Unnecessary Triangles

[Jon Perry]

Interesting problem. Your non-essential sequence is limited by taking only
integers to 1000 - integers around 1000 are only using triangular numbers of
around 40.

Your post made me think of this extension - which triangular numbers are the
sums of consecutive triangular numbers. Testing to n=59000 gives several
lengths of consecutive numbers which never equal a triangular number, i.e.
4,6,12,14,15 and 19 (tested to 20). A consecutive length of 9 yields only
one result, S(29)=S(5)+S(6)+S(7)+...+S(13).

The rest do yield results, with a strange multiplicative connection
throughout. For example, at a depth of 2, the next result will be around
331940 (multiply by 5.8ish). The depth 3 sequence has a factor of 3.9ish.
Depth 5 uses an alternating sequence of 7ish, 2ish, etc...

A final point is that all the numbers in the depth 2 sequence are powerful.

The full data is here, followed by the Delphi code I used:

---------------------------------------

depth 2
8:36(5)
49:1225(34)
288:41616(203)
1681:1413721(1188)
9800:48024900(6929)
57121:1631432881(40390)
depth 3
4:10(1)
16:136(8)
61:1891(34)
229:26335(131)
856:366796(493)
3196:5108806(1844)
11929:71156485(6886)
44521:991081981(25703)
depth 4
depth 5
10:55(2)
70:2485(29)
184:17020(80)
1264:799480(563)
3310:5479705(1478)
22690:257429395(10145)
depth 6
depth 7
20:210(4)
83:3486(28)
321:51681(118)
1329:883785(499)
5123:13125126(1933)
21188:224476266(8005)
depth 8
15:120(1)
23:276(4)
48:1176(13)
240:28920(81)
503:126756(174)
783:306936(273)
1640:1345620(576)
8168:33362196(2884)
17103:146264856(6043)
26615:354192420(9406)
55728:1552832856(19699)
depth 9
29:435(5)
depth 10
54:1485(12)
120:7260(33)
240:28920(71)
534:142845(164)
2055:2112540(645)
4569:10440165(1440)
9129:41673885(2882)
20295:205953660(6413)
depth 11
65:2145(14)
88:3916(21)
121:7381(31)
164:13530(44)
1298:843051(386)
1759:1547920(525)
2419:2926990(724)
3278:5374281(983)
25904:335521560(7805)
35101:616057651(10578)
48268:1164924046(14548)
depth 12
depth 13
64:2080(11)
910:414505(246)
7591:28815436(2099)
9931:49317346(2748)
depth 14
depth 15
depth 16
336:56616(76)
depth 17
51:1326(3)
135:9180(24)
221:24531(45)
1989:1979055(474)
3263:5325216(783)
8891:39529386(2148)
14585:106368405(3529)
depth 18
158:12561(28)
255:32640(51)
543:147696(119)
878:385881(198)
5358:14356761(1254)
8663:37528116(2033)
18455:170302740(4341)
29838:445168041(7024)
depth 19
depth 20
55:1540(1)
59:1770(2)
84:3570(8)
644:207690(134)
939:441330(200)
1015:515620(217)
1480:1095940(321)
11560:66822580(2575)
16855:142053940(3759)
18219:165975090(4064)
26564:352836330(5930)

---------------------------------------

procedure TForm1.Button1Click(Sender: TObject);
var tri:array[0..60000] of integer;
i,j,ic,tj,k:integer;
begin
tri[1]:=1;
for i:=2 to 60000 do tri[i]:=tri[i-1]+i;

for i:=2 to 50 do begin

memo1.lines.add('depth '+inttostr(i));

ic:=1;
for j:=1 to 59000 do begin
tj:=0;
for k:=0 to i-1 do inc(tj,tri[j+k]);
while ((tri[ic]<=tj) and (ic<59000)) do begin
if (tj=tri[ic]) then
memo1.lines.add(inttostr(ic)+':'+inttostr(tri[ic])+'('+inttostr(j)+')');
inc(ic);
end;
end;

end;

statusbar1.simpletext:='finished';
end;

-------------------------------------

Jon Perry
perry at globalnet.co.uk
http://www.users.globalnet.co.uk/~perry
http://www.users.globalnet.co.uk/~perry/maths
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com


-----Original Message-----
From: Ed Pegg [mailto:edp at wolfram.com]
Sent: 14 March 2002 23:30
To: seqfan at ext.jussieu.fr
Subject: Unnecessary Triangles


 From my mathpuzzle.com page.

  I pondered a possible game this weekend, where cards or dominoes would
represent <snip>