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 A305838 Triangle read by rows: T(0,0)= 1; T(n,k)= T(n-1,k) + 5*T(n-2,k-1) for k = 0..floor(n/2); T(n,k)=0 for n or k < 0. 1
 1, 1, 1, 5, 1, 10, 1, 15, 25, 1, 20, 75, 1, 25, 150, 125, 1, 30, 250, 500, 1, 35, 375, 1250, 625, 1, 40, 525, 2500, 3125, 1, 45, 700, 4375, 9375, 3125, 1, 50, 900, 7000, 21875, 18750, 1, 55, 1125, 10500, 43750, 65625, 15625, 1, 60, 1375, 15000, 78750, 175000, 109375 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The numbers in rows of the triangle are along skew diagonals pointing top-right in center-justified triangle given in A013612 ((1+5*x)^n). The coefficients in the expansion of 1/(1-x-5x^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 2.7913327..., when n approaches infinity. The row sums are A015440 (generalized Fibonacci numbers). REFERENCES Shara Lalo and Zagros Lalo, Polynomial Expansion Theorems and Number Triangles, Zana Publishing, 2018, ISBN: 978-1-9995914-0-3, pp. 70, 72, 380, 381. LINKS Table of n, a(n) for n=0..55. Shara Lalo, Skew diagonals in triangle A013612 Shara Lalo, Right-justified triangle FORMULA G.f.: 1/(1 - t*x - 5*t^2). EXAMPLE Triangle begins: 1; 1; 1, 5; 1, 10; 1, 15, 25; 1, 20, 75; 1, 25, 150, 125; 1, 30, 250, 500; 1, 35, 375, 1250, 625; 1, 40, 525, 2500, 3125; 1, 45, 700, 4375, 9375, 3125; 1, 50, 900, 7000, 21875, 18750; 1, 55, 1125, 10500, 43750, 65625, 15625; 1, 60, 1375, 15000, 78750, 175000, 109375; 1, 65, 1650, 20625, 131250, 393750, 437500, 78125; 1, 70, 1950, 27500, 206250, 787500, 1312500, 625000; 1, 75, 2275, 35750, 309375, 1443750, 3281250, 2812500, 390625; 1, 80, 2625, 45500, 446875, 2475000, 7218750, 9375000, 3515625; MATHEMATICA t[0, 0] = 1; t[n_, k_] := If[n < 0 || k < 0, 0, t[n - 1, k] + 5 t[n - 2, k - 1]]; Table[t[n, k], {n, 0, 12}, {k, 0, Floor[n/2]}] // Flatten CROSSREFS Row sums give A015440. Cf. A000012 (column 0), A008587 (column 1), A123296 (column 2), A141480 (column 3). Cf. A013612. Sequence in context: A308504 A040029 A105086 * A329373 A147362 A147225 Adjacent sequences: A305835 A305836 A305837 * A305839 A305840 A305841 KEYWORD tabf,nonn,easy AUTHOR Shara Lalo, Jun 11 2018 STATUS approved

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Last modified April 20 21:11 EDT 2024. Contains 371845 sequences. (Running on oeis4.)