A038791 -id:A038791 - cp's OEIS frontend

A038787 An intermediate sequence for nonisomorphic circulant self-complementary directed p^2-graphs, indexed by odd primes p.

2, 6, 12, 104, 356, 4134, 14572, 190652, 9588156, 35791472, 1908889156, 27487843256, 104715393912, 1529755308212, 86607687722856, 4969489243995032, 19215358445940816, 1117984489315857512, 16865594581677305360, 65588423375098872068, 3874762242354582408912

Offset: 1

N. J. A. Sloane, May 04 2000

M. Klin, V. A. Liskovets and R. Poeschel, Analytical enumeration of circulant graphs with prime-squared vertices, Sem. Lotharingien de Combin., B36d, 1996, 36 pages.

Cf. A038785.

a(p^2) = A038791(p^2) for p=4k-1.

a(p^2) = A038786(p^2) - A038785(p^2) + A038788(p^2).

More terms from Valery A. Liskovets, May 09 2001

More terms and offset corrected by Sean A. Irvine, Feb 14 2021

A054246 Non-Cayley-isomorphic circulant p^2-tournaments, indexed by odd primes p.

Original entry on oeis.org

1, 1, 4, 16, 36, 256, 900, 8836, 343396, 1201216, 53085796

Offset: 3

N. J. A. Sloane, May 04 2000

V. A. Liskovets and R. Poeschel, Non-Cayley-isomorphic self-complementary circulant graphs, J. Graph Th., 34, 2000, 128-141.

M. Klin, V. A. Liskovets and R. Poeschel, Analytical enumeration of circulant graphs with prime-squared vertices, Sem. Lotharingien de Combin., B36d, 1996, 36 pages.

Cf. A038788, A038791.

a(p^2)=A049288(p)^2

More terms from Valery A. Liskovets, May 09 2001

A238446 Let B be a nonempty and proper subset of A_n = {1,2,...,p_n-1}, where p_n is the n-th prime. Let C be the complement of B, so that the union B and C is A_n. a(n) is half the number of sums of products of elements of B and elements of C which are divisible by p_n, when B runs through all such subsets of A_n.

Original entry on oeis.org

0, 1, 3, 11, 103, 343, 4095, 14571, 190651, 9586983, 35791471, 1908874583, 27487790719, 104715393911, 1529755308211, 86607685141743, 4969489243995031, 19215358410149343, 1117984489315857511, 16865594581677305359, 65588423373189982911

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Feb 26 2014

nonn

			Take A_3 ={1,2,3,4}.  The nonempty and proper subsets are: {{1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}}.
Sums of products of elements of B and elements of C are: 1+2*3*4=25, and analogously 14,11,10,14,11,10,10,11,14,10,11,14,25.
We have 6 numbers divisible by 5. So a(3)=6/2=3.

Cf. A238444, A038791.

a(n) = A038791(n) - 1. - Ridouane Oudra, Jul 08 2025

Name edited and more terms from Ridouane Oudra, Jul 08 2025

A177352 The triangle t(n,k) of the binomial sum as in A177351 in the column index range -floor(n/2)-1 <=k <= floor(n/2)-1.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 3, 2, 5, 5, 5, 4, 1, 8, 8, 8, 7, 3, 13, 13, 13, 13, 12, 7, 1, 21, 21, 21, 21, 20, 14, 4, 34, 34, 34, 34, 34, 33, 26, 11, 1, 55, 55, 55, 55, 55, 54, 46, 25, 5, 89, 89, 89, 89, 89, 89, 88, 79, 51, 16, 1

Offset: 0

Roger L. Bagula, Dec 10 2010

Row sums are 1, 1, 5, 8, 20, 34, 72, 122, 241, 405, 769, 1284, 2375, 3947, 7165,
11866, 21238, 35078, 62094, 102340, 179561,.... which apparently is (n+1)*Fibonacci(n+1)- A129722(n) for even n, and n*Fibonacci(n+1)-A129722(n) for odd n.
The first column is A000045 by construction. The change in the column index range adds the Fibonacci numbers as a first column and removes the trailing zero in the rows compared to A177351.
Comment R. J. Mathar, Dec 20 2010 (Start):
If we construct the complements of each row's entries with respect to the Fibonacci number of that row, an array
1;  # complement to 2
1,4;  # complement to 4,1
1,5   # complement to 7,3
1,6,12 # complement to 12,7,1
1,7,17 # complement to 20,14,4
1,8,23,33 # complement to 33,26,11,1
emerges which appears to be related to A038791. (End).

			1
1;
2, 2, 1;
3, 3, 2;
5, 5, 5, 4, 1;
8, 8, 8, 7, 3;
13, 13, 13, 13, 12, 7, 1;
21, 21, 21, 21, 20, 14, 4;
34, 34, 34, 34, 34, 33, 26, 11, 1;
55, 55, 55, 55, 55, 54, 46, 25, 5;
89, 89, 89, 89, 89, 89, 88, 79, 51, 16, 1;

Cf. A177351, A000045

w[n_, m_, k_] = Binomial[n - (m + k), m + k];
t[n_, k_] := Sum[w[n, m, k], {m, 1, Floor[n/2 - k]}];
Table[Table[t[n, k], {k, -Floor[n/2 + 1], Floor[n/2 + 1] - 2}], {n, 0,
   10}]
Flatten[%]

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A038787 An intermediate sequence for nonisomorphic circulant self-complementary directed p^2-graphs, indexed by odd primes p.

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A054246 Non-Cayley-isomorphic circulant p^2-tournaments, indexed by odd primes p.

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A177352 The triangle t(n,k) of the binomial sum as in A177351 in the column index range -floor(n/2)-1 <=k <= floor(n/2)-1.

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