A329502 -id:A329502 - cp's OEIS frontend

A329501 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width n squares cut from the square grid by cuts parallel to grid lines.

1, 1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 4, 6, 4, 2, 1, 4, 7, 6, 4, 2, 1, 4, 8, 8, 6, 4, 2, 1, 4, 8, 10, 8, 6, 4, 2, 1, 4, 8, 11, 10, 8, 6, 4, 2, 1, 4, 8, 12, 12, 10, 8, 6, 4, 2, 1, 4, 8, 12, 14, 12, 10, 8, 6, 4, 2

Offset: 1

N. J. A. Sloane, Nov 19 2019

For the case when the cuts are at 45 degrees to the grid lines, see A329504.
See A329508, A329512, and A329515 for coordination sequences for cylinders formed by rolling up the hexagonal grid ("carbon nanotubes").
The g.f.s for the rows can easily be found using the "trunks and branches" method (see Goodman-Strauss and Sloane). In the illustration for n=5, there are two trunks (blue) and ten branches (red).

			Array begins:
  1, 2, 2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
  1, 3, 4,  4,  4,  4,  4,  4,  4,  4,  4,  4, ...
  1, 4, 6,  6,  6,  6,  6,  6,  6,  6,  6,  6, ...
  1, 4, 7,  8,  8,  8,  8,  8,  8,  8,  8,  8, ...
  1, 4, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, ...
  1, 4, 8, 11, 12, 12, 12, 12, 12, 12, 12, 12, ...
  1, 4, 8, 12, 14, 14, 14, 14, 14, 14, 14, 14, ...
  1, 4, 8, 12, 15, 16, 16, 16, 16, 16, 16, 16, ...
  1, 4, 8, 12, 16, 18, 18, 18, 18, 18, 18, 18, ...
  1, 4, 8, 12, 16, 19, 20, 20, 20, 20, 20, 20, ...
  ...
The initial antidiagonals are:
  1;
  1,  2;
  1,  3,  2;
  1,  4,  4,  2;
  1,  4,  6,  4,  2;
  1,  4,  7,  6,  4,  2;
  1,  4,  8,  8,  6,  4,  2;
  1,  4,  8, 10,  8,  6,  4,  2;
  1,  4,  8, 11, 10,  8,  6,  4,  2;
  1,  4,  8, 12, 12, 10,  8,  6,  4,  2;
  1,  4,  8, 12, 14, 12, 10,  8,  6,  4,  2;
  ...

Chaim Goodman-Strauss and N. J. A. Sloane, A Coloring Book Approach to Finding Coordination Sequences, Acta Cryst. A75 (2019), 121-134, also on NJAS's home page. Also arXiv:1803.08530.
N. J. A. Sloane, Illustration for rows 1 through 5, showing vertices of cylinder labeled with distance from base point (c = n is the width (or circumference)). The cylinders are formed by identifying the black lines.
Index entries for coordination sequences

Rows 1,2,3,4,5 are A040000, A113311, A329502, A115291, A329503.

Cf. A008574, A329504-A329517.

Let theta = (1+x)/(1-x).
If n = 2*k, the g.f. for the coordination sequence for row n is theta*(1+2*x+2*x^2+...+2*x^(k-1)+x^k).
If n = 2*k+1, the g.f. for the coordination sequence for row n is theta*(1+2*x+2*x^2+...+2*x^k).

A101328 Recurring numbers in the count of consecutive composite numbers between balanced primes and their lower or upper prime neighbors.

Original entry on oeis.org

1, 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 251, 257, 263, 269, 275, 281, 287, 293, 299, 305, 311, 317, 323, 329

Offset: 2

Cino Hilliard, Jan 26 2005

nonn

Except for the initial term, these numbers appear to differ by 6. Proof?
Numbers that occur in A101597. - David Wasserman, Mar 26 2008
There is no proof (yet). Heuristic evidence (Hardy-Littlewood 1923) and extensive computations indicates that the balanced-prime structure is not accidental. A theorem of van der Carput (1939) already guarantees infinitely many 3-term arithmetic progressions of primes exist, although not all of those progressions are consecutive primes. A full proof that every such 6k gap occurs infinitely often (and thus infinitely many balanced primes) remains elusive. - Hilko Koning, Apr 15 2025

Cf. A016969, A054342, A101597.

Conjectured partial sums of A329502.

balancedPrimes = {};compositeGaps = {}; Do[pPrev = Prime[i];p = Prime[i + 1]; pNext = Prime[i + 2]; If[p == (pPrev + pNext)/2, AppendTo[balancedPrimes, p];
gap1 = p - pPrev - 1; gap2 = pNext - p - 1; AppendTo[compositeGaps, gap1]; AppendTo[compositeGaps, gap2];], {i, 1, 50000}];recurringCounts = Select[Tally[compositeGaps], #[[2]] > 1 &][[All, 1]]; Sort[recurringCounts](* Hilko Koning, Apr 15 2025 *)
(* or with balanced primes *)
targetGaps = {1, 5, 11, 17, 23, 29, 35, 41, 47}; gapToBalancedPrimes = Association @@ (Rule[#, {}] & /@ targetGaps); Do[pPrev = Prime[i]; p = Prime[i + 1]; pNext = Prime[i + 2]; If[p == (pPrev + pNext)/2, gap1 = p - pPrev - 1; gap2 = pNext - p -1;uniqueGaps = DeleteDuplicates[{gap1, gap2}]; Do[If[KeyExistsQ[gapToBalancedPrimes, gap], gapToBalancedPrimes[gap] = Append[gapToBalancedPrimes[gap], p]], {gap,uniqueGaps}];], {i, 1, 50000}]; gapToBalancedPrimes (* Hilko Koning, Apr 15 2025 *)

If the numbers continue to differ by 6, then this is the sum of paired terms of 3n+1: (1, 4, 7, 10, 13, ...); and binomial transform of [1, 4, 2, -2, 2, -2, 2, ...]. - Gary W. Adamson, Sep 13 2007

a(n) = nextprime(A054342(n)+1)-A054342(n)-1. - David Wasserman, Mar 26 2008

More terms from David Wasserman, Mar 26 2008

cp's OEIS Frontend

A329501 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width n squares cut from the square grid by cuts parallel to grid lines.

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