login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A160326 Number of ways to express n=0,1,2,... as the sum of two squares and a pentagonal number. 10
1, 3, 3, 1, 2, 5, 4, 1, 1, 5, 6, 2, 1, 5, 5, 2, 4, 6, 5, 1, 3, 6, 5, 3, 1, 8, 8, 4, 2, 4, 8, 4, 5, 1, 4, 5, 4, 10, 6, 6, 5, 8, 6, 1, 3, 6, 6, 4, 6, 4, 7, 8, 8, 8, 5, 7, 4, 4, 6, 5, 6, 8, 7, 4, 8, 8, 6, 5, 4, 7, 7, 8, 7, 7, 8, 8, 8, 7, 3, 4, 12, 4, 4, 7, 3, 13, 12, 12, 5, 2, 12, 4, 5, 6, 6, 8, 10, 8, 3, 5, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

In April 2009, Zhi-Wei Sun conjectured that a(n)>0 for every n=0,1,2,3,... Note that pentagonal numbers are more sparse than squares. The Gauss-Legendre theorem asserts that n is the sum of three squares if and only if it is not of the form 4^a(8b+7) (a,b=0,1,2,...).

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 0..50000

B. K. Oh and Z. W. Sun, Mixed sums of squares and triangular numbers (III), J. Number Theory 129(2009), 964-969.

Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Zhi-Wei Sun, Various new conjectures involving polygonal numbers and primes (a message to Number Theory List), 2009.

Zhi-Wei Sun, Mixed Sums of Primes and Other Terms (a webpage).

Z. W. Sun, On universal sums of polygonal numbers, preprint, arXiv:0905.0635 [math.NT], 2009-2015.

FORMULA

a(n) = |{<x,y,z>: x,y=0,1,2,... & x^2+y^2+(3z^2-z)/2=n}|.

EXAMPLE

For n=5 the a(5)=5 solutions are 0+0+5, 1+4+5, 4+1+5, 0+4+1, 4+0+1.

MATHEMATICA

SQ[x_]:=x>-1&&IntegerPart[Sqrt[x]]^2==x RN[n_]:=Sum[If[SQ[n-y^2-(3z^2-z)/2], 1, 0], {y, 0, Sqrt[n]}, {z, 0, Sqrt[n-y^2]}] Do[Print[n, " ", RN[n]], {n, 0, 50000}]

CROSSREFS

Cf. A000290, A000326, A160324, A160325.

Sequence in context: A089680 A309507 A306690 * A213662 A213657 A215596

Adjacent sequences:  A160323 A160324 A160325 * A160327 A160328 A160329

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, May 08 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 1 05:04 EDT 2020. Contains 333155 sequences. (Running on oeis4.)