cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A195028 a(n) = n*(14*n + 13).

Original entry on oeis.org

0, 27, 82, 165, 276, 415, 582, 777, 1000, 1251, 1530, 1837, 2172, 2535, 2926, 3345, 3792, 4267, 4770, 5301, 5860, 6447, 7062, 7705, 8376, 9075, 9802, 10557, 11340, 12151, 12990, 13857, 14752, 15675, 16626, 17605, 18612, 19647, 20710, 21801, 22920, 24067, 25242
Offset: 0

Views

Author

Omar E. Pol, Oct 13 2011

Keywords

Comments

Sequence found by reading the line from 0, in the direction 0, 27, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. Numbers opposite to the semi-diagonal A195024 in the same square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].

Links

Crossrefs

Cf. A144555, A152760, A185019, A193053, A195019, A195020, A195021, A195023, A195024, A195025, A195026, A195027, A195320, A198017.

Programs

Formula

a(n) = 14*n^2 + 13*n.
G.f.: x*(27+x)/(1-x)^3. - Colin Barker, Jan 09 2012
From Elmo R. Oliveira, Dec 30 2024: (Start)
E.g.f.: exp(x)*x*(27 + 14*x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)

Extensions

Name suggested by Bruno Berselli, Oct 13 2011

A195029 a(n) = n*(14*n + 13) + 3.

Original entry on oeis.org

3, 30, 85, 168, 279, 418, 585, 780, 1003, 1254, 1533, 1840, 2175, 2538, 2929, 3348, 3795, 4270, 4773, 5304, 5863, 6450, 7065, 7708, 8379, 9078, 9805, 10560, 11343, 12154, 12993, 13860, 14755, 15678, 16629, 17608, 18615, 19650, 20713, 21804, 22923, 24070, 25245
Offset: 0

Views

Author

Omar E. Pol, Sep 07 2011

Keywords

Comments

Sequence found by reading the line from 3, in the direction 3, 30, ..., in the Pythagorean spiral whose edges have length A195019 and whose vertices are the numbers A195020. This is the semi-diagonal parallel to A195024 and also parallel to A195028 in the same square spiral, which is related to the primitive Pythagorean triple [3, 4, 5].
56*a(n) + 1 is a perfect square. - Bruno Berselli, Feb 14 2017

Links

Crossrefs

Bisection of A022264.
Cf. A144555, A152760, A195019, A195020, A195021, A195023, A195024, A195025, A195026, A195027, A195028, A195320.
Cf. A005408, A017017, A185019, A193053, A198017.

Programs

Formula

a(n) = 14*n^2 + 13*n + 3 = A195028(n) + 3 = (2*n + 1)*(7*n + 3).
From Colin Barker, Apr 09 2012: (Start)
G.f.: (3 + 21*x + 4*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
From Elmo R. Oliveira, Dec 29 2024: (Start)
E.g.f.: exp(x)*(3 + 27*x + 14*x^2).
a(n) = A005408(n)*A017017(n) = A022264(2*n+1). (End)

Extensions

Edited by Bruno Berselli, Feb 14 2017
Showing 1-2 of 2 results.

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