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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
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
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.)