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 A317055 Triangle read by rows: T(0,0) = 1; T(n,k) = 10 T(n-1,k) + T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. 2
 1, 10, 100, 1, 1000, 20, 10000, 300, 1, 100000, 4000, 30, 1000000, 50000, 600, 1, 10000000, 600000, 10000, 40, 100000000, 7000000, 150000, 1000, 1, 1000000000, 80000000, 2100000, 20000, 50, 10000000000, 900000000, 28000000, 350000, 1500, 1, 100000000000, 10000000000, 360000000, 5600000, 35000, 60 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The numbers in rows of the triangle are along skew diagonals pointing top-left in center-justified triangle given in A013617 ((1+10*x)^n) and  along skew diagonals pointing top-right in center-justified triangle given in A038303 ((10+x)^n). The coefficients in the expansion of 1/(1-10x-x^2) are given by the sequence generated by the row sums. The row sums are Denominators of continued fraction convergents to sqrt(26), see A041041. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 10.09901951359278483002... (a metallic mean) when n approaches infinity (see A176537: (5+sqrt(26))). REFERENCES Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 102 LINKS Zagros Lalo, Left-justified triangle EXAMPLE Triangle begins: 1; 10; 100, 1; 1000, 20; 10000, 300, 1; 100000, 4000, 30; 1000000, 50000, 600, 1; 10000000, 600000, 10000, 40; 100000000, 7000000, 150000, 1000, 1; 1000000000, 80000000, 2100000, 20000, 50; 10000000000, 900000000, 28000000, 350000, 1500, 1; 100000000000, 10000000000, 360000000, 5600000, 35000, 60; 1000000000000, 110000000000, 4500000000, 84000000, 700000, 2100, 1; 10000000000000, 1200000000000, 55000000000, 1200000000, 12600000, 56000, 70; 100000000000000, 13000000000000, 660000000000, 16500000000, 210000000, 1260000, 2800, 1; MATHEMATICA t[0, 0] = 1; t[n_, k_] := t[n, k] = If[n < 0 || k < 0, 0, 10 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 11}, {k, 0, Floor[n/2]}] // Flatten. CROSSREFS Row sums give A041041. Cf. A013617 Cf. A038303 Cf. A176537 Cf. A011557 (column 0), A053541 (column 1), A081140 (column 2). Sequence in context: A187019 A292429 A145644 * A284200 A228410 A119589 Adjacent sequences:  A317052 A317053 A317054 * A317056 A317057 A317058 KEYWORD tabf,nonn,easy AUTHOR Zagros Lalo, Jul 21 2018 STATUS approved

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Last modified July 12 15:09 EDT 2020. Contains 335665 sequences. (Running on oeis4.)