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 A137931 Sum of the principal diagonals of a 2n X 2n square spiral. 5

%I #8 May 06 2019 05:22:15

%S 0,10,56,170,384,730,1240,1946,2880,4074,5560,7370,9536,12090,15064,

%T 18490,22400,26826,31800,37354,43520,50330,57816,66010,74944,84650,

%U 95160,106506,118720,131834,145880,160890,176896,193930,212024,231210,251520,272986,295640

%N Sum of the principal diagonals of a 2n X 2n square spiral.

%C This is concerned with 2n X 2n square spirals of the form illustrated in the Example section.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = -1 + n + Sum_{k=0..2n} (2k^2 - k + 1) = n -1 +(2*n+1)*(8*n^2-n+3)/3.

%F a(n) = 2*n^2 + 2*n + (16*n^3 + 2*n)/3 = 2*n*(8*n^2+3*n+4)/3.

%F G.f.: 2*x*(3*x+5)*(x+1)/(x-1)^4. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009

%e Example with n = 2:

%e .

%e 7---8---9--10

%e | |

%e 6 1---2 11

%e | | |

%e 5---4---3 12

%e |

%e 16--15--14--13

%e .

%e a(0) = 2(0)^2 + 2(0) + (16(0)^3 + 2(0))/3 = 0;

%e a(2) = 2(2)^2 + 2(2) + (16(2)^3 + 2(2))/3 = 56.

%o (Python) f = lambda n: -1 + n + sum(2*k**2 - k + 1 for k in range(0,2*n+1))

%o (Python) a = lambda n: 2*n**2 + 2*n + (16*n**3 + 2*n)/3

%Y Cf. A137928, A002061. A bisection of A137930.

%K nonn,easy

%O 0,2

%A _William A. Tedeschi_, Feb 29 2008

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Last modified June 9 13:49 EDT 2023. Contains 363180 sequences. (Running on oeis4.)