A165513 Trapezoidal numbers.

5, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset

1,1

Comments

Trapezoidal numbers are polite numbers (A138591) that have a runsum representation which excludes one, and hence that can be depicted graphically by a trapezoid. Jones and Lord have shown that this is the sequence of integers excluding the powers of 2, the perfect numbers and integers of the form 2^(k-1)*(2^k+1) where k is necessarily a power of 2 and 2^k+1 is a Fermat prime (A019434).

References

Smith, Jim: Trapezoidal numbers, Mathematics in School (November 1997).

Links

Peter Kagey, Table of n, a(n) for n = 1..10000

Chris Jones and Nick Lord, Characterizing Non-Trapezoidal Numbers, The Mathematical Gazette, Vol. 83, No. 497, July 1999, pp. 262-263.

Ron Knott, Introducing runsums

Melvyn B. Nathanson, Trapezoidal numbers, divisor functions, and a partition theorem of Sylvester, arXiv:1601.07058 [math.NT], 2016.

T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Example

As 12=3+4+5 is the fifth integer with a runsum representation which excludes one, then a(5)=12.

Mathematica

Trapezoidal[n_]:=Module[{result}, result={}; Do[sum=0; start=i; lis={}; m=i; While[sum<n, sum=sum+m; lis=AppendTo[lis, m]; If[sum==n, AppendTo[result, lis]]; m++ ], {i, 2, Floor[n/2]}]; result]; Select[Range[100], Length[Trapezoidal[ # ]]>0 &]

Crossrefs

Cf. A138591, A165514, A019434.

Sequence in context: A212191 A336122 A241853 * A002342 A080353 A184108

Adjacent sequences: A165510 A165511 A165512 * A165514 A165515 A165516

Keyword

easy,nonn

Author

Ant King, Sep 23 2009

Status

approved