,f},f=Times@@@Partition[Fibonacci[Range[0,nn]],2,1];Table[Take[f,n],{n,nn}]]//Flatten (* Harvey P. Dale, Nov 26 2022 *)

PARI

T(n,k)=fibonacci(k)*fibonacci(k+1) \\ Charles R Greathouse IV, Nov 07 2016

Sage

flatten([[fibonacci(k)*fibonacci(k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 25 2021

Formula

T(n, k) = F(k)*F(k+1) with F(n) = A000045(n), for n>=0 and 0<=k<=n.

From Johannes W. Meijer, Jun 22 2015: (Start)

Kn1p(n) = Sum_{k=0..floor(n/2)} T(n-k+p-1, k+p-1), p >= 1.

Kn1p(n) = Kn11(n+2*p-2) - Sum_{k=0..p-2} T(n-k+2*p-2, k), p >= 2.

Kn2p(n) = Sum_{k=0..floor(n/2)} T(n-k+p-1, n-2*k), p >= 1.

Kn2p(n) = Kn21(n+2*p-2) - Sum_{k=0..p-2} T(n+k+p, n+2*k+2), p >= 2. (End)

G.f. as triangle: xy/((1-x)(1+xy)(1-3xy+x^2 y^2)). - Robert Israel, Sep 06 2015