A137930 - OEIS

%I #13 Jun 26 2019 03:04:58

%S 0,1,10,25,56,101,170,261,384,537,730,961,1240,1565,1946,2381,2880,

%T 3441,4074,4777,5560,6421,7370,8405,9536,10761,12090,13521,15064,

%U 16717,18490,20381,22400,24545,26826,29241,31800,34501,37354,40357,43520,46841,50330

%N The sum of the principal diagonals of an n X n spiral.

%C n X n spirals of the form:

%C (Examples of n = 3, 4)

%C 7...8...9

%C 6...1...2

%C 5...4...3

%C and

%C 7...8...9...10

%C 6...1...2...11

%C 5...4...3...12

%C 16..15..14..13

%H Robert Israel, <a href="/A137930/b137930.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = mod(n^(n+1),n+1) + floor(n/2)^2*(6-4(-1)^n) + [16*floor(n/2)^3 + floor(n/2)*(14-12(-1)^n)]/3

%F Interweave A114254 and A137931.

%F Empirical G.f.: x*(1+7*x-3*x^2+3*x^3)/((1-x)^4*(1+x)). [_Colin Barker_, Jan 12 2012]

%F From _Robert Israel_, Jun 25 2019: (Start)

%F Empirical G.f. confirmed using G.f.'s of A114254 and A137931.

%F a(n) = 2*n^3/3 + n^2/2 + 4*n/3 + 3*((-1)^n -1)/4. (End)

%e a(1) = mod(1^(1+1),1+1) + floor(1/2)^2*(6-4(-1)^1) + [16*floor(1/2)^3 + floor(1/2)*(14-12(-1)^1)]/3 = 1

%e a(2) = mod(2^(2+1),2+1) + floor(2/2)^2*(6-4(-1)^2) + [16*floor(2/2)^3 + floor(2/2)*(14-12(-1)^2)]/3 = 10

%p f:= n -> 2*n^3/3 + n^2/2 + 4*n/3 + 3*((-1)^n -1)/4:

%p map(f, [$0..100]); # _Robert Israel_, Jun 25 2019

%Y Cf. A114254, A137931.

%K nonn

%O 0,3

%A _William A. Tedeschi_, Feb 29 2008