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
Keywords
Examples
a(7) = T(7,1) + T(6,2) + T(5,3) + T(4,4) = 7 + 2 + 3 + 4 = 16.
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).
Programs
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Magma
R
:=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 -
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 *) -
SageMath
def A079810_list(prec): P.
= 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
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)
Extensions
Edited by David Wasserman, May 11 2004