A079825 Sum of numbers in n-th upward diagonal of triangle in A079826.

1, 3, 6, 15, 26, 41, 62, 92, 129, 169, 224, 287, 366, 443, 548, 656, 793, 919, 1090, 1255, 1466, 1653, 1906, 2140, 2441, 2701, 3052, 3367, 3774, 4119, 4584, 4992, 5521, 5963, 6558, 7071, 7738, 8289, 9030, 9660, 10481, 11153, 12056, 12815, 13806, 14611, 15692, 16592

Offset: 1

Author

Amarnath Murthy, Feb 11 2003

Keywords

• nonn
• easy

Links

• Bruno Berselli, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1,2,-2,-2,2,-1,1,1,-1).

Crossrefs

Cf. A056011, A079826.

Programs

Magma

m:=35; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+2*x+2*x^2+7*x^3+6*x^4+2*x^5+6*x^6+x^7+x^8)/((x^2+1)^2*(1+x)^3*(x-1)^4))); // Bruno Berselli, Sep 05 2012

Magma

[((2*n+1)*(7*n^2+7*n+12) -3*(-1)^n*n*(n-7) +6*(-1)^Binomial(n,2)*(2*n-3*(-1)^n+1))/96: n in [1..41]]; // G. C. Greubel, Dec 10 2023

Maple

A079825 := proc(n)
        local a, k;
        a := 0 ;
        for k from 1 to (n+1)/2 do
                a := a+A056011(n-k+1,k) ;
        end do:
        a ;
end proc: # R. J. Mathar, Sep 05 2012

Mathematica

LinearRecurrence[{1, 1, -1, 2, -2, -2, 2, -1, 1, 1, -1}, {1, 3, 6, 15, 26, 41, 62, 92, 129, 169, 224}, 35] (* Bruno Berselli, Sep 05 2012 *)

Maxima

makelist(expand(((2*n+1)*(7*n^2+7*n+12)-3*n*(n-7)*(-1)^n+6*(2*n-3*(-1)^n+1)*%i^(n*(n-1)))/96), n, 1, 35); /* Bruno Berselli, Sep 05 2012 */

SageMath

[((2*n+1)*(7*n^2+7*n+12)-3*(-1)^n*n*(n-7)+6*(-1)^binomial( n,2)*(2*n-3*(-1)^n+1))/96 for n in range(1,41)] # G. C. Greubel, Dec 10 2023

Formula

G.f.: x*( 1+2*x+2*x^2+7*x^3+6*x^4+2*x^5+6*x^6+x^7+x^8 ) / ( (1+x)^3*(1-x)^4*(1+x^2)^2 ). - R. J. Mathar, Sep 05 2012

a(n) = ( (2*n+1)*(7*n^2+7*n+12) -3*n*(n-7)*(-1)^n +6*(2*n-3*(-1)^n+1)*i^(n*(n-1)) )/96, where i=sqrt(-1). - Bruno Berselli, Sep 05 2012

E.g.f.: (1/48)*( (6+24*x+30*x^2+7*x^3)*cosh(x) + (6+42*x+33*x^2+7*x^3)* sinh(x) + 6*(x-1)*cos(x) - 6*(x-2)*sin(x) ). - G. C. Greubel, Dec 10 2023