A079810 Sums of diagonals (upward from left to right) of the triangle shown in A079809.

1, 1, 5, 3, 8, 8, 16, 12, 21, 21, 33, 27, 40, 40, 56, 48, 65, 65, 85, 75, 96, 96, 120, 108, 133, 133, 161, 147, 176, 176, 208, 192, 225, 225, 261, 243, 280, 280, 320, 300, 341, 341, 385, 363, 408, 408, 456, 432, 481, 481, 533, 507, 560, 560, 616, 588, 645, 645

Offset

1,3

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

Formula

a(4k) = 3k^2. a(4k+1) = a(4k+2) = 3k^2+4k+1. a(4k+3) = 3k^2+8k+5.

From Chai Wah Wu, Feb 03 2021: (Start)

a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n > 9.

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

From G. C. Greubel, Dec 12 2023: (Start)

a(n) = (1/32)*( (6*n^2 + 14*n + 5) - (-1)^n*(10*n + 9) + 2*((3 - i)*(-i)^n + (3 + i)*i^n) - 8*(-1)^floor(n/2)*floor((n+2)/2) ).

E.g.f.: 4*(1-x)*cos(x) - 4*(2-x)*sin(x) + 2*(3*x^2 + 15*x - 2)*cosh(x) 2*(3*x^2 + 5*x + 7)*sinh(x). (End)

Example

a(7) = T(7,1) + T(6,2) + T(5,3) + T(4,4) = 7 + 2 + 3 + 4 = 16.

Mathematica

LinearRecurrence[{1, 0, 0, 2, -2, 0, 0, -1, 1}, {1, 1, 5, 3, 8, 8, 16, 12, 21}, 70] (* G. C. Greubel, Dec 12 2023 *)

Prog

(Magma) R<x>:=PowerSeriesRing(Integers(), 71); Coefficients(R!( x*(1+4*x^2-2*x^3+3*x^4)/((1-x)*(1-x^4)^2) )); // G. C. Greubel, Dec 12 2023
(SageMath)
def A079810_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( x*(1+4*x^2-2*x^3+3*x^4)/((1-x)*(1-x^4)^2) ).list()
a=A079810_list(71); a[1:] # G. C. Greubel, Dec 12 2023

Crossrefs

Cf. A079808, A079809, A079811, A092542.

Sequence in context: A010484 A174930 A179452 * A300884 A337440 A092748

Adjacent sequences: A079807 A079808 A079809 * A079811 A079812 A079813

Keyword

nonn

Author

Amarnath Murthy, Feb 10 2003

Extensions

Edited by David Wasserman, May 11 2004

Status

approved