A271625 a(n) = = 2*(n+1)^2 - 5.

3, 13, 27, 45, 67, 93, 123, 157, 195, 237, 283, 333, 387, 445, 507, 573, 643, 717, 795, 877, 963, 1053, 1147, 1245, 1347, 1453, 1563, 1677, 1795, 1917, 2043, 2173, 2307, 2445, 2587, 2733, 2883, 3037, 3195, 3357, 3523, 3693, 3867, 4045, 4227, 4413, 4603, 4797, 4995, 5197, 5403, 5613, 5827

Offset: 1

Author

Juri-Stepan Gerasimov, Apr 11 2016

Keywords

• nonn
• easy

Comments

Numbers n such that 2*n + 10 is a perfect square.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000290, A201713.

Numbers h such that 2*h + k is a perfect square: A294774 (k=-9), A255843 (k=-8), A271649 (k=-7), A093328 (k=-6), A097080 (k=-5), A271624 (k=-4), A051890 (k=-3), A058331 (k=-2), A001844 (k=-1), A001105 (k=0), A046092 (k=1), A056222 (k=2), A142463 (k=3), A054000 (k=4), A090288 (k=5), A268581 (k=6), A059993 (k=7), (-1)*A147973 (k=8), A139570 (k=9), this sequence (k=10), A222182 (k=11), A152811 (k=12), A181510 (k=13), A161532 (k=14), no sequence (k=15).

Programs

Magma

[ 2*n^2 + 4*n - 3: n in [1..60]];

Magma

[ n: n in [1..6000] | IsSquare(2*n+10)];

Mathematica

Table[2 n^2 + 4 n - 3, {n, 53}] (* Michael De Vlieger, Apr 11 2016 *)
LinearRecurrence[{3,-3,1},{3,13,27},60] (* Harvey P. Dale, Jun 08 2023 *)
2*Range[2,60]^2 -5 (* G. C. Greubel, Jan 21 2025 *)

PARI

x='x+O('x^99); Vec(x*(3+4*x-3*x^2)/(1-x)^3) \\ Altug Alkan, Apr 11 2016

Python

def A271625(n): return 2*pow(n+1,2) - 5
print([A271625(n) for n in range(1,61)]) # G. C. Greubel, Jan 21 2025

Formula

G.f.: x*(3 + 4*x - 3*x^2)/(1 - x)^3. - Ilya Gutkovskiy, Apr 11 2016

Sum_{n>=1} 1/a(n) = 13/30 - Pi*cot(sqrt(5/2)*Pi)/(2*sqrt(10)) = 0.5627678459924... . - Vaclav Kotesovec, Apr 11 2016

From Elmo R. Oliveira, Nov 17 2024: (Start)

E.g.f.: exp(x)*(2*x^2 + 6*x - 3) + 3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

a(n) = 2*A000290(n+1) - 5. - G. C. Greubel, Jan 21 2025

Extensions

Name simplified by G. C. Greubel, Jan 21 2025