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 A305837 Triangle read by rows: T(0,0) = 1; T(n,k) = 5*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, 5, 25, 1, 125, 10, 625, 75, 1, 3125, 500, 15, 15625, 3125, 150, 1, 78125, 18750, 1250, 20, 390625, 109375, 9375, 250, 1, 1953125, 625000, 65625, 2500, 25, 9765625, 3515625, 437500, 21875, 375, 1, 48828125, 19531250, 2812500, 175000, 4375, 30, 244140625, 107421875, 17578125, 1312500, 43750, 525, 1 (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 A013612 ((1+5*x)^n). The coefficients in the expansion of 1/(1-5x-x^2) are given by the sequence generated by the row sums. If s(n) is the row sum at n, then the ratio s(n)/s(n-1) is approximately 5.1925824035..., a metallic mean (see A098318), when n approaches infinity. REFERENCES Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 92, 380, 382. LINKS Table of n, a(n) for n=0..48. Shara Lalo, Left-justified triangle Shara Lalo, Skew diagonals in triangle A013612 FORMULA G.f.: 1/(1 - 5*t*x - t^2). EXAMPLE Triangle begins: 1; 5; 25, 1; 125, 10; 625, 75, 1; 3125, 500, 15; 15625, 3125, 150, 1; 78125, 18750, 1250, 20; 390625, 109375, 9375, 250, 1; 1953125, 625000, 65625, 2500, 25; 9765625, 3515625, 437500, 21875, 375, 1; 48828125, 19531250, 2812500, 175000, 4375, 30; 244140625, 107421875, 17578125, 1312500, 43750, 525, 1; 1220703125, 585937500, 107421875, 9375000, 393750, 7000, 35; 6103515625, 3173828125, 644531250, 64453125, 3281250, 78750, 700, 1; 30517578125, 17089843750, 3808593750, 429687500, 25781250, 787500, 10500, 40; MATHEMATICA t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, 5 t[n - 1, k] + t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten CROSSREFS Row sums give A052918. Cf. A000351 (column 0), A053464 (column 1), A081135 (column 2), A081143 (column 3), A036071 (column 4). Cf. A013612. Cf. A098318. Sequence in context: A142725 A270590 A271378 * A175555 A346994 A070387 Adjacent sequences: A305834 A305835 A305836 * A305838 A305839 A305840 KEYWORD tabf,nonn,easy AUTHOR Shara Lalo, Jun 11 2018 STATUS approved

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Last modified April 20 07:43 EDT 2024. Contains 371799 sequences. (Running on oeis4.)