This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A346872 #43 Mar 22 2025 04:39:58 %S A346872 1,2,3,1,5,2,1,9,3,2,1,1,17,6,3,2,2,1,1,33,11,6,4,2,2,2,1,2,1,65,22, %T A346872 11,7,5,3,3,2,2,2,1,2,1,1,1,129,43,22,13,9,7,5,4,3,3,3,2,2,1,2,1,2,1, %U A346872 1,1,1,1,257,86,43,26,18,12,10,8,6,5,4,4,3,3,3,2,3 %N A346872 Irregular triangle read by rows in which row n lists the row 2^(n-1) of A237591, n >= 1. %C A346872 The characteristic shape of the symmetric representation of sigma(2^(n-1)) consists in that the diagram contains exactly one region (or part) and that region has width 1. %C A346872 So knowing this characteristic shape we can know if a number is power of 2 or not just by looking at the diagram, even ignoring the concept of power of 2. %C A346872 Therefore we can see a geometric pattern of the distribution of the powers of 2 in the stepped pyramid described in A245092. %C A346872 For the definition of "width" see A249351. %C A346872 T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(2^(n-1)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000079(n-1). %C A346872 T(n,k) is also the difference between the total number of partitions of all positive integers <= 2^(n-1) into exactly k consecutive parts, and the total number of partitions of all positive integers <= 2^(n-1) into exactly k + 1 consecutive parts. %e A346872 Triangle begins: %e A346872 1; %e A346872 2; %e A346872 3, 1; %e A346872 5, 2, 1; %e A346872 9, 3, 2, 1, 1; %e A346872 17, 6, 3, 2, 2, 1, 1; %e A346872 33, 11, 6, 4, 2, 2, 2, 1, 2, 1; %e A346872 65, 22, 11, 7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1; %e A346872 129, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1; %e A346872 ... %e A346872 Illustration of initial terms: %e A346872 . %e A346872 Row 1: _ %e A346872 |_| Semilength = 1 %e A346872 1 %e A346872 Row 2: _ %e A346872 _| | %e A346872 |_ _| %e A346872 2 Semilength = 2 %e A346872 . %e A346872 Row 3: _ %e A346872 | | %e A346872 _| | %e A346872 _ _| _| %e A346872 |_ _ _|1 Semilength = 4 %e A346872 3 %e A346872 . %e A346872 Row 4: _ %e A346872 | | %e A346872 | | %e A346872 | | %e A346872 _ _| | %e A346872 _| _ _| %e A346872 | _| %e A346872 _ _ _ _| | 1 Semilength = 8 %e A346872 |_ _ _ _ _|2 %e A346872 5 %e A346872 . %e A346872 Row 5: _ %e A346872 | | %e A346872 | | %e A346872 | | %e A346872 | | %e A346872 | | %e A346872 | | %e A346872 | | %e A346872 _ _ _| | %e A346872 | _ _ _| %e A346872 _| | %e A346872 _| _| %e A346872 _ _| _| Semilength = 16 %e A346872 | _ _|1 1 %e A346872 | | 2 %e A346872 _ _ _ _ _ _ _ _| |3 %e A346872 |_ _ _ _ _ _ _ _ _| %e A346872 9 %e A346872 . %e A346872 The area (also the number of cells) of the successive diagrams gives the nonzero Mersenne numbers A000225. %Y A346872 Row sums give A000079. %Y A346872 Column 1 gives A094373. %Y A346872 For the characteristic shape of sigma(A000040(n)) see A346871. %Y A346872 For the characteristic shape of sigma(A000217(n)) see A346873. %Y A346872 For the visualization of Mersenne numbers A000225 see A346874. %Y A346872 For the characteristic shape of sigma(A000384(n)) see A346875. %Y A346872 For the characteristic shape of sigma(A000396(n)) see A346876. %Y A346872 For the characteristic shape of sigma(A008588(n)) see A224613. %Y A346872 Cf. A000203, A000225, A237591, A237593, A245092, A249351, A262626. %K A346872 nonn,tabf %O A346872 1,2 %A A346872 _Omar E. Pol_, Aug 06 2021