A130130 -id:A130130 - cp's OEIS frontend

A181375 Total number of positive integers below 10^n requiring 2 positive cubes in their representation as sum of cubes.

2, 9, 41, 202, 938, 4354, 20330, 94625, 439959, 2045048, 9500746, 44124084, 204883131, 951202028, 4415710979, 20497646229, 95146359635

Offset: 1

Martin Renner, Jan 28 2011

A061439(n) + a(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A003325.

iscube:=proc(n) if root(n,3)=trunc(root(n,3)) then true; else false; fi; end:
isA003325:=proc(n) local x,y3; if iscube(n) then false; else for x from 1 do y3:=n-x^3; if y3A003325(k) then i:=i+1; fi; od: return(i); end:
for n from 1 do print(a(n)); od;

a(n)=my(N=10^n,v=List(),x3);sum(x=1,sqrtnint(N-1,3),x3=x^3;sum(y=1, min(sqrtnint(N-x3,3),x), !ispower(x3+y^3,3) && listput(v,x3+y^3))); #vecsort(v,,8) \\ Charles R Greathouse IV, Oct 16 2013

a(6)-a(12) from Lars Blomberg, May 04 2011

a(13)-a(17) from Hiroaki Yamanouchi, Jul 12 2014

A181377 Total number of positive integers below 10^n requiring 3 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 15, 122, 1128, 10678, 103421, 1017326, 10077684, 100294216

Offset: 1

Martin Renner, Jan 28 2011

A061439(n) + A181375(n) + a(n) + A181379(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A047702.

a(5)-a(9) from Lars Blomberg, May 04 2011

A181379 Total number of positive integers below 10^n requiring 4 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 18, 242, 3343, 46683, 605489, 7221246, 80884939, 865304098

Offset: 1

Martin Renner, Jan 28 2011

A061439(n) + A181375(n) + A181377(n) + a(n) + A181381(n) + A181400(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n).

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A047703.

a(5)-a(9) from Lars Blomberg, May 04 2011

A181381 Total number of positive integers below 10^n requiring 5 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 21, 293, 3842, 38076, 282579, 1736822, 8938227, 33956667

Offset: 1

Martin Renner, Jan 28 2011

nonn

A061439(n) + A181375(n) + A181377(n) + A181379(n) + a(n) + A181400(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n)

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A047704.

a(5)-a(7) from Lars Blomberg, May 04 2011

a(8)-a(9) from Hiroaki Yamanouchi, Sep 23 2014

A181400 Total number of positive integers below 10^n requiring 6 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 18, 202, 1325, 3440, 3919, 3922, 3922, 3922

Offset: 1

Martin Renner, Jan 28 2011

A061439(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + a(n) + A181402(n) + A181404(n) + A130130(n) = A002283(n)

Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A046040.

a(5)-a(7) from Lars Blomberg, May 04 2011

a(8)-a(9) from Hiroaki Yamanouchi, Sep 23 2014

A181402 Total number of positive integers below 10^n requiring 7 positive cubes in their representation as sum of cubes.

Original entry on oeis.org

1, 10, 73, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121

Offset: 1

Martin Renner, Jan 28 2011

An unpublished result of Deshouillers-Hennecart-Landreau, combined with Lemma 3 from Bertault, Ramaré, & Zimmermann implies that a(4)-a(34) are all 121. Probably a(n) = 121 for all n > 3. - Charles R Greathouse IV, Jan 23 2014

F. Bertault, O. Ramaré, and P. Zimmermann, On sums of seven cubes, Math. Comp. 68 (1999), pp. 1303-1310.
Eric Weisstein's World of Mathematics, Waring's Problem.

Cf. A018890.

Cf. A002283, A061439, A130130, A181375, A181377, A181379, A181381, A181400, A181404.

A061439(n) + A181375(n) + A181377(n) + A181379(n) + A181381(n) + A181400(n) + a(n) + A181404(n) + A130130(n) = A002283(n).
Conjectured g.f.: x*(1+9*x+63*x^2+48*x^3)/(1-x). - Colin Barker, May 04 2012
Conjectured e.g.f.: 121*(exp(x) - 1) - 120*x - 111*x^2/2 - 8*x^3. - Stefano Spezia, May 21 2024

a(5)-a(7) from Lars Blomberg, May 04 2011

a(8)-a(34) from Charles R Greathouse IV, Jan 23 2014

A278317 Number of neighbors of each new term in a right triangle read by rows.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 2, 4, 3, 2, 2, 4, 4, 3, 2, 2, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 2

Offset: 1

Omar E. Pol, Nov 18 2016

To evaluate T(n,k) consider only the neighbors of T(n,k) that are present in the triangle when T(n,k) should be a new term in the triangle.
Apart from the first column and the first two diagonals the rest of the elements are 4's.
For the same idea but for an isosceles triangle see A275015; for a square array see A278290, for a square spiral see A278354; and for a hexagonal spiral see A047931.

			Triangle begins:
0;
1, 2;
2, 3, 2;
2, 4, 3, 2;
2, 4, 4, 3, 2;
2, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 4, 4, 3, 2;
2, 4, 4, 4, 4, 4, 4, 4, 3, 2;
...

Apart from the initial zero, row sums give A004767.
Column 1 is A130130.
Columns > 1 give the terms greater than 1 of A158411.
Right border gives 0 together with A007395, also twice A057427.
Second right border gives A122553.
Cf. A047931, A274912, A274913, A275015, A278290, A278317, A278354,

A066150 Maximal number of divisors of any n-digit number.

Original entry on oeis.org

4, 12, 32, 64, 128, 240, 448, 768, 1344, 2304, 4032, 6720, 10752, 17280, 26880, 41472, 64512, 103680, 161280, 245760, 368640, 552960, 860160, 1290240, 1966080, 2764800, 4128768, 6193152, 8957952, 13271040, 19660800, 28311552, 41287680, 59719680, 88473600, 127401984, 181665792, 264241152, 382205952, 530841600

Offset: 1

Joseph L. Pe, Dec 12 2001

			a(1) = 4 since 8 has 4 divisors and that is the record for 1-digit numbers.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000005, A066151, A069650,

Cf. A130130 (minimal number of divisors of any n-digit number). [Jaroslav Krizek, Jul 18 2010]

a066150(m,n) = local(d,a,k,b); for(d=m,n,a=0; for(k=10^d,10^(d+1)-1,b =numdiv(k); if(b>a,a=b)); print1(a,","))
a066150(0,6)

a(n) = largest integer m such that A005179(m) < 10^n. - Max Alekseyev, Apr 29 2010

a(n) = A000005(A066151(n)). - Amiram Eldar, Jul 02 2019

One more term from Klaus Brockhaus, Dec 13 2001
Further terms from Vladeta Jovovic and Vladimir Baltic, Dec 16 2001
Extended further by David Wasserman, Jan 25 2002

A137891 Number of (directed) Hamiltonian paths in the graph join C_n + C_n of two cycles.

Original entry on oeis.org

720, 13824, 383000, 14804640, 764340024, 50913153536, 4256161751448, 436618291524000, 53955264479804600, 7908071556041000064, 1356709951589099693976, 269380212536429979520928, 61297096735652845698099000, 15847986814197933588682229760, 4620315237160994963528810238104

Offset: 3

Eric W. Weisstein, Feb 20 2008

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 3..200 (terms 3..50 from Andrew Howroyd)
Eric Weisstein's World of Mathematics, Graph Join.
Eric Weisstein's World of Mathematics, Hamiltonian Path.
Index entries for sequences related to graphs, Hamiltonian

Cf. A130130, A234628, A266213, A270047.

b[n_, k_] := If[k == 0, 0, Sum[j*Min[2, j] * Sum[ Binomial[n - j - k, kk - 1]*Binomial[k - 1, kk]*2^kk, {kk, 0, Min[k - 1, n - j - k + 1]}], {j, 1, n - k + 1}]];
Flatten[{{2, 24}, Table[Sum[2*k!*b[n, k]*(k!*b[n, k] + (k - 1)!*b[n, k - 1]), {k, 1, n}], {n, 3, 20}]}] (* Vaclav Kotesovec, Mar 08 2016, after Andrew Howroyd *)

B(n)=polcoef(1/(1 - x*y*(2/(1 - x) - 1)) + O(x*x^n), n)
a(n)={my(v=Vecrev(B(n))); 2*n^2*sum(k=1, n, my(t=v[1+k]*(k-1)!); t*(t + if(k>1, v[k]*(k-2)!)))} \\ Andrew Howroyd, Jan 10 2025

a(n) = Sum_ { k=1..n } 2*k!*b(n,k)*(k!*b(n,k)+(k-1)!*b(n,k-1)) where b(n,0)=0, b(n,k)=Sum_{ j=1..n-k+1 } j*A130130(j)*A266213(k-1,n-j-k+1) for k>0, n<>2. - Andrew Howroyd, Feb 14 2016

a(n) ~ c * n!^2, where c = A270047 = 42.12277421168156081166292550105956... . - Vaclav Kotesovec, Mar 08 2016

a(6)-a(7) from Eric W. Weisstein, Dec 16 2013
a(8)-a(10) from Eric W. Weisstein, Dec 24 2013
a(1)=2 and a(2)=24 prepended by Andrew Howroyd, Feb 14 2016
a(11)-a(16) from Andrew Howroyd, Feb 14 2016
a(1)-a(2) removed by Andrew Howroyd, Jan 10 2025

A294619 a(0) = 0, a(1) = 1, a(2) = 2 and a(n) = 1 for n > 2.

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Nov 05 2017

Continued fraction expansion of (sqrt(5) + 1)/(2*sqrt(5)).
Inverse binomial transform is {0, 1, 4, 10, 21, 41, 78, 148, ...}, A132925 with one leading zero.
Also the main diagonal in the expansion of (1 + x)^n - 1 + x^2 (A300453).
The partial sum of this sequence is A184985.
a(n) is the number of state diagrams having n components that are obtained from an n-foil [(2,n)-torus knot] shadow. Let a shadow diagram be the regular projection of a mathematical knot into the plane, where the under/over information at every crossing is omitted. A state for the shadow diagram is a diagram obtained by merging either of the opposite areas surrounding each crossing.
a(n) satisfies the identities a(n)^a(n+k) = a(n), 2^a(k) = 2*a(k) and a(k)! = a(k), k > 0.
Also the number of non-isomorphic simple connected undirected graphs with n+1 edges and a longest path of length 2. - Nathaniel Gregg, Nov 02 2021

			For n = 2, the shadow of the Hopf link yields 2 two-component state diagrams (see example in A300453). Thus a(2) = 2.

V. I. Arnold, Topological Invariants of Plane Curves and Caustics, American Math. Soc., 1994.
L. H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.
V. Manturov, Knot Theory, CRC Press, 2004.

I. Altintas, An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput. 194 (2007) 168-178.
J. A. Baldwin and A. S. Levine, A combinatorial spanning tree model for knot Floer homology, Advances in Mathematics, Vol. 231 (2012), 1886-1939.
A. Banerjee, Knot theory [Foil knot family].
D. Denton and P. Doyle, Shadow movies not arising from knots, arXiv preprint, arXiv:1106.3545 [math.GT], 2011.
L. H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.
Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A000007, A000045, A004278, A005096, A005803, A010701, A063524, A103775, A107583, A130130, A132925, A141044, A157928, A158411, A163985, A171386, A179184, A184985, A185012, A300453.

CoefficientList[Series[(x + x^2 - x^3)/(1 - x), {x, 0, 100}], x] (* Wesley Ivan Hurt, Nov 05 2017 *)
f[n_] := If[n > 2, 1, n]; Array[f, 105, 0] (* Robert G. Wilson v, Dec 27 2017 *)
PadRight[{0,1,2},120,{1}] (* Harvey P. Dale, Feb 20 2023 *)

makelist((1 + (-1)^((n + 1)!))/2 + kron_delta(n, 2), n, 0, 100);

PARI
```
a(n) = if(n>2, 1, n);
```

a(n) = ((-1)^2^(n^2 + 3*n + 2) + (-1)^2^(n^2 - n) - (-1)^2^(n^2 - 3*n + 2) + 1)/2.
a(n) = (1 + (-1)^((n + 1)!))/2 + Kronecker(n, 2).
a(n) = min(n, 3) - 2*(max(n - 2, 0) - max(n - 3, 0)).
a(n) = floor(F(n+1)/F(n)) for n > 0, with a(0) = 0, where F(n) = A000045(n) is the n-th Fibonacci number.
a(n) = a(n-1) for n > 3, with a(0) = 0, a(1) = 1, a(2) = 2 and a(3) = 1.
A005803(a(n)) = A005096(a(n)) = A000007(n).
A107583(a(n)) = A103775(n+5).
a(n+1) = 2^A185012(n+1), with a(0) = 0.
a(n) = A163985(n) mod A004278(n+1).
a(n) = A157928(n) + A171386(n+1).
a(n) = A063524(n) + A157928(n) + A185012(n).
a(n) = A010701(n) - A141044(n) - A179184(n).
G.f.: (x + x^2 - x^3)/(1 - x).
E.g.f.: (2*exp(x) - 2 + x^2)/2.

cp's OEIS Frontend

A181375 Total number of positive integers below 10^n requiring 2 positive cubes in their representation as sum of cubes.

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A181377 Total number of positive integers below 10^n requiring 3 positive cubes in their representation as sum of cubes.

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A181379 Total number of positive integers below 10^n requiring 4 positive cubes in their representation as sum of cubes.

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A181381 Total number of positive integers below 10^n requiring 5 positive cubes in their representation as sum of cubes.

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A181400 Total number of positive integers below 10^n requiring 6 positive cubes in their representation as sum of cubes.

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A181402 Total number of positive integers below 10^n requiring 7 positive cubes in their representation as sum of cubes.

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A278317 Number of neighbors of each new term in a right triangle read by rows.

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A066150 Maximal number of divisors of any n-digit number.

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A137891 Number of (directed) Hamiltonian paths in the graph join C_n + C_n of two cycles.

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