A294774 a(n) = 2*n^2 + 2*n + 5.

5, 9, 17, 29, 45, 65, 89, 117, 149, 185, 225, 269, 317, 369, 425, 485, 549, 617, 689, 765, 845, 929, 1017, 1109, 1205, 1305, 1409, 1517, 1629, 1745, 1865, 1989, 2117, 2249, 2385, 2525, 2669, 2817, 2969, 3125, 3285, 3449, 3617, 3789, 3965, 4145, 4329, 4517, 4709, 4905

Offset

0,1

Comments

This is the case k = 9 of 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 (similar sequences are listed in Crossrefs section). Note that:

2*( 2*n^2 + (1-(-1)^k)*n + (2*k-(-1)^k+1)/4 ) - k = ( 2*n + (1-(-1)^k)/2 )^2. From this follows an alternative definition for the sequence: Numbers h such that 2*h - 9 is a square. Therefore, if a(n) is a square then its base is a term of A075841.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

O.g.f.: (5 - 6*x + 5*x^2)/(1 - x)^3.

E.g.f.: (5 + 4*x + 2*x^2)*exp(x).

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

a(n) = 5*A000217(n+1) - 6*A000217(n) + 5*A000217(n-1).

n*a(n) - Sum_{j=0..n-1} a(j) = A002492(n) for n>0.

a(n) = Integral_{x=0..2n+4} |3-x| dx. - Pedro Caceres, Dec 29 2020

Maple

seq(2*n^2 + 2*n + 5, n=0..100); # Robert Israel, Nov 10 2017

Mathematica

Table[2n^2+2n+5, {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {5, 9, 17}, 50] (* Harvey P. Dale, Sep 18 2023 *)

Prog

(PARI) Vec((5 - 6*x + 5*x^2) / (1 - x)^3 + O(x^50)) \\ Colin Barker, Nov 13 2017

Crossrefs

1st diagonal of A154631, 3rd diagonal of A055096, 4th diagonal of A070216.

Second column of Mathar's array in A016813 (Comments section).

Subsequence of A001481, A001983, A004766, A020668, A046711 and A057653 (because a(n) = (n+2)^2 + (n-1)^2); A097268 (because it is also a(n) = (n^2+n+3)^2 - (n^2+n+2)^2); A047270; A243182 (for y=1).

Similar sequences (see the first comment): A161532 (k=-14), A181510 (k=-13), A152811 (k=-12), A222182 (k=-11), A271625 (k=-10), A139570 (k=-9), (-1)*A147973 (k=-8), A059993 (k=-7), A268581 (k=-6), A090288 (k=-5), A054000 (k=-4), A142463 or A132209 (k=-3), A056220 (k=-2), A046092 (k=-1), A001105 (k=0), A001844 (k=1), A058331 (k=2), A051890 (k=3), A271624 (k=4), A097080 (k=5), A093328 (k=6), A271649 (k=7), A255843 (k=8), this sequence (k=9).

Sequence in context: A175543 A200078 A190806 * A192746 A081295 A180565

Adjacent sequences: A294771 A294772 A294773 * A294775 A294776 A294777

Keyword

nonn,easy

Author

Bruno Berselli, Nov 08 2017

Status

approved