login
Sums of rows of the triangle in A153125.
6

%I #22 Aug 23 2022 09:50:22

%S 1,6,18,33,55,80,112,147,189,234,286,341,403,468,540,615,697,782,874,

%T 969,1071,1176,1288,1403,1525,1650,1782,1917,2059,2204,2356,2511,2673,

%U 2838,3010,3185,3367,3552,3744,3939,4141,4346,4558,4773,4995,5220,5452

%N Sums of rows of the triangle in A153125.

%C Sequence found by reading the line from 1, in the direction 1, 6,..., and the same line from 1, in the direction 1, 18,..., in the square spiral whose edges have length A195013 and whose vertices are the numbers A195014. Line perpendicular to the main axis A195015 in the same spiral. - _Omar E. Pol_, Oct 14 2011

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).

%F a(n) = n*(5*n+7)/2 + 1 - n mod 2.

%F a(n) = Sum_{k=1..n+1} A153125(n+1,k).

%F a(2*n) = A033571(n); a(2*n+1) = A153127(n).

%F a(n) = A000566(n+1) - n mod 2.

%F From _Colin Barker_, Jul 07 2012: (Start)

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

%F G.f.: (1+4*x+6*x^2-x^3)/((1-x)^3*(1+x)). (End)

%F Sum_{n>=0} 1/a(n) = 5/7 + 2*sqrt(1+2/sqrt(5))*Pi/21 + 2*sqrt(5)*log(phi)/21 + 5*log(5)/21 - 8*log(2)/21, where phi is the golden ratio (A001622). - _Amiram Eldar_, Aug 23 2022

%t LinearRecurrence[{2,0,-2,1},{1,6,18,33},50] (* _Harvey P. Dale_, Apr 13 2014 *)

%o (PARI) a(n)=n*(5*n+7)/2 + 1 - n%2 \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A000566, A001622, A033571, A153125, A153127.

%K nonn,easy

%O 0,2

%A _Reinhard Zumkeller_, Dec 20 2008