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 A055830 Triangle T read by rows: diagonal differences of triangle A037027. 31
 1, 1, 0, 2, 1, 0, 3, 3, 1, 0, 5, 7, 4, 1, 0, 8, 15, 12, 5, 1, 0, 13, 30, 31, 18, 6, 1, 0, 21, 58, 73, 54, 25, 7, 1, 0, 34, 109, 162, 145, 85, 33, 8, 1, 0, 55, 201, 344, 361, 255, 125, 42, 9, 1, 0, 89, 365, 707, 850, 701, 413, 175, 52, 10, 1, 0, 144, 655, 1416, 1918, 1806, 1239, 630, 236, 63, 11, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Or, coefficients of a generalized Lucas-Pell polynomial read by rows. - Philippe Deléham, Nov 05 2006 Equals A046854(shifted) * Pascal's triangle; where A046854 is shifted down one row and "1" inserted at (0,0). - Gary W. Adamson, Dec 24 2008 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 3D. FORMULA G.f.: (1-y*z) / (1-y*(1+y+z)). T(i, j) = R(i-j, j), where R(0, 0)=1, R(0, j)=0 for j >= 1, R(1, j)=1 for j >= 0, R(i, j) = Sum_{k=0..j} (R(i-2, k) + R(i-1, k)) for i >= 1, j >= 1. Sum_{k=0..n} x^k*T(n,k) = A039834(n-2), A000012(n), A000045(n+1), A001333(n), A003688(n), A015448(n), A015449(n), A015451(n), A015453(n), A015454(n), A015455(n), A015456(n), A015457(n) for x= -2,-1,0,1,2,3,4,5,6,7,8,9,10. - Philippe Deléham, Oct 22 2006 Sum_{k=0..floor(n/2)} T(n-k,k) = A011782(n). - Philippe Deléham, Oct 22 2006 Triangle T(n,k), 0 <= k <= n, given by [1, 1, -1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 05 2006 T(n,0) = Fibonacci(n+1) = A000045(n+1). Sum_{k=0..n} T(n,k) = A001333(n). T(n,k)=0 if k > n or if k < 0, T(0,0)=1, T(1,1)=0, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k). - Philippe Deléham, Nov 05 2006 EXAMPLE Triangle begins: 1 1, 0 2, 1, 0 3, 3, 1, 0 5, 7, 4, 1, 0 8, 15, 12, 5, 1, 0 13, 30, 31, 18, 6, 1, 0 21, 58, 73, 54, 25, 7, 1, 0 34, 109, 162, 145, 85, 33, 8, 1, 0 55, 201, 344, 361, 255, 125, 42, 9, 1, 0 ... MAPLE with(combinat); T:= proc(n, k) option remember; if k<0 or k>n then 0 elif k=0 then fibonacci(n+1) elif n=1 and k=1 then 0 else T(n-1, k-1) + T(n-1, k) + T(n-2, k) fi; end: seq(seq(T(n, k), k=0..n), n=0..12); # G. C. Greubel, Jan 21 2020 MATHEMATICA T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, Fibonacci[n+1], If[n==1 && k==1, 0, T[n-1, k-1] + T[n-1, k] + T[n-2, k]]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 19 2017 *) PROG (PARI) T(n, k) = if(k<0 || k>n, 0, if(k==0, fibonacci(n+1), if(n==1 && k==1, 0, T(n-1, k-1) + T(n-1, k) + T(n-2, k) ))); for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jan 21 2020 (Magma) function T(n, k) if k lt 0 or k gt n then return 0; elif k eq 0 then return Fibonacci(n+1); elif n eq 1 and k eq 1 then return 0; else return T(n-1, k-1) + T(n-1, k) + T(n-2, k); end if; return T; end function; [T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 21 2020 (Sage) @CachedFunction def T(n, k): if (k<0 or k>n): return 0 elif (k==0): return fibonacci(n+1) elif (n==1 and k==1): return 0 else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 21 2020 CROSSREFS Left-hand columns include A000045, A023610. Right-hand columns include A055831, A055832, A055833, A055834, A055835, A055836, A055837, A055838, A055839, A055840. Row sums: A001333 (numerators of continued fraction convergents to sqrt(2)). Cf. A122075 (another version). Cf. A046854. - Gary W. Adamson, Dec 24 2008 Sequence in context: A253829 A107238 A258170 * A293109 A233530 A079123 Adjacent sequences: A055827 A055828 A055829 * A055831 A055832 A055833 KEYWORD nonn,tabl AUTHOR Clark Kimberling, May 28 2000 EXTENSIONS Edited by Ralf Stephan, Jan 12 2005 STATUS approved

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Last modified November 26 20:27 EST 2022. Contains 358362 sequences. (Running on oeis4.)