A153126 Sums of rows of the triangle in A153125.

1, 6, 18, 33, 55, 80, 112, 147, 189, 234, 286, 341, 403, 468, 540, 615, 697, 782, 874, 969, 1071, 1176, 1288, 1403, 1525, 1650, 1782, 1917, 2059, 2204, 2356, 2511, 2673, 2838, 3010, 3185, 3367, 3552, 3744, 3939, 4141, 4346, 4558, 4773, 4995, 5220, 5452

Offset: 0

Author

Reinhard Zumkeller, Dec 20 2008

Keywords

• nonn
• easy

Comments

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

Links

• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

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

Programs

Mathematica

LinearRecurrence[{2,0,-2,1},{1,6,18,33},50] (* Harvey P. Dale, Apr 13 2014 *)

PARI

a(n)=n*(5*n+7)/2 + 1 - n%2 \\ Charles R Greathouse IV, Oct 07 2015

Formula

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

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

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

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

From Colin Barker, Jul 07 2012: (Start)

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

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

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