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%I #68 Nov 26 2022 17:24:54
%S 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,
%T 6,15,40,104,273,0,1,2,6,15,40,104,273,714,0,1,2,6,15,40,104,273,714,
%U 1870,0,1,2,6,15,40,104,273,714,1870,4895
%N The Golden Triangle: T(n,k) = A001654(k) for n>=0 and 0<=k<=n.
%C 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.
%C 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.
%C 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.
%C 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.
%C 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).
%C #..Name....Type..Code....Definition of triangle sums.
%C 1. Row......1....Row1.. a(n) = Sum_{k=0..n} T(n, k).
%C 2. Row Alt..2....Row2.. a(n) = Sum_{k=0..n} (-1)^(n+k)*T(n, k).
%C 3. Knight...1....Kn11.. a(n) = Sum_{k=0..floor(n/2)} T(n-k, k).
%C 4. Knight...1....Kn12.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, k+1).
%C 5. Knight...1....Kn13.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+2, k+2).
%C 6. Knight...2....Kn21.. a(n) = Sum_{k=0..floor(n/2)} T(n-k, n-2*k).
%C 7. Knight...2....Kn22.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, n-2*k).
%C 8. Knight...2....Kn23.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+2, n-2*k).
%C 9. Knight...3....Kn3... a(n) = Sum_{k=0..n} T(n+k, 2*k).
%C 10. Knight...4....Kn4... a(n) = Sum_{k=0..n} T(n+k, n-k).
%C 11. Fil......1....Fi1... a(n) = Sum_{k=0..floor(n/2)} T(n, 2*k).
%C 12. Fil......2....Fi2... a(n) = Sum_{k=0..floor(n/2)} T(n, n-2*k).
%C 13. Camel....1....Ca1... a(n) = Sum_{k=0..floor(n/3)} T(n-2*k, k).
%C 14. Camel....2....Ca2... a(n) = Sum_{k=0..floor(n/3)} T(n-2*k, n-3*k).
%C 15. Camel....3....Ca3... a(n) = Sum_{k=0..n} T(n+2*k, 3*k).
%C 16. Camel....4....Ca4... a(n) = Sum_{k=0..n} T(n+2*k, n-k).
%C 17. Giraffe..1....Gi1... a(n) = Sum_{k=0..floor(n/4)} T(n-3*k, k).
%C 18. Giraffe..2....Gi2... a(n) = Sum_{k=0..floor(n/4)} T(n-3*k, n-4*k).
%C 19. Giraffe..3....Gi3... a(n) = Sum_{k=0..n} T(n+3*k, 4*k).
%C 20. Giraffe..4....Gi4... a(n) = Sum_{k=0..n} T(n+3*k, n-k).
%C 21. Zebra....1....Ze1... a(n) = Sum_{k=0..floor(n/2)} T(n+k, 3*k).
%C 22. Zebra....2....Ze2... a(n) = Sum_{k=0..floor(n/2)} T(n+k, n-2*k).
%C 23. Zebra....3....Ze3... a(n) = Sum_{k=0..floor(n/3)} T(n-k, 2*k).
%C 24. Zebra....4....Ze4... a(n) = Sum_{k=0..floor(n/3)} T(n-k, n-3*k).
%D David Hooper and Kenneth Whyld, The Oxford Companion to Chess, p. 221, 1992.
%H Reinhard Zumkeller, <a href="/A180662/b180662.txt">Rows n = 0..120 of triangle, flattened</a>
%H Verner E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/6-4/hoggatt.pdf">A new angle on Pascal’s triangle</a>, The Fibonacci Quarterly, Vol. 6, Number 4, pp. 228-230, Oct. 1968.
%H Edouard Lucas, <a href="http://edouardlucas.free.fr/oeuvres/leonard de pise.pdf">Recherches sur plusieurs ouvrages de Léonard de Pise</a>, Ch. 1, pp. 12-14, 1877.
%H Johannes W. Meijer, <a href="http://jnsilva.ludicum.org/TJ/TJ1920/FamousNumbers.pdf">Famous numbers on a chessboard</a>, Acta Nova, Volume 4, No.4, December 2010; pp. 589-598.
%H Johannes W. Meijer, <a href="/A180662/a180662.jpg">Illustrations of the triangle sums</a>, Mar 07 2013.
%H S. Northshield, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.530.857">Sums across Pascal's triangle modulo 2</a>, Congressus Numerantium, 200, pp. 35-52, 2010.
%F T(n, k) = F(k)*F(k+1) with F(n) = A000045(n), for n>=0 and 0<=k<=n.
%F From _Johannes W. Meijer_, Jun 22 2015: (Start)
%F Kn1p(n) = Sum_{k=0..floor(n/2)} T(n-k+p-1, k+p-1), p >= 1.
%F Kn1p(n) = Kn11(n+2*p-2) - Sum_{k=0..p-2} T(n-k+2*p-2, k), p >= 2.
%F Kn2p(n) = Sum_{k=0..floor(n/2)} T(n-k+p-1, n-2*k), p >= 1.
%F Kn2p(n) = Kn21(n+2*p-2) - Sum_{k=0..p-2} T(n+k+p, n+2*k+2), p >= 2. (End)
%F G.f. as triangle: xy/((1-x)(1+xy)(1-3xy+x^2 y^2)). - _Robert Israel_, Sep 06 2015
%e The first few rows of the Golden Triangle are:
%e 0;
%e 0, 1;
%e 0, 1, 2;
%e 0, 1, 2, 6;
%e 0, 1, 2, 6, 15;
%e 0, 1, 2, 6, 15, 40;
%p F:= combinat[fibonacci]:
%p T:= (n, k)-> F(k)*F(k+1):
%p seq(seq(T(n, k), k=0..n), n=0..10); # revised _Johannes W. Meijer_, Sep 13 2012
%t Table[Times @@ Fibonacci@ {k, k + 1}, {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Aug 18 2016 *)
%t 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 *)
%o (Haskell)
%o import Data.List (inits)
%o a180662 n k = a180662_tabl !! n !! k
%o a180662_row n = a180662_tabl !! n
%o a180662_tabl = tail $ inits a001654_list
%o -- _Reinhard Zumkeller_, Jun 08 2013
%o (PARI) T(n,k)=fibonacci(k)*fibonacci(k+1) \\ _Charles R Greathouse IV_, Nov 07 2016
%o (Magma) [Fibonacci(k)*Fibonacci(k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 25 2021
%o (Sage) flatten([[fibonacci(k)*fibonacci(k+1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 25 2021
%Y Cf. A180663 (Mirror), A001654 (Golden Rectangle), A000045 (F(n)).
%Y 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).
%K easy,nonn,tabl
%O 0,6
%A _Johannes W. Meijer_, Sep 21 2010
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