A195309 Row sums of an irregular triangle read by rows in which row n lists the next A026741(n+1) natural numbers A000027.

1, 9, 11, 45, 39, 126, 94, 270, 185, 495, 321, 819, 511, 1260, 764, 1836, 1089, 2565, 1495, 3465, 1991, 4554, 2586, 5850, 3289, 7371, 4109, 9135, 5055, 11160, 6136, 13464, 7361, 16065, 8739, 18981, 10279, 22230, 11990, 25830, 13881

Offset: 1

Author

Omar E. Pol, Sep 21 2011

Keywords

• nonn

Comments

The integers in same rows of the source triangle have a property related to Euler's Pentagonal Theorem.

Note that the column 1 of the mentioned triangle gives the positive terms of A001318 (see example).

Examples

			a(1) = 1
a(2) = 2+3+4 = 9
a(3) = 5+6 = 11
a(4) = 7+8+9+10+11 = 45
a(5) = 12+13+14 = 39
a(6) = 15+16+17+18+19+20+21 = 126
a(7) = 22+23+24+25 = 94
a(8) = 26+27+28+29+30+31+32+33+34 = 270
a(9) = 35+36+37+38+39 = 185
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Crossrefs

Cf. A026741, A195310, A195311, A004188 (bisection).

Programs

Maple

A195309 := proc(n)
        (n+1)*(9*n^2+18*n-1+(3*n^2+6*n+1)*(-1)^n)/32
end proc:
seq(A195309(n),n=1..60) ; # R. J. Mathar, Oct 08 2011

Mathematica

LinearRecurrence[{0,4,0,-6,0,4,0,-1},{1,9,11,45,39,126,94,270},80] (* Harvey P. Dale, Jun 22 2015 *)

Formula

a(n) = (n+1)*(9*n^2+18*n-1+(3*n^2+6*n+1)*(-1)^n)/32 . - R. J. Mathar, Oct 08 2011

G.f. x*(1+9*x+7*x^2+9*x^3+x^4) / ( (x-1)^4*(1+x)^4 ). - R. J. Mathar, Oct 08 2011