On Numerical Triangles Generated by 2x + 1 and 3x + 2

On Numerical Triangles Generated by 2x+ 1 and 3x+ 2 Miguel Cerda Bennassar October 26, 2025 Abstract This paper introduces and studies a family of numerical triangles T(a) generated by two fundamental linear transformations: f(x) = 2x+ 1 (intra-column generation) and g(x) = 3x+ 2 (inter-row jump). We establish that the non-redundant triangles—those not repeating rows from other triangles—correspond exactly to initial values a≡0,4 (mod 6). Furthermore, we prove that the union of all these triangles exhaustively covers the set N0of non-negative integers, with each number appearing in a unique position within the family. Among our main results are: a closed formula for row endpoints, a unique localization algorithm, an absorption theorem showing convergence to the cycle 2 ↔1 in T(0), and the structural equivalence of all non-redundant triangles under affine normalization. Contents 1 Introduction 3 2 Definition and construction of the family T(a)3 3 Example: the triangle T(0) 4 4 Fundamental properties of triangle T(0) 4 5 Characteristic examples: T(4) and T(6) 6 5.1 Definition ................................... 6 5.2 Example: triangle T(4)............................ 6 5.3 Example: triangle T(6)............................ 7 6 Structural comparison 7 7 General structure of the family T(a)7 7.1 General formula for row endpoints . . . . . . . . . . . . . . . . . . . . . . 8 7.2 Modular classification of a.......................... 8 8 Equivalence and normalization 11 1 A Internal formulas of the table: entries a(a) n,j in T(a)11 B Inverse problem: localization of a number 11 C Practical recipe 12 D Table of symbols 12 2 1 Introduction Discrete dynamical systems generated by simple affine transformations have been a continuous subject of study in number theory and dynamical systems, notable for their ability to generate complex structures from elementary rules. Among these, the celebrated Collatz conjecture—involving the operations 3x+ 1 for odd numbers and x/2 for even ones—represents perhaps the best-known example, but many variants and related systems remain unexplored. In this work, we introduce and study a family of numerical triangles T(a) generated by two fundamental linear transformations: f(x) = 2x+ 1 (intra-column generation) g(x) = 3x+ 2 (inter-row jump) The construction proceeds recursively: each row is obtained by applying fto the elements of the previous row, while the last term of each row is calculated by applying g to the last odd element of the preceding row. This seemingly simple mechanism generates a rich triangular structure with notable algebraic and combinatorial properties. We demonstrate that the non-redundant triangles—those not repeating rows from other triangles—correspond exactly to initial values a≡0,4 (mod 6). Furthermore, we establish that the union of all these triangles exhaustively covers the set N0of non-negative integers, with each number appearing in a unique position within the family. Among our main results are: •A closed formula for row endpoints: L(a) n= (a+ 1)3n−1−1 •A unique localization algorithm that determines the position (a, n, j) of any natural number •An absorption theorem showing how every trajectory in the triangle network converges to the cycle 2 ↔1inT(0) •The structural equivalence of all non-redundant triangles under affine normalization This work not only develops the complete theory of these numerical triangles but also suggests deep connections with classical problems in number theory and dynamical systems, offering a unified framework for studying numerical generation patterns through simple linear transformations. 2 Definition and construction of the family T(a) The idea is to construct a numerical triangle T(n) from two very simple linear transformations: f(x) = 2x+ 1 (generation within the same column) g(x) = 3x+ 2 (jump between rows/columns) The triangle is organized in columns Ckand rows Fm: - Column C1starts at row F1;C2at F2;C3at F3; etc. - The first column begins with a base value (for example, 0 in T(0)). Iterating 2x+ 1 yields the subsequent terms of that same column. 3 - The first number of the next column (and of the next row) is calculated with 3x+ 2 from the last number of the previous row. This mechanism completely determines the triangle from the initial value of C1. Not all initial values produce distinct triangles: if the first value of C1is 0, 4, 6, 10, 12, 16, 18, 22, 24, . . . , that is, if n≡0,4 (mod 6), we obtain a non-redundant triangle; other values reproduce rows already present in one of these triangles. Corollary 1 (Complete coverage of N0).The union of all non-redundant triangles T(a) with a≡0,4(mod 6) contains all non-negative integers. •Every odd number Nappears uniquely as N=a(a) n,j = (a+ 1)2n−j3j−1−1 for a unique triple (a, n, j)with a≡0,4(mod 6). •Every even number Eappears in one of the following forms: –if E≡0or 4(mod 6), as initial value ain some triangle T(a); –if E≡2(mod 6), as last term L(a) nof a row in some triangle. Consequently, [ a≡0,4 (mod 6) T(a) = N0. 3 Example: the triangle T(0) Below we show the first rows. In each row, on the left we indicate the starting equality (where the first term comes from), and on the right the terms that complete the row. Row 1: 0 0 Row 2: 2·0 + 1 = 1 1 2 Row 3: 2·1 + 1 = 3 3 5 8 Row 4: 2·3 + 1 = 7 7 11 17 26 Row 5: 2·7 + 1 = 15 15 23 35 53 80 Row 6: 2·15 + 1 = 31 31 47 71 107 161 242 Row 7: 2·31 + 1 = 63 63 95 143 215 323 485 728 Row 8: 2·63 + 1 = 127 127 191 287 431 647 971 1457 2186 Row 9: 2·127 + 1 = 255 255 383 575 863 1295 1943 2915 4373 6560 4 Fundamental properties of triangle T(0) In this section we collect properties that help understand the structure of triangle T(0) without resorting to heavy formalism. We denote by Lnthe last term of row n. 4 Proposition 2 (Recurrence and closed formula of the last term).From the construction it follows that Ln= 3 Ln−1+ 2, L1= 0, and therefore Ln= 3n−1−1 (n≥1) . Proof idea. The last term of row nis obtained by closing the row with 3(·)+2 applied to the previous last odd; solving the affine recurrence xn= 3xn−1+ 2 with condition x1= 0 gives xn+ 1 = 3n−1. Corollary 3 (Modular class of the last term).For n≥2, Ln≡2 (mod 6). Proof idea. Since Ln= 3n−1−1 and 3m≡3 (mod 6) for m≥1, we obtain Ln≡3−1=2 (mod 6). Proposition 4 (Parity and mod 4 pattern in each row).In row n, all terms except the last are odd; the last is even. Moreover, the penultimate is of the form 4k+ 1 and the terms before the penultimate are of the form 4k+ 3. Proof idea. 2x+1 preserves oddness when iterating within the row; the closure 3(·)+2 applied to an odd produces the final even. The 1/3 mod 4 pattern is verified by congruence inspection when backtracking from the last odd. Observation 1 (Row length).Row ncontains exactly nnumerical terms (counting the last Ln). Proof idea. Each new row adds a column to the active front of the triangle, increasing the number of listed terms by one. Proposition 5 (Unreachability of row endpoint by divisions in powers of 2).Let Lnbe the last term of row nin T(0). Then, for all k≥1, Ln 2k/∈Fn. Proof idea. The terms of row nhave the form {Hn, f(Hn), f[2](Hn), . . . , Ln}, where Hn is the first odd and f(x) = 2x+ 1. By construction, Lnis the only even in the row and Ln= 2f[n−2](Hn)+1. Dividing Lnby 2kwith k≥1 can only yield: either an odd number smaller than Hn, or an even number not belonging to the internal sequence generated by f. In neither case do we obtain any of the nelements of Fn. Corollary 6 (Multiplicative isolation of the row endpoint).The endpoint Lnof each row is unique in its multiplicative class by powers of 2: there is no jsuch that Ln/2k=an,j for some k≥1. Observation 2 (Growth and dominant bound).The maximum of row nis Ln= 3n−1−1; therefore, any aggregate sum or magnitude of the row is asymptotically dominated by 3n. Observation 3 (Digital roots by columns).The following pattern is observed: •C1: digital roots in cycle (1,3,7,6,4,9) 5 •C2: digital roots alternate (2,5) •Ck(k≥3): constant digital root 8 Proof idea. The digital root respects simple linear combinations: rd(2x+1) = rd(2 rd(x)+ 1). When iterating 2x+ 1 in each column, stable cycles appear: for k≥3, the steady state is 8. Proposition 7 (Non-redundant initial values).The new triangles (not duplicated by rows of others) appear when the first value of C1satisfies n≡0,4 (mod 6), which generates the sequence 0,4,6,10,12,16,18,22,24, . . .. Observation 4 (Linking between rows and columns).If Hnis the first odd listed in row n, then within the row we iterate by 2x+ 1, and the closure jump is achieved by applying 3x+ 2 to the last odd to obtain Lnand prepare the start of the next column. This view clarifies the complementary role of 2x+ 1 (internal progression) and 3x+ 2 (closure and jump). 5 Characteristic examples: T(4) and T(6) In the previous section we analyzed in detail the structure of triangle T(0). Now we consider the next two non-redundant triangles, T(4) and T(6), to observe which properties are preserved and which are modified when varying the initial value of the first column. 5.1 Definition The construction procedure is exactly the same: f(x) = 2x+1 (generation within column), g(x) = 3x+2 (jump between rows/columns). The only difference is that the first value of C1is now 4 or 6 instead of 0. 5.2 Example: triangle T(4) Row 1: 4 4 Row 2: 2·4 + 1 = 9 9 14 Row 3: 2·9 + 1 = 19 19 29 44 Row 4: 2·19 + 1 = 39 39 59 89 134 Row 5: 2·39 + 1 = 79 79 119 179 269 404 Row 6: 2·79 + 1 = 159 159 239 359 539 809 1214 Row 7: 2·159 + 1 = 319 319 479 719 1079 1619 2429 3644 Row 8: 2·319 + 1 = 639 639 959 1439 2159 3239 4859 7289 10934 6 5.3 Example: triangle T(6) Row 1: 6 6 Row 2: 2·6 + 1 = 13 13 20 Row 3: 2·13 + 1 = 27 27 41 62 Row 4: 2·27 + 1 = 55 55 83 125 188 Row 5: 2·55 + 1 = 111 111 167 251 377 566 Row 6: 2·111 + 1 = 223 223 335 503 755 1133 1700 Row 7: 2·223 + 1 = 447 447 671 1007 1511 2267 3401 5102 Row 8: 2·447 + 1 = 895 895 1343 2015 3023 4547 6839 10259 15386 6 Structural comparison Proposition 8 (Relative growth).The three triangles grow exponentially at rate 3n−1, but T(4) and T(6) are shifted relative to T(0) by a constant factor: L(4) n+1 L(0) n+1 = 5 and L(6) n+1 L(0) n+1 = 7. This shift is exact for all rows. Observation 5 (Digital roots).In T(0), the digital roots of C1form a cycle (1,3,7,6,4,9). In T(4), the digital roots of C1start from 5 and follow the same shifted cycle. In T(6), the digital roots of C1start from 7 and also follow the cycle. From C3onward, the digital root stabilizes again at 8 in all three cases. Observation 6 (Mod 4 pattern and parity).In T(4) and T(6) the same modular structure is maintained: all terms except the last are odd, with pattern 4k+ 3 in the interior and 4k+ 1 in the penultimate; the last is always even. Observation 7 (Invariant row length).For every triangle T(a)with a≡0,4 (mod 6), row ncontains nnumerical terms. Proposition 9 (Structural equivalence).The triangles T(0),T(4), and T(6) are structurally equivalent: their column-row trajectories behave in the same way except for a translation factor in the numerical domain. Observation 8 (Final remark).The family of triangles T(a)with a≡0,4 (mod 6) can be seen as the same dynamic structure shifted. This property is useful for studying general behaviors (parity, digital roots, growth rates, modular patterns) without needing to repeat all proofs for each initial case. 7 General structure of the family T(a) In the previous sections we have analyzed the triangles T(0), T(4), and T(6), which represent the non-redundant initial cases of the family. Below we present a unified summary of the general properties of triangles T(a), covering all initial values congruent with 0 or 4 (mod 6). 7 7.1 General formula for row endpoints Observation 9 (Last terms and closed formula).Let L(a) nbe the last term of row nin triangle T(a). The recurrence holds: L(a) n= 3 L(a) n−1+ 2, L(a) 1=a, whose closed solution is L(a) n= (a+ 1) 3n−1−1 (n≥1) . This expression shows that all triangles T(a)are structurally equivalent to the base model T(0), except for a shift factor a+ 1. Observation 10 (Normalization).The transformation Na:x7−→ x+ 1 a+ 1 sends the last terms of T(a)to those of T(0): Na(L(a) n) = 3n−1. Therefore, the row endpoints of the entire family {T(a)}can be seen as scaled copies of the base case. 7.2 Modular classification of a Observation 11 (Non-redundant values).The new triangles (not duplicated by rows of others) appear when the first value of C1satisfies n≡0,4 (mod 6), which generates the sequence 0,4,6,10,12,16,18,22,24, . . .. Observation 12 (General network and infinite chains of triangles).The union of triangles T(a)with a≡0,4 (mod 6) forms an infinite network in which each node represents a triangle and each edge corresponds to an odd number of class 0 or 4 (mod 6) that serves as the start of another triangle. Although the even endpoint of each row does not generate new connections (except in row 1), the intermediate odd numbers can do so, so it is possible to construct infinite chains T(a0)−→ T(a1)−→ T(a2)−→ · · · by successively choosing new nodes. Observation 13 (Comment).The existence of infinite chains does not imply uniqueness: the same trajectory can branch at different points, and cycles can also appear if repeated numbers are allowed in different trajectories. The resulting network is infinite and not acyclic. Definition 1 (Chaining of triangles).Given a row nof T(a)with even last term L(a) n, we define the deterministic chaining as M=L(a) n 2ν2(L(a) n), which is odd. By the inverse problem, Mbelongs uniquely to another row (possibly in another triangle T(b)). We say that row nof T(a)chains with the row containing Min T(b). 8 Proposition 10 (Triangle network and deterministic connection).The union of triangles T(a)with a≡0,4 (mod 6) forms an infinite network in which each edge corresponds to the chaining between rows: given a row of T(a)with endpoint L(a) n, repeated division by 2 produces an odd Mthat belongs uniquely to another row, possibly in a different triangle T(b). (a) (General network). It is possible to construct infinite chains T(a0)→T(a1)→T(a2)→ · · · by freely choosing odd numbers of class 0or 4 (mod 6) that serve as starts of new triangles; cycles can also occur if a start already visited is reused. (b) (Deterministic connection by row endpoint). If L(a) n= (a+ 1)3n−1−1and k= ν2(L(a) n), then M=L(a) n/2kis odd and belongs uniquely to a non-redundant triangle T(b), determined by M+ 1 = 2m3rqwith gcd(q, 6) = 1 and b=q−1. This defines a deterministic mapping T(a)→T(b). Observation 14 (Local structure of deterministic dynamics).The rule L(a) n7→ L(a) n 2ν2(L(a) n) presents the following elementary properties: •There are no intra-row cycles, except in T(0), row 2. •There are no cycles between different triangles. •The only possible cycle is the fixed loop in T(0), row 2. •Each deterministic chain is infinite, unless a row endpoint that is a power of 2 appears. Observation 15 (Unique intra-row cycle).The only exception to the impossibility of intra-row cycles occurs in T(0), row 2:L(0) 2= 2 and ν2(2) = 1, so that L(0) 2/2 = 1 belongs to the same row. The deterministic dynamics T(0) 7→ T(0) thus forms a trivial loop. No other cases with this property exist for n>2or for other T(a). Proposition 11 (Entry gates to the T(0) trap).If in the deterministic dynamics an endpoint appears of the form L=2 3(4n−1) (n≥1), then the associated odd is M=L 2=4n−1 3, which belongs to row 2of the triangle The last term of that row is L(a) 2= 22n−1, a power of 2, so that the trajectory enters the unique cycle of T(0) (via 1). Consequently, every deterministic chaining of triangles is infinite until the appearance of an entry gate of this type, at which point the trajectory becomes trapped in T(0), row 2. The deterministic dynamics possesses an absorbing structure and a unique global cycle, as established by the results presented below. 9