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 A262626 #215 Nov 09 2022 19:05:27 %S A262626 1,1,1,3,2,2,2,2,2,1,1,2,7,3,1,1,3,3,3,3,2,2,3,12,4,1,1,1,1,4,4,4,4,2, %T A262626 1,1,2,4,15,5,2,1,1,2,5,5,3,5,5,2,2,2,2,5,9,9,6,2,1,1,1,1,2,6,6,6,6,3, %U A262626 1,1,1,1,3,6,28,7,2,2,1,1,2,2,7,7,7,7,3,2,1,1,2,3,7,12,12,8,3,1,2,2,1,3,8,8,8,8,8,3,2,1,1 %N A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593. %C A262626 Also the rows of both triangles A237270 and A237593 interleaved. %C A262626 Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section). %C A262626 The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270. %C A262626 The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n. %C A262626 Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels. %C A262626 Odd-indexed rows are also the rows of the irregular triangle A237270. %C A262626 Even-indexed rows are also the rows of the triangle A237593. %C A262626 Lengths of the odd-indexed rows are in A237271. %C A262626 Lengths of the even-indexed rows give 2*A003056. %C A262626 Row sums of the odd-indexed rows gives A000203, the sum of divisors function. %C A262626 Row sums of the even-indexed rows give the positive even numbers (see A005843). %C A262626 Row sums give A245092. %C A262626 From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers. %C A262626 The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence. %H A262626 Robert Price, <a href="/A262626/b262626.txt">Table of n, a(n) for n = 1..16048</a> (n=1..412 rows) %H A262626 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr01.jpg">An infinite stepped pyramid (A237593, A237270, A262626)</a> %H A262626 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)</a> %H A262626 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (levels: 1..16)</a> %H A262626 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a> %e A262626 Irregular triangle begins: %e A262626 1; %e A262626 1, 1; %e A262626 3; %e A262626 2, 2; %e A262626 2, 2; %e A262626 2, 1, 1, 2; %e A262626 7; %e A262626 3, 1, 1, 3; %e A262626 3, 3; %e A262626 3, 2, 2, 3; %e A262626 12; %e A262626 4, 1, 1, 1, 1, 4; %e A262626 4, 4; %e A262626 4, 2, 1, 1, 2, 4; %e A262626 15; %e A262626 5, 2, 1, 1, 2, 5; %e A262626 5, 3, 5; %e A262626 5, 2, 2, 2, 2, 5; %e A262626 9, 9; %e A262626 6, 2, 1, 1, 1, 1, 2, 6; %e A262626 6, 6; %e A262626 6, 3, 1, 1, 1, 1, 3, 6; %e A262626 28; %e A262626 7, 2, 2, 1, 1, 2, 2, 7; %e A262626 7, 7; %e A262626 7, 3, 2, 1, 1, 2, 3, 7; %e A262626 12, 12; %e A262626 8, 3, 1, 2, 2, 1, 3, 8; %e A262626 8, 8, 8; %e A262626 8, 3, 2, 1, 1, 1, 1, 2, 3, 8; %e A262626 31; %e A262626 9, 3, 2, 1, 1, 1, 1, 2, 3, 9; %e A262626 ... %e A262626 Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid: %e A262626 . %e A262626 n A000203 A237270 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ %e A262626 1 1 = 1 |_| | | | | | | | | | | | | | | | %e A262626 2 3 = 3 |_ _|_| | | | | | | | | | | | | | %e A262626 3 4 = 2 + 2 |_ _| _|_| | | | | | | | | | | | %e A262626 4 7 = 7 |_ _ _| _|_| | | | | | | | | | %e A262626 5 6 = 3 + 3 |_ _ _| _| _ _|_| | | | | | | | %e A262626 6 12 = 12 |_ _ _ _| _| | _ _|_| | | | | | %e A262626 7 8 = 4 + 4 |_ _ _ _| |_ _|_| _ _|_| | | | %e A262626 8 15 = 15 |_ _ _ _ _| _| | _ _ _|_| | %e A262626 9 13 = 5 + 3 + 5 |_ _ _ _ _| | _|_| | _ _ _| %e A262626 10 18 = 9 + 9 |_ _ _ _ _ _| _ _| _| | %e A262626 11 12 = 6 + 6 |_ _ _ _ _ _| | _| _| _| %e A262626 12 28 = 28 |_ _ _ _ _ _ _| |_ _| _| %e A262626 13 14 = 7 + 7 |_ _ _ _ _ _ _| | _ _| %e A262626 14 24 = 12 + 12 |_ _ _ _ _ _ _ _| | %e A262626 15 24 = 8 + 8 + 8 |_ _ _ _ _ _ _ _| | %e A262626 16 31 = 31 |_ _ _ _ _ _ _ _ _| %e A262626 ... %e A262626 The above diagram arises from a simpler diagram as shown below. %e A262626 Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid: %e A262626 . %e A262626 . A237593 %e A262626 Level _ _ %e A262626 1 _|1|1|_ %e A262626 2 _|2 _|_ 2|_ %e A262626 3 _|2 |1|1| 2|_ %e A262626 4 _|3 _|1|1|_ 3|_ %e A262626 5 _|3 |2 _|_ 2| 3|_ %e A262626 6 _|4 _|1|1|1|1|_ 4|_ %e A262626 7 _|4 |2 |1|1| 2| 4|_ %e A262626 8 _|5 _|2 _|1|1|_ 2|_ 5|_ %e A262626 9 _|5 |2 |2 _|_ 2| 2| 5|_ %e A262626 10 _|6 _|2 |1|1|1|1| 2|_ 6|_ %e A262626 11 _|6 |3 _|1|1|1|1|_ 3| 6|_ %e A262626 12 _|7 _|2 |2 |1|1| 2| 2|_ 7|_ %e A262626 13 _|7 |3 |2 _|1|1|_ 2| 3| 7|_ %e A262626 14 _|8 _|3 _|1|2 _|_ 2|1|_ 3|_ 8|_ %e A262626 15 _|8 |3 |2 |1|1|1|1| 2| 3| 8|_ %e A262626 16 |9 |3 |2 |1|1|1|1| 2| 3| 9| %e A262626 ... %e A262626 The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n. %e A262626 The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n). %e A262626 The total number of vertical line segments in the n-th level of the diagram equals A131507(n). %e A262626 The diagram represents the first 16 levels of the pyramid. %e A262626 The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - _Omar E. Pol_, Aug 28 2018 %e A262626 The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - _Omar E. Pol_, Nov 09 2022 %Y A262626 Famous sequences that are visible in the stepped pyramid: %Y A262626 Cf. A000040 (prime numbers)......., for the characteristic shape see A346871. %Y A262626 Cf. A000079 (powers of 2)........., for the characteristic shape see A346872. %Y A262626 Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level. %Y A262626 Cf. A000217 (triangular numbers).., for the characteristic shape see A346873. %Y A262626 Cf. A000225 (Mersenne numbers)...., for a visualization see A346874. %Y A262626 Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875. %Y A262626 Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876. %Y A262626 Cf. A000668 (Mersenne primes)....., for a visualization see A346876. %Y A262626 Cf. A001097 (twin primes)........., for a visualization see A346871. %Y A262626 Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level. %Y A262626 Cf. A002378 (oblong numbers)......, for a visualization see A346873. %Y A262626 Cf. A008586 (multiples of 4)......, perimeters of the successive levels. %Y A262626 Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613. %Y A262626 Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo. %Y A262626 Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864. %Y A262626 Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level. %Y A262626 Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others). %Y A262626 Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238. %Y A262626 Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508. %K A262626 nonn,tabf,look %O A262626 1,4 %A A262626 _Omar E. Pol_, Sep 26 2015