A087503 -id:A087503 - cp's OEIS frontend

A357303 Records in the numbers of representations of k^2 as x^2 - xy + y^2, x > 2y >= 0, corresponding to the numbers of grid points with squared radius A357302(n)^2 in an angular sector 0 <= phi < Pi/6 of the triangular lattice.

1, 2, 3, 5, 8, 14, 23, 41, 68, 122, 203, 365, 608, 851, 1094, 1823, 2552, 3281, 5468, 7655, 9842, 16403, 22964, 29525, 49208, 68891, 82013, 88574, 114818, 147623, 206672, 246038, 265721

Offset: 1

Hugo Pfoertner, Sep 25 2022

The total number of possible representations of A357302(n)^2 = x^2 + x*y + y^2 with integers x,y is 12*a(n) - 6.

			See A357302.

Cf. A002324, A003136, A004016, A230655, A357302.
A246360 seems to be a subsequence.
Cf. A087503.

records[n_] := Module[{ri = n, m = 0, rcs = {}, len}, len = Length[ri];
   While[len > 0, If[First[ri] > m, m = First[ri]; AppendTo[rcs, m]];
    ri = Rest[ri]; len--]; rcs]; (* after Harvey P. Dale in A336679 *)
records[Table[
  Length[FindInstance[
    x^2 - x*y + y^2 == k^2 && x > 2*y && y >= 0, {x, y}, Integers,
    1000]], {k, 0, 2000}]]

A372782 Least number m for which there exists some positive k < m where the sum of the integers from k + 1 to m inclusive is an n-th power > 1.

Original entry on oeis.org

2, 4, 7, 13, 22, 40, 67, 121, 202, 364, 607, 1093, 1822, 3280, 5467, 9841, 16402, 29524, 49207, 88573, 147622, 265720, 442867, 797161, 1328602, 2391484, 3985807, 7174453, 11957422, 21523360, 35872267, 64570081, 107616802, 193710244, 322850407, 581130733, 968551222, 1743392200

Offset: 1

Jean-Marc Rebert, May 14 2024

With triangular sum T(i) = i*(i+1)/2 = Sum_{j=1..i} i, the aim is T(m) - T(k) = b^n for some b. T(m) - T(k) = (m+k+1)*(m-k)/2 so if m-k is odd then use m-k = b^x to eliminate k in the other factor (m+k+1)/2 = b^(n-x), so that m = b^(n-x) + (b^x - 1)/2. If instead m+k+1 is odd then the resulting expression is the same. This form is an integer and minimized at b=3 and x = ceiling(n/2). - Kevin Ryde, May 18 2024

			a(2) = 4 because the sum of all integers from 3 + 1 to 4 inclusive is 4 = 2^2, a square.
a(3) = 7 as we have 2 + 3 + 4 + 5 + 6 + 7 = 27 = 3^3, i.e., m = 7 and k = 1. - _David A. Corneth_, May 15 2024

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A002452, A052993, A087503, A372631.

LinearRecurrence[{1, 3, -3}, {2, 4, 7}, 50] (* Paolo Xausa, Jun 09 2024 *)

isok(m, n) = my(s = m*(m+1)/2); for (k=1, m-1, s -= k; if (ispower(s, n), return(k)););
a(n) = my(m=1); while (! isok(m, n), m++); m; \\ Michel Marcus, May 16 2024

a(n) = {
	my(res, c, l, u);
	res = 2^n; c = 8*3^n;
	l = (sqrt(c) - 1)\2; u = res;
	for(i = l, u,
		if(issquare(4*(i + 1)*i + 1 - c),
			return(i);
		)
	);
	return(2^n)
} \\ David A. Corneth, May 16 2024

a(n) = A087503(n-1) + 1.
a(n) = 3*a(n-2) + 1.
G.f.: x*(2 + 2*x - 3*x^2)/((1 - x)*(1 - 3*x^2)). - Stefano Spezia, May 18 2024

More terms from David A. Corneth, May 16 2024

cp's OEIS Frontend

A357303 Records in the numbers of representations of k^2 as x^2 - xy + y^2, x > 2y >= 0, corresponding to the numbers of grid points with squared radius A357302(n)^2 in an angular sector 0 <= phi < Pi/6 of the triangular lattice.

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A372782 Least number m for which there exists some positive k < m where the sum of the integers from k + 1 to m inclusive is an n-th power > 1.

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cp's OEIS Frontend

A357303 Records in the numbers of representations of k^2 as x^2 - x*y + y^2, x > 2*y >= 0, corresponding to the numbers of grid points with squared radius A357302(n)^2 in an angular sector 0 <= phi < Pi/6 of the triangular lattice.

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Examples

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Programs

Mathematica

A372782 Least number m for which there exists some positive k < m where the sum of the integers from k + 1 to m inclusive is an n-th power > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A357303 Records in the numbers of representations of k^2 as x^2 - xy + y^2, x > 2y >= 0, corresponding to the numbers of grid points with squared radius A357302(n)^2 in an angular sector 0 <= phi < Pi/6 of the triangular lattice.