cp's OEIS Frontend

A304249 Triangle T(n,k) = 3*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1 and T(n,k) = 0 for n < 0 or k < 0, read by rows.

%I A304249 #35 Sep 08 2022 08:46:21
%S A304249 1,3,9,1,27,6,81,27,1,243,108,9,729,405,54,1,2187,1458,270,12,6561,
%T A304249 5103,1215,90,1,19683,17496,5103,540,15,59049,59049,20412,2835,135,1,
%U A304249 177147,196830,78732,13608,945,18,531441,649539,295245,61236,5670,189,1
%N A304249 Triangle T(n,k) = 3*T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2), with T(0,0) = 1 and T(n,k) = 0 for n < 0 or k < 0, read by rows.
%C A304249 The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013610 ((1+3*x)^n), cf. formula.
%C A304249 The coefficients in the expansion of 1/(1-3x-x^2) are given by the sequence generated by the row sums.
%C A304249 The sequence of the row sums are the "Bronze Fibonacci numbers" A006190, and the limit of their ratio is 3.30277563773... (Bronze ratio), see A098316.
%D A304249 Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 86, 363.
%H A304249 Zagros Lalo, <a href="/A304249/a304249.pdf">Left justified triangle</a>
%H A304249 Zagros Lalo, <a href="/A304249/a304249_1.pdf">Skew diagonals in center-justified triangle of coefficients in expansion of (1+3x)^n</a>
%F A304249 T(n,k) = A013610(n-k, n-2k). - _M. F. Hasler_, Jun 01 2018
%e A304249 Triangle begins:
%e A304249           1;
%e A304249           3;
%e A304249           9,         1;
%e A304249          27,         6;
%e A304249          81,        27,         1;
%e A304249         243,       108,         9;
%e A304249         729,       405,        54,        1;
%e A304249        2187,      1458,       270,       12;
%e A304249        6561,      5103,      1215,       90,        1;
%e A304249       19683,     17496,      5103,      540,       15;
%e A304249       59049,     59049,     20412,     2835,      135,       1;
%e A304249      177147,    196830,     78732,    13608,      945,      18;
%e A304249      531441,    649539,    295245,    61236,     5670,     189,      1;
%e A304249     1594323,   2125764,   1082565,   262440,    30618,    1512,     21;
%e A304249     4782969,   6908733,   3897234,  1082565,   153090,   10206,    252,    1;
%e A304249    14348907,  22320522,  13817466,  4330260,   721710,   61236,   2268,   24;
%e A304249    43046721,  71744535,  48361131, 16888014,  3247695,  336798,  17010,  324,  1;
%e A304249   129140163, 229582512, 167403915, 64481508, 14073345, 1732104, 112266, 3240, 27;
%t A304249 T[0, 0] = 1; T[n_, k_]:= If[n<0 || k<0, 0, 3T[n-1, k] + T[n-2, k-1]];
%t A304249 Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}]//Flatten
%t A304249 With[{q=2}, Table[3^(n-q*k)*Binomial[n-(q-1)*k, k], {n,0,24}, {k,0,Floor[n/q]}] ]//Flatten (* _G. C. Greubel_, May 12 2021 *)
%o A304249 (PARI) T(n,k)=if( n>0 && k>0, 3*T(n-1, k) + T(n-2, k-1), !n && !k)
%o A304249 tabf(nn) = for (n=0, nn, for (k=0, n\2, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, May 10 2018
%o A304249 (Magma) [3^(n-2*k)*Binomial(n-k,k): k in [0..Floor(n/2)], n in [0..24]]; // _G. C. Greubel_, May 12 2021
%o A304249 (Sage) flatten([[3^(n-2*k)*binomial(n-k,k) for k in (0..n//2)] for n in (0..24)]) # _G. C. Greubel_, May 12 2021
%Y A304249 Row sums give A006190.
%Y A304249 Cf. A013610, A098316.
%Y A304249 Sequences of the form 3^(n-q*k)*binomial(n-(q-1)*k, k): A027465 (q=1), this sequence (q=2), A317497 (q=3), A318773 (q=4).
%K A304249 tabf,nonn,easy
%O A304249 0,2
%A A304249 _Zagros Lalo_, May 08 2018