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.
%I A180662 #72 Jan 05 2025 19:51:39
%S A180662 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 A180662 6,15,40,104,273,0,1,2,6,15,40,104,273,714,0,1,2,6,15,40,104,273,714,
%U A180662 1870,0,1,2,6,15,40,104,273,714,1870,4895
%N A180662 The Golden Triangle: T(n,k) = A001654(k) for n>=0 and 0<=k<=n.
%C A180662 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 A180662 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 A180662 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 A180662 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 A180662 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 A180662 #..Name....Type..Code....Definition of triangle sums.
%C A180662 1. Row......1....Row1.. a(n) = Sum_{k=0..n} T(n, k).
%C A180662 2. Row Alt..2....Row2.. a(n) = Sum_{k=0..n} (-1)^(n+k)*T(n, k).
%C A180662 3. Knight...1....Kn11.. a(n) = Sum_{k=0..floor(n/2)} T(n-k, k).
%C A180662 4. Knight...1....Kn12.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, k+1).
%C A180662 5. Knight...1....Kn13.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+2, k+2).
%C A180662 6. Knight...2....Kn21.. a(n) = Sum_{k=0..floor(n/2)} T(n-k, n-2*k).
%C A180662 7. Knight...2....Kn22.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, n-2*k).
%C A180662 8. Knight...2....Kn23.. a(n) = Sum_{k=0..floor(n/2)} T(n-k+2, n-2*k).
%C A180662 9. Knight...3....Kn3... a(n) = Sum_{k=0..n} T(n+k, 2*k).
%C A180662 10. Knight...4....Kn4... a(n) = Sum_{k=0..n} T(n+k, n-k).
%C A180662 11. Fil......1....Fi1... a(n) = Sum_{k=0..floor(n/2)} T(n, 2*k).
%C A180662 12. Fil......2....Fi2... a(n) = Sum_{k=0..floor(n/2)} T(n, n-2*k).
%C A180662 13. Camel....1....Ca1... a(n) = Sum_{k=0..floor(n/3)} T(n-2*k, k).
%C A180662 14. Camel....2....Ca2... a(n) = Sum_{k=0..floor(n/3)} T(n-2*k, n-3*k).
%C A180662 15. Camel....3....Ca3... a(n) = Sum_{k=0..n} T(n+2*k, 3*k).
%C A180662 16. Camel....4....Ca4... a(n) = Sum_{k=0..n} T(n+2*k, n-k).
%C A180662 17. Giraffe..1....Gi1... a(n) = Sum_{k=0..floor(n/4)} T(n-3*k, k).
%C A180662 18. Giraffe..2....Gi2... a(n) = Sum_{k=0..floor(n/4)} T(n-3*k, n-4*k).
%C A180662 19. Giraffe..3....Gi3... a(n) = Sum_{k=0..n} T(n+3*k, 4*k).
%C A180662 20. Giraffe..4....Gi4... a(n) = Sum_{k=0..n} T(n+3*k, n-k).
%C A180662 21. Zebra....1....Ze1... a(n) = Sum_{k=0..floor(n/2)} T(n+k, 3*k).
%C A180662 22. Zebra....2....Ze2... a(n) = Sum_{k=0..floor(n/2)} T(n+k, n-2*k).
%C A180662 23. Zebra....3....Ze3... a(n) = Sum_{k=0..floor(n/3)} T(n-k, 2*k).
%C A180662 24. Zebra....4....Ze4... a(n) = Sum_{k=0..floor(n/3)} T(n-k, n-3*k).
%D A180662 David Hooper and Kenneth Whyld, The Oxford Companion to Chess, p. 221, 1992.
%H A180662 Reinhard Zumkeller, <a href="/A180662/b180662.txt">Rows n = 0..120 of triangle, flattened</a>
%H A180662 Verner E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://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 A180662 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 A180662 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 A180662 Johannes W. Meijer, <a href="/A180662/a180662.jpg">Illustrations of the triangle sums</a>, Mar 07 2013.
%H A180662 S. Northshield, <a href="http://citeseerx.ist.psu.edu/pdf/a765e1f8064266bf3e36c34bf5c60bb8bb32d392">Sums across Pascal's triangle modulo 2</a>, Congressus Numerantium, 200, pp. 35-52, 2010.
%F A180662 T(n, k) = F(k)*F(k+1) with F(n) = A000045(n), for n>=0 and 0<=k<=n.
%F A180662 From _Johannes W. Meijer_, Jun 22 2015: (Start)
%F A180662 Kn1p(n) = Sum_{k=0..floor(n/2)} T(n-k+p-1, k+p-1), p >= 1.
%F A180662 Kn1p(n) = Kn11(n+2*p-2) - Sum_{k=0..p-2} T(n-k+2*p-2, k), p >= 2.
%F A180662 Kn2p(n) = Sum_{k=0..floor(n/2)} T(n-k+p-1, n-2*k), p >= 1.
%F A180662 Kn2p(n) = Kn21(n+2*p-2) - Sum_{k=0..p-2} T(n+k+p, n+2*k+2), p >= 2. (End)
%F A180662 G.f. as triangle: xy/((1-x)(1+xy)(1-3xy+x^2 y^2)). - _Robert Israel_, Sep 06 2015
%e A180662 The first few rows of the Golden Triangle are:
%e A180662 0;
%e A180662 0, 1;
%e A180662 0, 1, 2;
%e A180662 0, 1, 2, 6;
%e A180662 0, 1, 2, 6, 15;
%e A180662 0, 1, 2, 6, 15, 40;
%p A180662 F:= combinat[fibonacci]:
%p A180662 T:= (n, k)-> F(k)*F(k+1):
%p A180662 seq(seq(T(n, k), k=0..n), n=0..10); # revised _Johannes W. Meijer_, Sep 13 2012
%t A180662 Table[Times @@ Fibonacci@ {k, k + 1}, {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Aug 18 2016 *)
%t A180662 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 A180662 (Haskell)
%o A180662 import Data.List (inits)
%o A180662 a180662 n k = a180662_tabl !! n !! k
%o A180662 a180662_row n = a180662_tabl !! n
%o A180662 a180662_tabl = tail $ inits a001654_list
%o A180662 -- _Reinhard Zumkeller_, Jun 08 2013
%o A180662 (PARI) T(n,k)=fibonacci(k)*fibonacci(k+1) \\ _Charles R Greathouse IV_, Nov 07 2016
%o A180662 (Magma) [Fibonacci(k)*Fibonacci(k+1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, May 25 2021
%o A180662 (Sage) flatten([[fibonacci(k)*fibonacci(k+1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 25 2021
%Y A180662 Cf. A180663 (Mirror), A001654 (Golden Rectangle), A000045 (F(n)).
%Y A180662 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 A180662 easy,nonn,tabl
%O A180662 0,6
%A A180662 _Johannes W. Meijer_, Sep 21 2010