The Androsova Universal Series Combinatorial and geometric structure

10.5281/zenodo.19002779 We investigate the quadratic sequence an = 3 +n(n + 9)2, n = 0, 1, 2, . . .It is discovered that the numbers 137 and 256 possess unique combinatorialproperties: 137 has exactly one representation as a sum of two,three, and four terms of the series, while 256 has 7 representationsas sums of three terms, no quadruples, and unique representations assums of five, six, seven, and eight terms. A system of bridges — numbersappearing in both the 137 and 256 representations—is identified.A geometric interpretation on the 360° circle and a DNA-like doublehelix structure are constructed. Powers of two behave as hierarchicalthresholds. Gaussian analysis reveals that 59 and 71 are Gaussianprimes, and the Gaussian sums for moduli 137 and 256 equal√137and 16, respectively. The statistical significance of all regularities isp < 10−58, excluding chance. Androsova universal series, number 137, number 256, bridges, DNA-like structure, powers of two, Gaussian primes, Gaussian sums, combinatorics, circle geometry, statistical significance.