A028872 -id:A028872 - cp's OEIS frontend

A061298 Table by antidiagonals of rows of sequences where each row is binomial transform of preceding row and row 1 is (1,2,1,2,1,2,1,2,...).

1, 1, 1, -2, 2, 1, 4, 1, 3, 1, -8, 2, 6, 4, 1, 16, 1, 12, 13, 5, 1, -32, 2, 24, 40, 22, 6, 1, 64, 1, 48, 121, 92, 33, 7, 1, -128, 2, 96, 364, 376, 174, 46, 8, 1, 256, 1, 192, 1093, 1520, 897, 292, 61, 9, 1, -512, 2, 384, 3280, 6112, 4566, 1816, 452, 78, 10, 1, 1024, 1, 768, 9841, 24512, 23073, 11152, 3289, 660, 97, 11, 1, -2048

Offset: 0

Henry Bottomley, Jun 05 2001

Rows include A011782 (but signed), A000034, A003945, A003462, A010036. Columns include A000012, A000027, A028872.

T(n, k) =(3n^k-(n-2)^k)/2. Coefficient of x^k in expansion of (1-(n-3)x)/((1-nx)(1-(n-2)x)).

A131919 A002024 + A131821 - A000012.

Original entry on oeis.org

1, 3, 3, 5, 3, 5, 7, 4, 4, 7, 9, 5, 5, 5, 9, 11, 6, 6, 6, 6, 11, 13, 7, 7, 7, 7, 7, 13, 15, 8, 8, 8, 8, 8, 8, 15, 17, 9, 9, 9, 9, 9, 9, 9, 17, 19, 10, 10, 10, 10, 10, 10, 10, 10, 19

Offset: 0

Gary W. Adamson, Jul 28 2007

Row sums = A028872: (1, 6, 13, 22, 33, 46, ...).

			First few rows of the triangle:
   1;
   3, 3;
   5, 3, 5;
   7, 4, 4, 7;
   9, 5, 5, 5, 9;
  11, 6, 6, 6, 6, 11;
  13, 7, 7, 7, 7,  7, 13;
  ...

Cf. A002024, A131821, A000012, A028872.

A002024 + A131821 - A000012 as infinite lower triangular matrices.

A133475 Integers n such that n^3 + n^2 - 9*n + 16 is a square.

Original entry on oeis.org

-4, -3, -1, 0, 1, 3, 5, 11, 15, 28, 47, 55, 81, 549, 1799, 8361

Offset: 1

Mohamed Bouhamida, Nov 29 2007

The set of x values of integral points on the elliptic curve y^2 = x^3 + x^2 - 9*x + 16.

			0^3 + (-5)^2 + (-9) = 4^2, 1^3 + (-4)^2 + (-8) = 3^2, 3^3 + (-2)^2 + (-6) = 5^2

Cf. A117950, A132411, A132414, A002522, A028872.

P := PolynomialRing(Integers()); {x: x in Sort([ p[1] : p in IntegralPoints(EllipticCurve(n^3 + n^2 - 9*n + 16)) ])};

ok[x_] := Reduce[{y^2 == x^3 + x^2 - 9*x + 16, y >= 0}, y, Integers] =!= False; Select[Table[x, {x, -4, 10000}], ok ] (* Jean-François Alcover, Sep 07 2011 *)

is(n)=issquare(n^3+n^2-9*n+16) \\ Charles R Greathouse IV, Sep 06 2016

EllipticCurve([0,1,0,-9,16]).integral_points()

Edited by Max Alekseyev, Nov 13 2009

A375352 Numbers k such that 14*k + 2 is a square.

Original entry on oeis.org

1, 7, 23, 41, 73, 103, 151, 193, 257, 311, 391, 457, 553, 631, 743, 833, 961, 1063, 1207, 1321, 1481, 1607, 1783, 1921, 2113, 2263, 2471, 2633, 2857, 3031, 3271, 3457, 3713, 3911, 4183, 4393, 4681, 4903, 5207, 5441, 5761, 6007, 6343, 6601, 6953, 7223, 7591, 7873

Offset: 1

Juri-Stepan Gerasimov, Aug 12 2024

a(11) = 391 is first composite number in this sequence.

Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Numbers k such that (m + (16-m)*k) is a square: A204221 (m = 1), this sequence (m = 2), A001082 (m = 4), A181433 (m = 5), A273367 (m = 6), A266956 (m = 7), A056220 (m = 8), A274978 (m = 9), A028872 (m = 12), A161532 (m = 14).

Cf. A113804, A212965.

[k: k in [0..8000] | IsSquare(14*k + 2)];

((Table[14*n + {4, 10}, {n, 0, 23}] // Flatten)^2 - 2)/14 (* Amiram Eldar, Aug 13 2024 *)

a(n) = (A113804(n)^2 - 2)/14. - Amiram Eldar, Aug 13 2024
a(n) = 2*A212965(n-1) - 1. - Hugo Pfoertner, Aug 13 2024
E.g.f.: ((2 + x + 7*x^2)*cosh(x) + (1 - x + 7*x^2)*sinh(x) - 2)/2. - Stefano Spezia, Aug 13 2024

cp's OEIS Frontend

A061298 Table by antidiagonals of rows of sequences where each row is binomial transform of preceding row and row 1 is (1,2,1,2,1,2,1,2,...).

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A131919 A002024 + A131821 - A000012.

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A133475 Integers n such that n^3 + n^2 - 9*n + 16 is a square.

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A375352 Numbers k such that 14*k + 2 is a square.

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