A093916 a(2*k-1) = (2*k-1)^2 + 2 - k, a(2*k) = 6*k^2 + 2 - k: First column of the triangle A093915.

2, 7, 9, 24, 24, 53, 47, 94, 78, 147, 117, 212, 164, 289, 219, 378, 282, 479, 353, 592, 432, 717, 519, 854, 614, 1003, 717, 1164, 828, 1337, 947, 1522, 1074, 1719, 1209, 1928, 1352, 2149, 1503, 2382, 1662, 2627, 1829, 2884, 2004, 3153, 2187, 3434, 2378, 3727, 2577, 4032, 2784, 4349, 2999, 4678, 3222, 5019, 3453, 5372

Offset

1,1

Comments

The sequence was initially defined as the first column of the triangle A093915, constructed by trial and error. It is however easy to prove that the sum of the r-th row of A093915, A093917(r), equals twice A006003(r) when r is odd, and three times A006003(r) when r is even. Given the expression of the row sum A093917(r) in terms of the first element a(r), one obtains the explicit formula for a(r). - M. F. Hasler, Apr 04 2009

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Formula

a(n) = ((n^2+1)*b(n) - n + 1)/2 where b(n) = 3 - (n mod 2) = 2 if n odd, = 3 if n even. - M. F. Hasler, Apr 04 2009

From Colin Barker, May 01 2012: (Start)

a(n) = (n*(5*n-2) + (n^2+1)*(-1)^n + 7)/4.

a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6).

G.f.: x*(2+7*x+3*x^2+3*x^3+3*x^4+2*x^5)/((1-x)^3*(1+x)^3). (End)

E.g.f.: (1/4)*( (7 +3*x +5*x^2)*exp(x) - 8 + (1 -x +x^2)*exp(-x) ). - G. C. Greubel, Dec 30 2021

Mathematica

LinearRecurrence[{0, 3, 0, -3, 0, 1}, {2, 7, 9, 24, 24, 53}, 80] (* Harvey P. Dale, Nov 24 2017 *)

Prog

(PARI)
A093916(n)=((n^2+1)*(3-n%2)-n+1)/2
/* or the "experimental" version, trying out all allowed values */
A093916(n)={ local( s=(n^3+n)/2, d=(n^2-n)/2, k=ceil((2*s-d)/n)); while( (n*k+d)%s, k++ ); k } \\ M. F. Hasler, Apr 04 2009
(Magma) [(n*(5*n-2) + (-1)^n*(n^2+1) + 7)/4: n in [1..70]]; // G. C. Greubel, Dec 30 2021
(SageMath) [(5*n^2 -2*n +7 +(-1)^n*(n^2 +1))/4 for n in (1..70)] # G. C. Greubel, Dec 30 2021

Crossrefs

Cf. A093915, A093917, A093918.

Sequence in context: A321322 A343495 A065139 * A042451 A042929 A082962

Adjacent sequences: A093913 A093914 A093915 * A093917 A093918 A093919

Keyword

nonn,easy

Author

Amarnath Murthy, Apr 25 2004

Extensions

Edited and extended by M. F. Hasler, Apr 04 2009

Status

approved