A075117 - OEIS

%I #8 Sep 08 2022 08:45:07

%S 2,1,2,1,1,2,1,3,1,2,1,4,5,1,2,1,7,7,7,1,2,1,11,17,10,9,1,2,1,18,31,

%T 31,13,11,1,2,1,29,65,61,49,16,13,1,2,1,47,127,154,101,71,19,15,1,2,1,

%U 76,257,337,297,151,97,22,17,1,2,1,123,511,799,701,506,211,127,25,19,1,2

%N Table by antidiagonals of generalized Lucas numbers: T(n,k) = T(n,k-1) + n*T(n,k-2) with T(n,0)=2 and T(n,1)=1.

%H G. C. Greubel, <a href="/A075117/b075117.txt">Antidiagonals n = 0..100, flattened</a>

%F T(n, k) = ((1+sqrt(4*n+1))/2)^k + ((1-sqrt(4*n+1))/2)^k = 2*A060959(n, k+1) - A060959(n, k).

%F T(n, k) = 2^(1-k)*Sum_{j=0..floor(k/2)} binomial(k, 2*j)*(1+4*n)^j. - _G. C. Greubel_, Jan 27 2020

%e Array starts as:

%e   2, 1,  1,  1,  1,   1, ...;

%e   2, 1,  3,  4,  7,  11, ...;

%e   2, 1,  5,  7, 17,  31, ...;

%e   2, 1,  7, 10, 31,  61, ...;

%e   2, 1,  9, 13, 49, 101, ...;

%e   2, 1, 11, 16, 71, 151, ...; etc.

%p seq(seq( 2^(1+k-n)*add( binomial(n-k, 2*j)*(1+4*k)^j, j=0..floor((n-k)/2)), k=0..n), n=0..13); # _G. C. Greubel_, Jan 27 2020

%t T[n_, k_]:= ((1 + Sqrt[1+4n])/2)^k + ((1 - Sqrt[1+4n])/2)^k; Table[If[n==0 && k==0, 2, T[k, n-k]]//Simplify, {n,0,13}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 27 2020 *)

%o (Magma) [2^(1+k-n)*(&+[Binomial(n-k,2*j)*(1+4*k)^j: j in [0..Floor((n-k)/2)]]): k in [0..n], n in [0..13]]; // _G. C. Greubel_, Jan 27 2020

%o (Sage)

%o def T(n, k): return 2^(1-k)*sum( binomial(k, 2*j)*(1+4*n)^j for j in (0..floor(k/2)) )

%o [[T(k,n-k) for k in (0..n)] for n in (0..13)] # _G. C. Greubel_, Jan 27 2020

%Y Cf. A060959.

%Y Rows include: A054977, A000032, A014551, A075118, A072265.

%Y Columns include: A007395, A000012, A005408, A016777, A056220, A062786.

%K nonn,tabl

%O 0,1

%A _Henry Bottomley_, Sep 02 2002