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, 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

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