Sums Of Triangular Numbers Research Articles

Preservice mathematics teachers’ proving skills in an incorrect statement: Sums of triangular numbers

Preservice mathematics teachers’ proving skills in an incorrect statement: Sums of triangular numbers

Sums of triangular numbers and sums of squares

For non-negative integers a,b, and n, let N(a,b;n) be the number of representations of n as a sum of squares with coefficients 1 or 3 (a of ones and b of threes). Let N⁎(a,b;n) be the number of representations of n as a sum of odd squares with coefficients 1 or 3 (a of ones and b of threes). We have that N⁎(a,b;8n+a+3b) is the number of representations of n as a sum of triangular numbers with coefficients 1 or 3 (a of ones and b of threes). It is known that for a and b satisfying 1≤a+3b≤7, we haveN⁎(a,b;8n+a+3b)=22+(a4)+abN(a,b;8n+a+3b) and for a and b satisfying a+3b=8, we haveN⁎(a,b;8n+a+3b)=22+(a4)+ab(N(a,b;8n+a+3b)−N(a,b;(8n+a+3b)/4)). Such identities are not known for a+3b>8. In this paper, for general a and b with a+b even, we prove asymptotic equivalence of formulas similar to the above, as n→∞. One of our main results extends a theorem of Bateman, Datskovsky, and Knopp where the case b=0 and general a was considered. Our approach is different from Bateman-Datskovsky-Knopp's proof where the circle method and singular series were used. We achieve our results by explicitly computing the Eisenstein components of the generating functions of N⁎(a,b;8n+a+3b) and N(a,b;8n+a+3b). The method we use is robust and can be adapted in studying the asymptotics of other representation numbers with general coefficients.

Automatic proofs of some conjectures of Sun on the relations between sums of squares and sums of triangular numbers

Automatic proofs of some conjectures of Sun on the relations between sums of squares and sums of triangular numbers

Proofs of Some Conjectures of Z. -H. Sun on Relations Between Sums of Squares and Sums of Triangular Numbers

Let N(a, b, c, d; n) be the number of representations of n as ax2+by2+cz2+dw2 and T(a, b, c, d, n) be the number of representations of n as $$a\frac{{X(X + 1)}}{2} + b\frac{{Y(Y + 1)}}{2} + c\frac{{Z(Z + 1)}}{2} + d\frac{{W(W + 1)}}{2}$$ , where a, b, c, d are positive integers, n, X, Y, Z, W are nonnegative integers, and x, y, z, w are integers. Recently, Z.-H. Sun found many relations between N(a, b, c, d, n) and T(a, b, c, d, n) and conjectured 23 more relations. Yao proved five of Sun’s conjectures by using (p, k)-parametrization of theta functions and stated that six more could be proved by using the same method. More recently, Sun himself confirmed two more conjectures by proving a general result whereas Xia and Zhong proved three more conjectures of Sun by employing theta function identities. In this paper, we prove the remaining seven conjectures. Six are proved by employing Ramanujan’s theta function identities and one is proved by elementary techniques.

Ramanujan’s theta function identities and the relations between sums of squares and sums of triangular numbers

For positive integers [Formula: see text], [Formula: see text] and [Formula: see text], let [Formula: see text] denote the number of representations of a nonnegative integer [Formula: see text] as [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are nonnegative integers, and let [Formula: see text] denote the number of representations of [Formula: see text] as [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are integers. Recently, Sun proved a number of relations between [Formula: see text] and [Formula: see text] along with numerous conjectures on such relations. In this work, we confirm several conjectures of Sun by using Ramanujan’s theta function identities.

Proofs of some conjectures of Sun on the relations between sums of squares and sums of triangular numbers

In a recent paper, Sun posed six conjectures on the relations between [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the number of representations of [Formula: see text] as [Formula: see text], where [Formula: see text] are positive integers, [Formula: see text] are arbitrary nonnegative integers, and [Formula: see text] denotes the number of representations of [Formula: see text] as [Formula: see text], where this time [Formula: see text] are integers. In this paper, we prove Sun’s six conjectures by using Ramanujan’s theta function identities.

Binary Complementary Code Structure via a Simple Necessary Condition

A complementary set of $K$ binary $\pm 1$ codes of length $N$ has the useful property that the sum of the autocorrelation sequences of the codes has only one nonvanishing element, its size- $KN$ peak. The $N\times K$ matrix whose columns are the codes of a binary complementary set, arranged in order, is the code set's complementary code matrix (CCM). One might ask for which $K$ and $N$ such matrices exist. A simple necessary condition for the existence of an $N\times K$ CCM is introduced, involving the sum of the squares of the imbalance values of the codes in the set (the imbalance of a binary $\pm 1$ code being the difference between the number of 1 and number of $-1$ s in the code). The one-to-one relationship between binary CCMs and binary complementary code sets allows the necessary condition for the $N\times K$ CCM existence to extend to a necessary condition for sets of $K$ complementary code sets of length $N$ . We examine the sum-of-squares condition for the separate cases of $N$ even and $N$ odd. The odd case is especially interesting, since representations in terms of sums of squares of odd integers are equivalent to representations as sums of triangular numbers, a subject of recent research in Number Theory. Furthermore, we exhibit a refinement of the existence condition in terms of the imbalance values of pairs of interlaced subcodes. This refinement is new, to the best of our knowledge. Finally, we investigate whether odd-length complementary code sets exist for $K>2$ , unlike the Golay case of $K=2$ where odd-length sets cannot exist. We easily find, by computer search, a four-code set of length 15, suggesting that these are likely not uncommon.

On a new transformation formula for bilateral $$_6{\psi }_6$$ 6 ψ 6 series and applications

In this paper, we derive a new transformation formula for bilateral \(_6\psi _6\) series using Roger’s \(_6\phi _5\) summation and then use it to deduce the well-known Bailey’s \(_6\psi _6\) summation formula. Also, we deduce some identities involving theta functions and new identities analogous to identities of Ramanujan. Further, we give formulas for finding the number of representations of an integer as sums of squares and sums of triangular numbers.

Proof Without Words: Sums of Triangular Numbers

Proof Without Words: Sums of Triangular Numbers

The triangular theorem of eight and representation by quadratic polynomials

We investigate here the representability of integers as sums of triangular numbers, where the n n -th triangular number is given by T n = T_n= n ( n + 1 ) / 2 n(n+1)/2 . In particular, we show that f ( x 1 , x 2 , … , x k ) = b 1 T x 1 + ⋯ + b k T x k f(x_1, x_2, \ldots , x_k)=b_1T_{x_1}+\cdots +b_kT_{x_k} , for fixed positive integers b 1 , b 2 , … , b k b_1, b_2, \ldots , b_k , represents every nonnegative integer if and only if it represents 1 1 , 2 2 , 4 4 , 5 5 , and 8 8 . Moreover, if ‘cross-terms’ are allowed in f f , we show that no finite set of positive integers can play an analogous role, in turn showing that there is no overarching finiteness theorem which generalizes the statement from positive definite quadratic forms to totally positive quadratic polynomials.

INDUCTIVE FORMULAS FOR SOME ARITHMETIC FUNCTIONS

We prove recursive formulas involving sums of divisors and sums of triangular numbers and give a variety of identities relating arithmetic functions to divisor functions providing inductive identities for such arithmetic functions.

Binary quadratic forms and sums of triangular numbers

Notation: Z the set of integers, N the set of positive integers, [x] the greatest integer not exceeding x, ( a m) the Legendre-JacobiKronecker symbol, ordpm the nonnegative integer α such that pα | m but pα+1 m, (a, b) the greatest common divisor of a and b, (a, b, c) the form ax2 + bxy + cy2, [a, b, c] the equivalence class containing the form (a, b, c), H(d) the form class group of discriminant d, h(d) the class number of discriminant d, R(K, n) the number of representations of n by the class K.

Representing Sets with Sums of Triangular Numbers

We investigate here sums of triangular numbers where T n is the nth triangular number. We show that for a set of positive integers S, there is a finite subset S 0 such that f represents S if and only if f represents S 0 . However, computationally determining S 0 is ineffective for many choices of S. We give an explicit and efficient algorithm to determine the set S 0 under certain generalized Riemann hypotheses, and implement the algorithm to determine S 0 when S is the set of all odd integers.

An asymptotic for the representation of integers as sums of triangular numbers

An asymptotic for the representation of integers as sums of triangular numbers

Mixed sums of squares and triangular numbers

By means of $q$-series, we prove that any natural number is a sum of an even square and two triangular numbers, and that each positive integer is a sum of a triangular number plus $x^2+y^2$ for some integers $x$ and $y$ with $x\not\equiv y (mod 2)$ or $x=y>0$. The paper also contains some other results and open conjectures on mixed sums of squares and triangular numbers.

Sums of Squares and Sums of Triangular Numbers

Abstract Motivated by two results of Ramanujan, we give a family of 15 results and 4 related ones. Several have interesting interpretations in terms of the number of representations of an integer by a quadratic form , where λ1 + . . . + λ𝑚 = 2, 4 or 8. We also give a new and simple combinatorial proof of the modular equation of order seven.

Sums of triangular numbers from the Frobenius determinant

We show that the denominator formula for the strange series of affine superalgebras, conjectured by Kac and Wakimoto and proved by Zagier, follows from a classical determinant evaluation of Frobenius. As a limit case, we obtain exact formulas for the number of representations of an arbitrary number as a sum of 4 m 2 / d triangles, whenever d | 2 m , and 4 m ( m + 1 ) / d triangles, when d | 2 m or d | 2 m + 2 . This extends recent results of Getz and Mahlburg, Milne, and Zagier.

Proof Without Words: Sums of Triangular Numbers

(2005). Proof Without Words: Sums of Triangular Numbers. Mathematics Magazine: Vol. 78, No. 3, pp. 231-231.

Proof without Words: Sums of Triangular Numbers

Proof without Words: Sums of Triangular Numbers

An Identity of Ramanujan and the Representation of Integers as Sums of Triangular Numbers

Let k be a positive number and t k(n) denote the number of representations of n as a sum of k triangular numbers. In this paper, we will calculate t 2k (n) in the spirit of Ramanujan. We first use the complex theory of elliptic functions to prove a theta function identity. Then from this identity we derive two Lambert series identities, one of them is a well-known identity of Ramanujan. Using a variant form of Ramanujan's identity, we study two classes of Lambert series and derive some theta function identities related to these Lambert series . We calculate t 12(n), t 16(n), t 20(n), t 24(n), and t 28(n) using these Lambert series identities. We also re-derive a recent result of H. H. Chan and K. S. Chua [6] about t 32(n). In addition, we derive some identities involving the Ramanujan function τ(n), the divisor function σ11(n), and t 24(n). Our methods do not depend upon the theory of modular forms and are somewhat more transparent.