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 A067990 Triangle A067979 with rows read backwards. 12
 1, 6, 3, 17, 13, 4, 38, 31, 19, 7, 80, 69, 48, 32, 11, 158, 140, 107, 79, 51, 18, 303, 274, 220, 176, 127, 83, 29, 566, 519, 432, 360, 283, 206, 134, 47, 1039, 963, 822, 706, 580, 459, 333, 217, 76, 1880, 1757, 1529, 1341, 1138, 940, 742, 539, 351, 123, 3364, 3165, 2796, 2492, 2163, 1844, 1520, 1201 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The column m (without leading 0's) gives the convolution of Lucas numbers {L(n+1) := A000204(n+1)}, n>=0, with those with m-shifted index: a(n+m,m)=sum(L(k+1)*L(m+n+1-k),k=0..n), n>=0,m=0,1,... The columns give A004799(n-1), A067980-7 for m= 0..8, respectively. Row sums give A067989. The row polynomials p(n,x) := sum(a(n,m)*x^m,m=0..n) are generated by A(z)*(A(z)-x*A(x*z))/(1-x), with A(x) := (1+2*x)/(1-x-x^2) (g.f. for Lucas {L(n+1)}). LINKS Michael De Vlieger, Table of n, a(n) for n = 0..10000 FORMULA a(n, m)=A067330(n, n-m), n>=m>=0, else 0. a(n, m)=(n-m+1)*L(m+1)*F(n-m)+((n-m+1)*L(m+1)+(n-m)*L(m))*F(n-m+1), n>=m>=0, else 0; with F(n) := A000045(n)(Fibonacci) and L(n) := A000032(n) (Lucas). G.f. for column m=0, 1, ...: (x^m)*(L(m+1)+L(m)*x)*(1+2*x)/(1-x-x^2)^2. a(n, m) = -(-1)^m*F(n-2*m+1)-m*L(n+2)+n*L(n+2)+F(n+3), with F(-n) = (-1)^(n+1)*F(n), hence a(n, m) = -5*A067418(n, m)+2*(n-m+1)*L(n+2), n>=m>=0. - Ehren Metcalfe, Apr 11 2016 EXAMPLE {1}; {6,3}; {17,13,4}; {38,31,19,7}; ...; p(2,x)=17+13*x+4*x^2. MATHEMATICA Reverse /@ Table[Sum[LucasL[k + 1] LucasL[n - k + 1], {k, 0, m}], {n, 0, 11}, {m, 0, n}] // Flatten (* Michael De Vlieger, Apr 11 2016 *) CROSSREFS Sequence in context: A328011 A231881 A229005 * A174012 A050008 A166450 Adjacent sequences: A067987 A067988 A067989 * A067991 A067992 A067993 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Feb 15 2002 STATUS approved

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Last modified June 19 18:58 EDT 2024. Contains 373507 sequences. (Running on oeis4.)