The 2-adic valuation of the Newman–Shanks sequence

We study the sequence of integers analysed by M. Newman and D. Shanks in 1984 in connection with some rapidly converging series for pi, listed in the OEIS as A005148. We prove that the highest power of 2 dividing each term is a power of 8 whose exponent is the number of ones in the binary expansion of the index minus one, as conjectured by B. Cloitre in 2002 in the OEIS entry of the sequence. As particular cases we obtain the theorem of Newman and Shanks stating that a term is odd if and only if its index is a power of 2, and the fact that the terms of sequence A071644 are odd integers, each being the smallest integer of its form, as stated in its OEIS entry. The proof proceeds by induction from a convolution identity with the cubes of the central binomial coefficients, due to Shanks and Beukers, together with Kummer’s theorem on the 2-adic valuation of binomial coefficients: the terms of minimal valuation in the convolution are exactly those whose indices have their binary ones contained in those of the given index, and their number is even. A Python program that computes the sequence in two independent ways and checks the result up to index 600 is included, together with its output. Version 2: the text of the note is unchanged; this version adds the verification program and its output.