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TWEET  # Triangular numbers in Mathematica Not-a-tree  Jul 19th, 2017 (edited) 60 Never
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1. Some functions for triangular numbers in Mathematica, sorted by byte count
2.
3. 15: ((2#+1)^2-1)/8&
4. 17: Tr[Range@#^3]^.5&
5. 17: ⌊(2#+1)^2/8⌋&
6. 18: ((#+1)^3-#^3-1)/6&
7. 19: Sqrt@Tr[Range@#^3]&
8. 19: -StirlingS1[#+1,#]&
9. 19: Sum[1,{i,#},{j,i}]&
10. 20: f@0=0;f@i_:=i+f[i-1]
11. 20: Tr[{#+1,-#,-1}^3/6]&
12. 22: ⌊(#+1)^3/(#+2)⌋/2&
13. 23: Tr@Table[i^3,{i,#}]^.5&
14. 26: Sqrt@Sum[i*j,{i,#},{j,#}]&
15. 27: Sum[i^2(-1)^i,{i,#}](-1)^#&
16. 28: D[x/(1-x)^3,{x,#}]/#!/.x->0&
17. 28: (For[i=n=0,i<#,n+=i,i++];n)&
18. 29: n=0;(Do[n+=i,{i,#}];Print@n)&
19. 29: EdgeCount@CompleteGraph[#+1]&
20. 29: Length@Expand[(x+y+z)^(#-1)]&   (note: doesn't work for 1)
21. 29: Coefficient[(1+x)^(#+1),x^2]&
22. 29: ⌊(#+1)/(Exp[2/(#+1)]-1)⌋&
23. 30: Nest[#+(1+Sqrt[1+8#])/2&,0,#]&
24. 31: Sum[EulerPhi@k⌊#/k⌋,{k,#}]&
25. 33: Series[x/(1-x)^3,{x,0,#}][[3,#]]&
26. 34: Boole[#2>=#]&~Array~{#,#}~Total~2&  (thanks to Mark S., https://codegolf.stackexchange.com/questions/133109/sum-of-all-integers-from-1-to-n/133160#comment339998_133160)
27. 37: SeriesCoefficient[x/(1-x)^3,{x,0,#}]&
28. 37: Det@Array[Binomial[1+##,#]&,{#,#}-1]&   (note: doesn't work for 1)
29. 39: Length@Solve[0<a<=b<=#,{a,b},Integers]&
30. 40: UpperTriangularize@Table[1,#,#]~Total~2&
31. 42: Last@LinearRecurrence[{3,-3,1},{1,3,6},#]&
32. 42: CellularAutomaton[50,{{1},0},#-1]~Total~2&
33. 45: 1/Integrate[4Sin@x^(2#-1)Cos@x^3,{x,0,Pi/2}]&
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