General relation between sums of figurate numbers
In this study, we seek to find relations between the number of representations of a nonnegative integer n as a sum of figurate numbers of different types. Firstly, we give a relation between the number of representations, ck(n), of n as the sum k cubes and the number of representations, pk(n), of...
| Main Author: | |
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| Format: | Thesis |
| Language: | English |
| Published: |
2013
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| Subjects: | |
| Online Access: | http://psasir.upm.edu.my/id/eprint/67683/1/IPM%202013%209%20IR.pdf |
| _version_ | 1796978563534028800 |
|---|---|
| author | Mohamat Johari, Mohamat Aidil |
| author_facet | Mohamat Johari, Mohamat Aidil |
| author_sort | Mohamat Johari, Mohamat Aidil |
| collection | UPM |
| description | In this study, we seek to find relations between the number of representations of a
nonnegative integer n as a sum of figurate numbers of different types.
Firstly, we give a relation between the number of representations, ck(n), of n as
the sum k cubes and the number of representations, pk(n), of n as the sum of k
triangular pyramidal numbers, namely under certain conditions
pk(n) = c k odd (v);
where c k odd denotes the number of representations as a sum of k odd cubes and the
integer v is derived from n. Then we extend this problem by considering sums of
s-th powers with s > 3 and the associated polytopic numbers of order s.
Next, we discuss the relation between ɸ(2;k)(n), the number of representations of
n as a sum of k fourth powers, and ψ(2;k)(n), the number of representations of n as a sum of k terms of the form 8γ2 + 2γ where γ is a triangular number. When
1 ≤ k ≤ 7, the relation is
ɸ(2;k)(8n + k) = 2kψ (2;k) (n).
We extend this result by considering the relation between the number of represen-
tations of n as a sum of k 2m-th powers and the number of representations of n as
a sum of k terms determined by an associated polynomial of degree m evaluated
at a triangular number.
Thirdly, we consider the relation between sk(n), the number of representations of
n as a sum of k squares, and ek(n), the number of representations of n as a sum
of k centred pentagonal numbers. When 1 ≤k ≤ 7, this relation is
αkek(n) = sk (8n -3k)÷5 ; where αk = 2k + 2k-1 (k4)
We extend the analysis to the number of representations induced by a partition γ
of k into m parts. If corresponding number of representations of n are respectively
sγ(n) and eγ(n), then
βγeγ(n) = sγ(8n - 3k)÷5
where
βγ = 2m + 2(m-1) (( i1/4) + (i1/2)(i2/1)+(i1/1)(i3/1)
and ij denotes the number of parts of γ which are equal to j.
We end this thesis with a short discussion and proposal of various open problems
for further research. |
| first_indexed | 2024-03-06T09:56:22Z |
| format | Thesis |
| id | upm.eprints-67683 |
| institution | Universiti Putra Malaysia |
| language | English |
| last_indexed | 2024-03-06T09:56:22Z |
| publishDate | 2013 |
| record_format | dspace |
| spelling | upm.eprints-676832019-03-26T04:14:11Z http://psasir.upm.edu.my/id/eprint/67683/ General relation between sums of figurate numbers Mohamat Johari, Mohamat Aidil In this study, we seek to find relations between the number of representations of a nonnegative integer n as a sum of figurate numbers of different types. Firstly, we give a relation between the number of representations, ck(n), of n as the sum k cubes and the number of representations, pk(n), of n as the sum of k triangular pyramidal numbers, namely under certain conditions pk(n) = c k odd (v); where c k odd denotes the number of representations as a sum of k odd cubes and the integer v is derived from n. Then we extend this problem by considering sums of s-th powers with s > 3 and the associated polytopic numbers of order s. Next, we discuss the relation between ɸ(2;k)(n), the number of representations of n as a sum of k fourth powers, and ψ(2;k)(n), the number of representations of n as a sum of k terms of the form 8γ2 + 2γ where γ is a triangular number. When 1 ≤ k ≤ 7, the relation is ɸ(2;k)(8n + k) = 2kψ (2;k) (n). We extend this result by considering the relation between the number of represen- tations of n as a sum of k 2m-th powers and the number of representations of n as a sum of k terms determined by an associated polynomial of degree m evaluated at a triangular number. Thirdly, we consider the relation between sk(n), the number of representations of n as a sum of k squares, and ek(n), the number of representations of n as a sum of k centred pentagonal numbers. When 1 ≤k ≤ 7, this relation is αkek(n) = sk (8n -3k)÷5 ; where αk = 2k + 2k-1 (k4) We extend the analysis to the number of representations induced by a partition γ of k into m parts. If corresponding number of representations of n are respectively sγ(n) and eγ(n), then βγeγ(n) = sγ(8n - 3k)÷5 where βγ = 2m + 2(m-1) (( i1/4) + (i1/2)(i2/1)+(i1/1)(i3/1) and ij denotes the number of parts of γ which are equal to j. We end this thesis with a short discussion and proposal of various open problems for further research. 2013-04 Thesis NonPeerReviewed text en http://psasir.upm.edu.my/id/eprint/67683/1/IPM%202013%209%20IR.pdf Mohamat Johari, Mohamat Aidil (2013) General relation between sums of figurate numbers. Doctoral thesis, Universiti Putra Malaysia. Mathematics Numbers, Polygonal |
| spellingShingle | Mathematics Numbers, Polygonal Mohamat Johari, Mohamat Aidil General relation between sums of figurate numbers |
| title | General relation between sums of figurate numbers |
| title_full | General relation between sums of figurate numbers |
| title_fullStr | General relation between sums of figurate numbers |
| title_full_unstemmed | General relation between sums of figurate numbers |
| title_short | General relation between sums of figurate numbers |
| title_sort | general relation between sums of figurate numbers |
| topic | Mathematics Numbers, Polygonal |
| url | http://psasir.upm.edu.my/id/eprint/67683/1/IPM%202013%209%20IR.pdf |
| work_keys_str_mv | AT mohamatjoharimohamataidil generalrelationbetweensumsoffiguratenumbers |