This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A027960 #32 Jun 16 2025 00:41:50
%S A027960 1,1,3,1,1,3,4,4,1,1,3,4,7,8,5,1,1,3,4,7,11,15,13,6,1,1,3,4,7,11,18,
%T A027960 26,28,19,7,1,1,3,4,7,11,18,29,44,54,47,26,8,1,1,3,4,7,11,18,29,47,73,
%U A027960 98,101,73,34,9,1,1,3,4,7,11,18,29,47,76,120,171,199,174,107,43,10,1
%N A027960 'Lucas array': triangular array T read by rows.
%C A027960 The k-th row contains 2k+1 numbers.
%C A027960 Columns in the right half consist of convolutions of the Lucas numbers with the natural numbers.
%C A027960 T(n,k) = number of strings s(0),...,s(n) such that s(n)=n-k. s(0) in {0,1,2}, s(1)=1 if s(0) in {1,2}, s(1) in {0,1,2} if s(0)=0 and for 1 <= i <= n, s(i) = s(i-1)+d, with d in {0,2} if s(i)=2i, in {0,1,2} if s(i)=2i-1, in {0,1} if 0 <= s(i) <= 2i-2.
%H A027960 Nathaniel Johnston, <a href="/A027960/b027960.txt">Table of n, a(n) for n = 0..10000</a>
%F A027960 T(n, k) = Lucas(k+1) for k <= n, otherwise the (2n-k)th coefficient of the power series for (1+2*x)/{(1-x-x^2)*(1-x)^(k-n)}.
%F A027960 Recurrence: T(n, 0)=T(n, 2n)=1 for n >= 0; T(n, 1)=3 for n >= 1; and for n >= 2, T(n, k) = T(n-1, k-2) + T(n-1, k-1) for 2 <= k <= 2*n-1.
%F A027960 From _G. C. Greubel_, Jun 08 2025: (Start)
%F A027960 T(n, k) = A000032(k+1) - f(n, k)*[k > n], where f(n, k) = Sum_{j=0..k-n-1} binomial(2*n -k+j, j)*A000032(2*(k-n-j)).
%F A027960 Sum_{k=0..A004396(n+1)} T(n-k, k) = A027975(n).
%F A027960 Sum_{k=0..n} T(n, k) = A027961(n).
%F A027960 Sum_{k=0..2*n} T(n, k) = A168616(n+2) + 2.
%F A027960 Sum_{k=n+1..2*n} (-1)^k*T(n, k) = A075193(n-1), n >= 1. (End)
%e A027960 1
%e A027960 1, 3, 1
%e A027960 1, 3, 4, 4, 1
%e A027960 1, 3, 4, 7, 8, 5, 1
%e A027960 1, 3, 4, 7, 11, 15, 13, 6, 1
%e A027960 1, 3, 4, 7, 11, 18, 26, 28, 19, 7, 1
%e A027960 1, 3, 4, 7, 11, 18, 29, 44, 54, 47, 26, 8, 1
%e A027960 1, 3, 4, 7, 11, 18, 29, 47, 73, 98, 101, 73, 34, 9, 1
%p A027960 T:=proc(n,k)option remember:if(k=0 or k=2*n)then return 1:elif(k=1)then return 3:else return T(n-1,k-2) + T(n-1,k-1):fi:end:
%p A027960 for n from 0 to 6 do for k from 0 to 2*n do print(T(n,k));od:od: # _Nathaniel Johnston_, Apr 18 2011
%t A027960 (* First program *)
%t A027960 t[_, 0] = 1; t[_, 1] = 3; t[n_, k_] /; (k == 2*n) = 1; t[n_, k_] := t[n, k] = t[n-1, k-2] + t[n-1, k-1]; Table[t[n, k], {n, 0, 8}, {k, 0, 2*n}] // Flatten (* _Jean-François Alcover_, Dec 27 2013 *)
%t A027960 (* Second program *)
%t A027960 f[n_, k_]:= f[n,k]= Sum[Binomial[2*n-k+j,j]*LucasL[2*(k-n-j)], {j,0,k-n-1}];
%t A027960 A027960[n_, k_]:= LucasL[k+1] - f[n,k]*Boole[k>n];
%t A027960 Table[A027960[n,k], {n,0,12}, {k,0,2*n}]//Flatten (* _G. C. Greubel_, Jun 08 2025 *)
%o A027960 (PARI) T(r,n)=if(r<0||n>2*r,return(0)); if(n==0||n==2*r,return(1)); if(n==1,3,T(r-1,n-1)+T(r-1,n-2)) /* _Ralf Stephan_, May 04 2005 */
%o A027960 (SageMath)
%o A027960 @CachedFunction
%o A027960 def T(n, k): # T = A027960
%o A027960 if (k>2*n): return 0
%o A027960 elif (k<n+1): return lucas_number2(k+1, 1,-1)
%o A027960 else: return T(n-1, k-2) + T(n-1, k-1)
%o A027960 [[T(n, k) for k in (0..2*n)] for n in (0..12)] # _G. C. Greubel_, Jun 01 2019; Jun 08 2025
%o A027960 (Magma)
%o A027960 function T(n,k) // T = A027960
%o A027960 if k le n then return Lucas(k+1);
%o A027960 elif k gt 2*n then return 0;
%o A027960 else return T(n-1, k-2) + T(n-1, k-1);
%o A027960 end if;
%o A027960 end function;
%o A027960 [T(n,k): k in [0..2*n], n in [0..12]]; // _G. C. Greubel_, Jun 08 2025
%Y A027960 Central column is the Lucas numbers without initial 2: A000204.
%Y A027960 Columns in the right half include A027961, A027962, A027963, A027964, A053298.
%Y A027960 Bisection triangles are in A026998 and A027011.
%Y A027960 Row sums: A036563, A153881 (alternating sign).
%Y A027960 Sums include: A023537, A027973, A027974, A027975, A027976, A027978, A027979, A027980, A027981, A027982, A027984, A027985, A027986, A027987, A053298.
%Y A027960 Diagonals of the form T(n, 2*n-p): A000012 (p=0), A000027 (p=1), A034856 (p=2), A027965 (p=3), A027966 (p=4), A027967 (p=5), A027968 (p=6), A027969 (p=7), A027970 (p=8), A027971 (p=9), A027972 (p=10).
%Y A027960 Diagonals of the form T(n, n+p): A000032 (p=0), A027961 (p=1), A023537 (p=2), A027963 (p=3), A027964 (p=4), A053298 (p=5), A027002 U A027018 (p=6), A027007 U A027014 (p=7), A027003 U A027019 (p=8).
%Y A027960 Cf. A000032, A004396, A026998, A027011, A027977, A075193, A168616.
%K A027960 nonn,easy,tabf
%O A027960 0,3
%A A027960 _Clark Kimberling_
%E A027960 Edited by _Ralf Stephan_, May 04 2005