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 A027960 'Lucas array': triangular array T read by rows. 33
 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, 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, 98, 101, 73, 34, 9, 1, 1, 3, 4, 7, 11, 18, 29, 47, 76, 120, 171, 199, 174, 107, 43, 10, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The k-th row contains 2k+1 numbers. Columns in the right half consist of convolutions of the Lucas numbers with the natural numbers. 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. LINKS Nathaniel Johnston, Table of n, a(n) for n = 0..10000 FORMULA 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)}. 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. EXAMPLE 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, 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, 98, 101, 73, 34, 9, 1 MAPLE 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: 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 MATHEMATICA 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 *) PROG (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 */ (Sage) def T(n, k):     if (k<0 or k>2*n): return 0     elif (k==0 or k==2*n): return 1     elif (k==1): return 3     else: return T(n-1, k-2) + T(n-1, k-1) [[T(n, k) for k in (0..2*n)] for n in (0..12)] # G. C. Greubel, Jun 01 2019 CROSSREFS Central column is the Lucas numbers without initial 2, cf. A000204. Row sums are A036563. Columns in the right half include A027961, A027962, A027963, A027964, A053298. Bisection triangles are in A026998 and A027011. Sequence in context: A206496 A097560 A218905 * A319182 A247282 A246685 Adjacent sequences:  A027957 A027958 A027959 * A027961 A027962 A027963 KEYWORD nonn,easy,tabf AUTHOR EXTENSIONS Edited by Ralf Stephan, May 04 2005 STATUS approved

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Last modified October 21 03:24 EDT 2019. Contains 328291 sequences. (Running on oeis4.)