A320259 - OEIS

%I #30 Nov 14 2018 07:19:48

%S 0,2,5,9,15,22,30,40,51,63,77,92,108,126,145,165,187,210,234,260,287,

%T 315,345,376,408,442,477,513,551,590,630,672,715,759,805,852,900,950,

%U 1001,1053,1107,1162,1218,1276,1335

%N Terms that are on the y-axis of the square spiral built with 2*k, 2*k+1, 2*k+1 for k >= 0.

%C a(n) mod 9 is of period 27.

%C The spiral:

%C      28--29--29--30--31--31--32

%C       |

%C      27  13--14--15--15--16--17

%C       |   |                   |

%C      27  13   4---5---5---6  17

%C       |   |   |           |   |

%C      26  12   3   0---1   7  18

%C       |   |   |       |   |   |

%C      25  11   3---2---1   7  19

%C       |   |               |   |

%C      25  11--10---9---9---8  19

%C       |                       |

%C      24--23--23--22--21--21--20

%H Colin Barker, <a href="/A320259/b320259.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1,1,-2,1).

%F a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5), a(0)=0, a(1)=2, a(2)=5, a(3)=9, a(4)=15.

%F a(n) = a(n-1) + A004772(n+1), a(0)=0, n>0.

%F a(n+15) = a(n-15) + 10*A004767(n).

%F a(-n-1) = ({0} U A000969(n)) = 0, 1, 3, 7, ... = b(n), the full x-axis terms.

%F a(-n-1) + a(n) = 0, 3, 8, 16, ... = A211480(n+1).

%F a(n) = b(n) + A004523(n+1).

%F G.f.: x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)). - _Colin Barker_, Oct 08 2018

%F a(n) = A143978(n) + A002264(n+2).

%F a(n) = A000969(-2-n) for all n in Z. - _Michael Somos_, Nov 13 2018

%e G.f. = 2*x + 5*x^2 + 9*x^3 + 15*x^4 + 22*x^5 + 30*x^6 + ... - _Michael Somos_, Nov 13 2018

%p seq(coeff(series(x^2*(2+x+x^2)/((1-x)^3*(1+x+x^2)),x,n+1), x, n), n = 1 .. 50); # _Muniru A Asiru_, Oct 08 2018

%t LinearRecurrence[{2,-1,1,-2,1}, {0, 2, 5, 9, 15}, 50] (* or *)

%t CoefficientList[Series[x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)), {x, 0, 50}], x] (* _Stefano Spezia_, Oct 09 2018 *)

%t a[ n_] := Quotient[(n + 1) (2 n + 1), 3]; (* _Michael Somos_, Nov 13 2018 *)

%o (PARI) concat(0, Vec(x*(2 + x + x^2) / ((1 - x)^3*(1 + x + x^2)) + O(x^60))) \\ _Colin Barker_, Oct 08 2018

%o (PARI) {a(n) = (n + 1) * (2*n + 1) \ 3}; /* _Michael Somos_, Nov 13 2018 */

%o (GAP) a:=[0,2,5,9,15];; for n in [6..50] do a[n]:=2*a[n-1]-a[n-2]+a[n-3]-2*a[n-4]+a[n-5]; od; a; # _Muniru A Asiru_, Oct 08 2018

%Y Cf. A000969, A004396, A004523, A004767, A004772 (first differences), A211480, A002264, A143978.

%K nonn

%O 0,2

%A _Paul Curtz_, Oct 08 2018