A002817 -id:A002817 - cp's OEIS frontend

A321246 Non-isomorphic proper colorings of the 5 X 5 grid graph using at most n colors under rotational and reflectional symmetries.

0, 2, 76332, 2557101612, 6352711134515, 2747239197568620, 378972203462839707, 23698347614119889312, 832593421909253876202, 18885862442806789810230, 304064344379602597321190, 3716359333313224494744012, 36226784801918510547852117, 292338319876651811428566992, 2009992643746035728869251645, 12045344786281525649136156960, 64072515057361294676198896292

Offset: 1

Marko Riedel, Nov 01 2018

nonn

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Tree graphs colorings, Math StackExchange, December 2017.
Marko Riedel et al., 3-colourings of a 3×3 table with one of 3 colors up to symmetries, Math StackExchange, October 2018.

Cf. A002817, A321244, A321245.

a(n) = (1/8)*n^25 - 5*n^24 + (195/2)*n^23-1233*n^22 + (45399/4)*n^21 - 80919*n^20 + (928545/2)*n^19 - (17590911/8)*n^18 + (69997383/8)*n^17 - (118477969/4)*n^16 + (172111059/2)*n^15 - (1726958987/8)*n^14 + (3754019329/8)*n^13 - (1770719251/2)*n^12 + (5797425049/4)*n^11 - 2053661272*n^10 + (20055169857/8)*n^9 - (20932696169/8)*n^8 + (9236896437/4)*n^7 - (6780818949/4)*n^6 + (8083053959/8)*n^5 - (3768579695/8)*n^4 + (1292510453/8)*n^3 - (145271789/4)*n^2 + 4017958*n.

A336502 Partial sums of A057003.

Original entry on oeis.org

1, 7, 127, 5167, 365527, 39435607, 6006997207, 1226103906007, 322796982334807, 106460296033918807, 42980408446129381207, 20846482682939051365207, 11959807608801430284133207, 8010447502346968140207973207, 6193994326661240674349352805207, 5476021766725276671842502543205207

Offset: 1

Seiichi Manyama, Jul 23 2020

Inspired by doubly triangular numbers (A002817).

			a(2) = 1 + 2*3 = 7.
a(3) = 1 + 2*3 + 4*5*6 = 127.
a(4) = 1 + 2*3 + 4*5*6 + 7*8*9*10 = 5167.

Seiichi Manyama, Table of n, a(n) for n = 1..226

Cf. A000217, A002817, A007489, A057003, A227364, A336513.

Accumulate @ Table[(n * (n + 1)/2)!/((n - 1) * n /2)!, {n, 1, 16}] (* Amiram Eldar, Jul 23 2020 *)

{a(n) = sum(i=1, n, prod(j=(i-1)*i/2+1, i*(i+1)/2, j))}

a(n) = Sum_{i=1..n} Product_{j=T(i-1)+1..T(i)} j where T(n) is n-th triangular number.
a(n) = A227364(T(n)) where T(n) is n-th triangular number.
a(n) ~ n^(2*n) / 2^n. - Vaclav Kotesovec, Nov 20 2021

A378198 Table T(n, k) read by upward antidiagonals. T(n,1) = A375602(n), T(n,2) = A375602(A375602(n)), T(n,3) = A375602(A375602(A375602(n))) and so on.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 3, 3, 2, 1, 5, 4, 4, 2, 1, 6, 5, 3, 3, 2, 1, 7, 6, 5, 4, 4, 2, 1, 10, 7, 6, 5, 3, 3, 2, 1, 13, 16, 7, 6, 5, 4, 4, 2, 1, 16, 14, 9, 7, 6, 5, 3, 3, 2, 1, 8, 9, 17, 13, 7, 6, 5, 4, 4, 2, 1, 11, 10, 13, 12, 14, 7, 6, 5, 3, 3, 2, 1, 14, 8, 16, 14, 11, 17, 7, 6, 5, 4, 4, 2, 1, 17, 17, 10, 9, 17, 8, 12, 7, 6, 5, 3, 3, 2, 1, 19, 12, 12, 16, 13, 12, 10, 11, 7

Offset: 1

Boris Putievskiy, Nov 19 2024

Sequence A375602 generates an infinite cyclic group under composition. The identity element is A000027.
Each column is triangle read by rows (blocks). Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. Row n  has length n(n^2 + 1)/2 = A006003(n).
Each column is an intra-block permutation of the positive integers.
For n > 1, each row combines n consecutive antidiagonals.
Generalization of the Cantor numbering method.

			Table begins:
  k =      1   2   3   4   5   6
--------------------------------------
  n =  1:  1,  1,  1,  1,  1,  1, ...
  n =  2:  2,  2,  2,  2,  2,  2, ...
  n =  3:  4,  3,  4,  3,  4,  3, ...
  n =  4:  3,  4,  3,  4,  3,  4, ...
  n =  5:  5,  5,  5,  5,  5,  5, ...
  n =  6:  6,  6,  6,  6,  6,  6, ...
  n =  7:  7,  7,  7,  7,  7,  7, ...
  n =  8: 10, 16,  9, 13, 14, 17, ...
  n =  9: 13, 14, 17, 12, 11,  8, ...
  n = 10: 16,  9, 13, 14, 17, 12, ...
    ...
Column k = 1 contains the start of A375602.
Ord(T(1,1),T(2,1), ... T(7,1)) = 2, ord(T(1,1),T(2,1), ... T(21,1)) = 18, ord(T(1,1),T(2,1), ... T(55,1)) = 1980, ord(T(1,1),T(2,1), ... T(120,1)) = 51480, where ord is order of permutation.
The first 6 antidiagonals are:
  1;
  2, 1;
  4, 2, 1;
  3, 3, 2, 1;
  5, 4, 4, 2, 1;
  6, 5, 3, 3, 2, 1;

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A006003 (row lengths), A002817, A370655, A373498, A374447, A374494, A374531, A375602, A375725.

a[n_]:=Module[{L,Ld,Rd,P,Result},L=Ceiling[(Sqrt[4*Sqrt[8*n+1]-3]-1)/2]; Ld=Ceiling[(Sqrt[8*n+1]-1)/2]; Rd=n-(Ld-1)*Ld/2; P=L*Rd+Ld-L*(L+1)/2-Max[Rd-(L^2-L+2)/2,0]*(Max[Rd-(L^2-L+2)/2,0]+1)/2; Result=P+(L-1)*L*(L^2-L+2)/8; Result] (*A375602*) composeSequence[a_,n_,k_]:=Nest[a,n,k]
Nmax=10; Kmax=6; T=Table[composeSequence[a,n,k],{n,1,Nmax},{k,1,Kmax}]

(T(1,k),T(2,k), ... T(A002817(n),k)) is permutation of the integers from 1 to A002817(n). (T(1,k),T(2,k), ... T(A002817(n),k)) = (T(1,1),T(2,1), ... T(A002817(n),1))^k.

A264894 a(n) = n(7n - 5)(49n^2 - 35*n - 10)/8.

Original entry on oeis.org

0, 1, 261, 1956, 7291, 19500, 42846, 82621, 145146, 237771, 368875, 547866, 785181, 1092286, 1481676, 1966875, 2562436, 3283941, 4148001, 5172256, 6375375, 7777056, 9398026, 11260041, 13385886, 15799375, 18525351, 21589686, 25019281, 28842066, 33087000

Offset: 0

Ilya Gutkovskiy, Nov 27 2015

Doubly 9-gonal (or nonagonal) numbers.

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Nonagonal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A001106, A002817, A000583, A232713, A063249.

[n*(7*n-5)*(49*n^2-35*n-10)/8: n in [0..30]]; // Vincenzo Librandi, Nov 28 2015

Table[n (7 n - 5) (49 n^2 - 35 n - 10)/8, {n, 0, 30}]
LinearRecurrence[{5,-10,10,-5,1},{0,1,261,1956,7291},40] (* Harvey P. Dale, Apr 29 2017 *)

vector(100, n, n--; n*(7*n-5)*(49*n^2-35*n-10)/8) \\ Altug Alkan, Nov 27 2015

G.f.: x*(1 + 256*x + 661*x^2 + 111*x^3)/(1 - x)^5.
a(n) = A001106(A001106(n)).
Sum_{n>0} 1/a(n) = (4*(sqrt(65)*gamma + sqrt(65)*polygamma(0, 2/7) - 5*polygamma(0, (1/14)*(9 - sqrt(65))) + 5*polygamma(0, (1/14)*(9 + sqrt(65)))))/(25*sqrt(65)) = 1.0045877861645573..., where gamma is the Euler-Mascheroni constant (A001620), and polygamma is the derivative of the logarithm of the gamma function.

A264895 a(n) = n(4n - 3)(16n^2 - 12*n - 3).

Original entry on oeis.org

0, 1, 370, 2835, 10660, 28645, 63126, 121975, 214600, 351945, 546490, 812251, 1164780, 1621165, 2200030, 2921535, 3807376, 4880785, 6166530, 7690915, 9481780, 11568501, 13981990, 16754695, 19920600, 23515225, 27575626, 32140395, 37249660, 42945085, 49269870

Offset: 0

Ilya Gutkovskiy, Nov 27 2015

Doubly 10-gonal (or decagonal) numbers.

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Decagonal Number
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1)

Cf. A001107, A002817, A000583, A232713, A063249.

[n*(4*n - 3)*(16*n^2 - 12*n - 3): n in [0..30]]; // Vincenzo Librandi, Nov 28 2015

Table[n (4 n - 3) (16 n^2 - 12 n - 3), {n, 0, 30}]
LinearRecurrence[{5,-10,10,-5,1}, {0, 1, 370, 2835, 10660}, 50] (* G. C. Greubel, Sep 07 2018 *)

vector(100, n, n--; n*(4*n-3)*(16*n^2-12*n-3)) \\ Altug Alkan, Nov 27 2015

G.f.: x*(1 + 365*x + 995*x^2 + 175*x^3)/(1 - x)^5.
a(n) = A001107(A001107(n)).
Sum_{n>0} 1/a(n) = (sqrt(21)*gamma + sqrt(21)*polygamma(0, 1/4)  - 3*polygamma(0, (1/8)*(5 - sqrt(21))) + 3*polygamma(0, (1/8)*(5 + sqrt(21))))/(9*sqrt(21))= 1.00322253307732984...., where gamma is the Euler-Mascheroni constant (A001620), and polygamma is the derivative of the logarithm of the gamma function.

A375764 a(n) is the sum of distinct sums of all subsets with two or more elements of {1, 2, ..., n}.

Original entry on oeis.org

0, 0, 3, 18, 52, 117, 228, 403, 663, 1032, 1537, 2208, 3078, 4183, 5562, 7257, 9313, 11778, 14703, 18142, 22152, 26793, 32128, 38223, 45147, 52972, 61773, 71628, 82618, 94827, 108342, 123253, 139653, 157638, 177307, 198762, 222108, 247453, 274908, 304587

Offset: 0

Darío Clavijo, Aug 26 2024

The cardinality of the set for n is A034856(n-1).

			For n = 3 the starting set is {1,2,3} and there are subsets {1,2}{1,3}{2,3}{1,2,3} that sum to 3,4,5 and 6 and the sum of distinct sums (3+4+5+6) is 18.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002817, A034856.

Join[{0, 0}, Nest[PolygonalNumber, Range[2, 50], 2] - 3] (* Paolo Xausa, Sep 13 2024 *)

a = lambda n: max(0,(n**4+2*n**3+3*n**2+2*n-24)//8)
print([a(n) for n in range(1,40)])

def A375764(n): return (m:=n*(n+1)-4)*(m+10)>>3 if n>1 else 0 # Chai Wah Wu, Aug 30 2024

a(n) = A002817(n) - 3 for n > 1.
From Alois P. Heinz, Aug 27 2024: (Start)
G.f.: x^2*(2*x^4-7*x^3+8*x^2-3*x-3)/(x-1)^5.
a(n) = max(0,(n^4+2*n^3+3*n^2+2*n-24)/8). (End)
E.g.f.: exp(x)*(x^4/8 + x^3 + 2*x^2 + x - 3) + 2*x + 3. - Stefano Spezia, Aug 28 2024

More terms from Alois P. Heinz, Aug 27 2024

A096662 Least nontrivial n-tuply triangular number.

Original entry on oeis.org

3, 6, 21, 231, 26796, 359026206, 64449908476890321, 2076895351339769460477611370186681, 2156747150208372213435450937462082366919951682912789656986079991221

Offset: 1

Robert G. Wilson v, Jul 02 2004

nonn

Cf. A000217, A002817, A064322, A066370.

f[n_] := n(n + 1)/2; Table[ Nest[f, 2, n], {n, 10}]

a(n)=A007501(n). [From R. J. Mathar, Sep 04 2008]

A128769 Number of inequivalent n-colorings of the 6D hypercube under the full orthogonal group of the cube (of order 2^6*6! = 46080).

Original entry on oeis.org

1, 400507806843728, 74515759884862073604656433, 7384600028168436080716029918923776, 11764346491956060465118857334844472390625, 1374572193221502774409273556832082839526247376

Offset: 1

Ricardo Perez-Aguila (ricardo.perez.aguila(AT)gmail.com), Apr 04 2007

I assume this refers to colorings of the vertices of the cube. - N. J. A. Sloane, Apr 06 2007

			a(2)=400507806843728 because there are 400507806843728 inequivalent 2-colorings of the 6D hypercube.

Banks, D. C.; Linton, S. A. & Stockmeyer, P. K. Counting Cases in Substitope Algorithms. IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371-384, 2004.
Perez-Aguila, Ricardo. Enumerating the Configurations in the n-Dimensional Orthogonal Polytopes Through Polya's Countings and A Concise Representation. Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 63-66.
Polya, G. & Read R. C. Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. Springer-Verlag, 1987.

Banks, D. C.; Linton, S. A. & Stockmeyer, P. K., Counting Cases in Substitope Algorithms, IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371-384. 2004.
Perez-Aguila, Ricardo, Orthogonal Polytopes: Study and Application, PhD Thesis. Universidad de las Americas, Puebla. November, 2006.
Perez-Aguila, Ricardo, Enumerating the Configurations in the n-Dimensional Orthogonal Polytopes Through Polya's Countings and A Concise Representation, Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 63-66.

Cf. A000616, A002817.

A[n_] := (1/46080)*(3840n^6 + 16512*n^8 + 1920*n^12 + 3840*n^14 + 12504*n^16 + 2160*n^20 + 1440*n^22 + 2320*n^24 + 1213*n^32 + 120*n^36 + 180*n^40 + 30*n^48 + n^64)

a(n) = (1/46080)*(3840*n^6 + 16512*n^8 + 1920*n^12 + 3840*n^14 + 12504*n^16 + 2160*n^20 + 1440*n^22 + 2320*n^24 + 1213*n^32 + 120*n^36 + 180*n^40 + 30*n^48 + n^64); \\ Joerg Arndt, Apr 15 2013

a(n) = (1/46080)*(3840*n^6 + 16512*n^8 + 1920*n^12 + 3840*n^14 + 12504*n^16 + 2160*n^20 + 1440*n^22 + 2320*n^24 + 1213*n^32 + 120*n^36 + 180*n^40 + 30*n^48 + n^64)

A129347 Number of inequivalent n-colorings of the 5D hypercube under the set of geometric transformations generated by all possible compositions of the 5 main reflections and the 10 main rotations and their inverses, in any order, with repetition of these geometric transformations allowed.

Original entry on oeis.org

1, 1228158, 484086357207, 4805323147589984, 6063609955178082875, 2072592733807533035358, 287612372569381586086269, 20632358601785638477436416, 894188910508179779377279557

Offset: 1

Ricardo Perez-Aguila (ricardo.perez.aguila(AT)gmail.com), Apr 10 2007

The formula was obtained by computing the cycle index of the group of geometric transformations, in 5D space, generated by all possible compositions of the 5 main reflections and the 10 main rotations and their inverses, in any order, with repetition of these geometric transformations allowed. The cycle index was obtained through the well known Polya's Enumeration Theorem.

			a(2)=1228158 because there are 1228158 inequivalent 2-colorings of the 5D hypercube.

Banks, D.C.; Linton, S.A. & Stockmeyer, P.K. Counting Cases in Substitope Algorithms. IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371-384, 2004.
Perez-Aguila, Ricardo. Enumerating the Configurations in the n-Dimensional Orthogonal Polytopes Through Polya's Countings and A Concise Representation. Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 63-66.
Polya, G. & Read, R. C., Combinatorial Enumeration of Groups, Graphs and Chemical Compounds. Springer-Verlag, 1987.

Banks, D.C.; Linton, S.A. & Stockmeyer, P.K., Counting Cases in Substitope Algorithms, IEEE Transactions on Visualization and Computer Graphics, Vol. 10, No. 4, pp. 371-384, 2004.
Perez-Aguila, Ricardo, Enumerating the Configurations in the n-Dimensional Orthogonal Polytopes Through Polya's Countings and A Concise Representation, Proceedings of the 3rd International Conference on Electrical and Electronics Engineering and XII Conference on Electrical Engineering ICEEE and CIE 2006, pp. 63-66.
Perez-Aguila, Ricardo, Orthogonal Polytopes: Study and Application, PhD Thesis. Universidad de las Americas, Puebla. November, 2006.

Cf. A000616, A002817.

A[n_] := (1/3840)*(1184*n^4 + 1624*n^8 + 240*n^10 + 400*n^12 + 311*n^16 + 60*n^20 + 20*n^24 + n^32)

a(n) = (1/3840)*(1184*n^4 + 1624*n^8 + 240*n^10 + 400*n^12 + 311*n^16 + 60*n^20 + 20*n^24 + n^32)

cp's OEIS Frontend

A321246 Non-isomorphic proper colorings of the 5 X 5 grid graph using at most n colors under rotational and reflectional symmetries.

Views

Author

Keywords

References

Links

Crossrefs

Formula

A336502 Partial sums of A057003.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A378198 Table T(n, k) read by upward antidiagonals. T(n,1) = A375602(n), T(n,2) = A375602(A375602(n)), T(n,3) = A375602(A375602(A375602(n))) and so on.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A264894 a(n) = n*(7*n - 5)*(49*n^2 - 35*n - 10)/8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A264895 a(n) = n*(4*n - 3)*(16*n^2 - 12*n - 3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A375764 a(n) is the sum of distinct sums of all subsets with two or more elements of {1, 2, ..., n}.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Python

Formula

Extensions

A096662 Least nontrivial n-tuply triangular number.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A128769 Number of inequivalent n-colorings of the 6D hypercube under the full orthogonal group of the cube (of order 2^6*6! = 46080).

Views

A264894 a(n) = n(7n - 5)(49n^2 - 35*n - 10)/8.

A264895 a(n) = n(4n - 3)(16n^2 - 12*n - 3).