A115730 -id:A115730 - cp's OEIS frontend

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

Johannes W. Meijer, Sep 21 2010

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.
Row......1....Row1..  a(n) = Sum_{k=0..n} T(n, k).
Row Alt..2....Row2..  a(n) = Sum_{k=0..n} (-1)^(n+k)*T(n, k).
Knight...1....Kn11..  a(n) = Sum_{k=0..floor(n/2)} T(n-k, k).
Knight...1....Kn12..  a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, k+1).
Knight...1....Kn13..  a(n) = Sum_{k=0..floor(n/2)} T(n-k+2, k+2).
Knight...2....Kn21..  a(n) = Sum_{k=0..floor(n/2)} T(n-k, n-2*k).
Knight...2....Kn22..  a(n) = Sum_{k=0..floor(n/2)} T(n-k+1, n-2*k).
Knight...2....Kn23..  a(n) = Sum_{k=0..floor(n/2)} T(n-k+2, n-2*k).
Knight...3....Kn3...  a(n) = Sum_{k=0..n} T(n+k, 2*k).
Knight...4....Kn4...  a(n) = Sum_{k=0..n} T(n+k, n-k).
Fil......1....Fi1...  a(n) = Sum_{k=0..floor(n/2)} T(n, 2*k).
Fil......2....Fi2...  a(n) = Sum_{k=0..floor(n/2)} T(n, n-2*k).
Camel....1....Ca1...  a(n) = Sum_{k=0..floor(n/3)} T(n-2*k, k).
Camel....2....Ca2...  a(n) = Sum_{k=0..floor(n/3)} T(n-2*k, n-3*k).
Camel....3....Ca3...  a(n) = Sum_{k=0..n} T(n+2*k, 3*k).
Camel....4....Ca4...  a(n) = Sum_{k=0..n} T(n+2*k, n-k).
Giraffe..1....Gi1...  a(n) = Sum_{k=0..floor(n/4)} T(n-3*k, k).
Giraffe..2....Gi2...  a(n) = Sum_{k=0..floor(n/4)} T(n-3*k, n-4*k).
Giraffe..3....Gi3...  a(n) = Sum_{k=0..n} T(n+3*k, 4*k).
Giraffe..4....Gi4...  a(n) = Sum_{k=0..n} T(n+3*k, n-k).
Zebra....1....Ze1...  a(n) = Sum_{k=0..floor(n/2)} T(n+k, 3*k).
Zebra....2....Ze2...  a(n) = Sum_{k=0..floor(n/2)} T(n+k, n-2*k).
Zebra....3....Ze3...  a(n) = Sum_{k=0..floor(n/3)} T(n-k, 2*k).
Zebra....4....Ze4...  a(n) = Sum_{k=0..floor(n/3)} T(n-k, n-3*k).

			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;

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

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.

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).

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

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

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

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 *)

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

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

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

A180664 Golden Triangle sums: a(n) = a(n-1) + A001654(n+1) with a(0)=0.

Original entry on oeis.org

0, 2, 8, 23, 63, 167, 440, 1154, 3024, 7919, 20735, 54287, 142128, 372098, 974168, 2550407, 6677055, 17480759, 45765224, 119814914, 313679520, 821223647, 2149991423, 5628750623, 14736260448, 38580030722, 101003831720

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n+1) (terms doubled) are the Kn13 sums of the Golden Triangle A180662. See A180662 for information about these knight and other chess sums.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Cf. A000032, A000045, A005248, A064831, A115730, A180662, A180664, A180665, A180666.

[(1/10)*((-1)^n - 15 + 2*Lucas(2*n+4)): n in [0..40]]; // G. C. Greubel, Jan 21 2022

nmax:=26: with(combinat): for n from 0 to nmax+1 do A001654(n):=fibonacci(n) * fibonacci(n+1) od: a(0):=0: for n from 1 to nmax do a(n) := a(n-1)+A001654(n+1) od: seq(a(n),n=0..nmax);

Table[Sum[Fibonacci[i+2]*Fibonacci[i+3], {i,0,n-1}], {n,0,40}] (* Rigoberto Florez, Jul 07 2020 *)
LinearRecurrence[{3,0,-3,1},{0,2,8,23},30] (* Harvey P. Dale, Mar 30 2023 *)

[(1/10)*((-1)^n - 15 + 2*lucas_number2(2*n+4,1,-1)) for n in (0..40)] # G. C. Greubel, Jan 21 2022

a(n+1) = Sum_{k=0..n} A180662(2*n-k+2, k+2).
a(n) = (-15 + (-1)^n + (6-2*A)*A^(-n-1) + (6-2*B)*B^(-n-1))/10 with A=(3+sqrt(5))/2 and B=(3-sqrt(5))/2.
G.f.: (2*x+2*x^2-x^3)/(1-3*x-x^4+3*x^3).
a(n) = Sum_{i=0..n-1} F(i+2)*F(i+3), where F(i) = A000045(i). - Rigoberto Florez, Jul 07 2020
a(n) = (1/10)*((-1)^n - 15 + 2*Lucas(2*n+4)). - G. C. Greubel, Jan 21 2022

A180665 Golden Triangle sums: a(n)=a(n-2)+A001654(n) with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 2, 7, 17, 47, 121, 320, 835, 2190, 5730, 15006, 39282, 102847, 269252, 704917, 1845491, 4831565, 12649195, 33116030, 86698885, 226980636, 594243012, 1555748412, 4073002212, 10663258237, 27916772486, 73087059235

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n) are the Kn21, Kn22, Kn23, Fi2, and Ze2 sums of the Golden Triangle A180662. Furthermore the a(2*n) are the Kn3, Fi1 (terms doubled) and Ze3 (terms tripled) sums. See A180662 for information about these and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (2, 3, -3, -2, 1).

Cf. A064831, A180664, A180665, A115730, A180666.

nmax:=27: with(combinat): for n from 0 to nmax do A001654(n):=fibonacci(n)*fibonacci(n+1) od: a(0):=0: a(1):=1: for n from 2 to nmax do a(n) := a(n-2) + A001654(n) od: seq(a(n),n=0..nmax);

a(n) = a(n-2)+A001654(n) with a(0)=0 and a(1)=1.
GF(x) = (-x)/((x-1)*(x+1)^2*(x^2-3*x+1)).
a(n) = ((-1)^(-n-1)*(15+10*n)-25+(16-4*A)*A^(-n-1)+(16-4*B)*B^(-n-1))/100 with A=(3+sqrt(5))/2 and B=(3-sqrt(5))/2.

A180666 Golden Triangle sums: a(n)=a(n-4)+A001654(n) with a(0)=0, a(1)=1, a(2)=2 and a(3)=6.

Original entry on oeis.org

0, 1, 2, 6, 15, 41, 106, 279, 729, 1911, 5001, 13095, 34281, 89752, 234971, 615165, 1610520, 4216400, 11038675, 28899630, 75660210, 198081006, 518582802, 1357667406, 3554419410, 9305590831, 24362353076, 63781468404

Offset: 0

Johannes W. Meijer, Sep 21 2010

The a(n) are the Gi2 sums of the Golden Triangle A180662. See A180662 for information about these giraffe and other chess sums.

Index entries for linear recurrences with constant coefficients, signature (2,2,-1,1,-2,-2,1).

Cf. A064831, A180664, A180665, A115730, A180666.

nmax:=27: with(combinat): for n from 0 to nmax do A001654(n):=fibonacci(n)*fibonacci(n+1) od: a(0):=0: a(1):=1: a(2):=2: a(3):=6: for n from 4 to nmax do a(n):=a(n-4)+A001654(n) od: seq(a(n),n=0..nmax);
A180666 := proc(n)
    option remember;
    if n <=3 then
        op(n+1,[0,1,2,6]) ;
    else
        procname(n-4)+A001654(n) ;
    end if;
end proc:
seq(A180666(n),n=0..100 ) ; # R. J. Mathar, Aug 18 2016

Take[Total@{#, PadLeft[Drop[#, -4], Length@ #]}, Length@ # - 4] &@ Table[Times @@ Fibonacci@ {n, n + 1}, {n, 0, 31}] (* or *)
CoefficientList[Series[(-x)/((x^2 - 3 x + 1) (x - 1) (x + 1)^2 (x^2 + 1)), {x, 0, 27}], x] (* Michael De Vlieger, Aug 18 2016 *)

a(n) = a(n-4)+A001654(n) with a(0)=0, a(1)=1, a(2)=2 and a(3)=6.
G.f.: (-x)/((x^2-3*x+1)*(x-1)*(x+1)^2*(x^2+1)).
a(n) = Sum_{k=0..floor(n/4)} A180662(n-3*k,n-4*k).
120*a(n) = 8*A001519(n) -10*A087960(n) -9*(-1)^n -15 -6*(n+1)*(-1)^n. - R. J. Mathar, Aug 18 2016

A180663 Mirror image of the Golden Triangle: T(n,k) = A001654(n-k) for n>=0 and 0<=k<=n.

Original entry on oeis.org

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

Offset: 0

Johannes W. Meijer, Sep 21 2010

This triangle is the mirror image of the Golden Triangle A180662. The terms in the n-th row of the triangle are the first (n+1) golden rectangle numbers in reversed order. The golden rectangle numbers are A001654(n)=F(n)*F(n+1), with F(n) the Fibonacci numbers.

The chess sums, see A180662 for their definitions, mirror those of the Golden Triangle: Row1 & Row1; Row 2 & Row2; Kn1 and Kn2; Kn3 and Kn4; Fi1 and Fi2; Ca1 and Ca2; Ca3 and Ca4; Gi1 and Gi2; Gi3 and Gi4; Ze1 and Ze2; Ze3 and Ze4.

			The first few rows of this triangle are:
0;
1, 0;
2, 1, 0;
6, 2, 1, 0;
15, 6, 2, 1, 0;
40, 15, 6, 2, 1, 0;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Cf. A180662 (Golden Triangle), A001654 (Golden Rectangle numbers), A000045 (F(n)).

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

a180663 n k = a180663_tabl !! n !! k
a180663_row n = a180663_tabl !! n
a180663_tabl = map reverse a180662_tabl
-- Reinhard Zumkeller, Jun 08 2013

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

Module[{nn=20,fb},fb=Times@@@Partition[Fibonacci[Range[0,(nn(nn+1))/2]],2,1];Table[ Reverse[Take[fb,n]],{n,nn}]]//Flatten (* Harvey P. Dale, Jan 30 2023 *)

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

cp's OEIS Frontend

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

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A180664 Golden Triangle sums: a(n) = a(n-1) + A001654(n+1) with a(0)=0.

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A180665 Golden Triangle sums: a(n)=a(n-2)+A001654(n) with a(0)=0 and a(1)=1.

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A180666 Golden Triangle sums: a(n)=a(n-4)+A001654(n) with a(0)=0, a(1)=1, a(2)=2 and a(3)=6.

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A180663 Mirror image of the Golden Triangle: T(n,k) = A001654(n-k) for n>=0 and 0<=k<=n.

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