A128619 -id:A128619 - cp's OEIS frontend

A128620 Row sums of A128619.

1, 1, 4, 6, 15, 24, 52, 84, 170, 275, 534, 864, 1631, 2639, 4880, 7896, 14373, 23256, 41810, 67650, 120406, 194821, 343884, 556416, 975325, 1578109, 2749852, 4449354, 7713435, 12480600, 21540304, 34852944, 59917826, 96949079, 166094370, 268746336

Offset: 0

Gary W. Adamson, Mar 14 2007

Diagonals sums of A199512. - Philippe Deléham, Dec 01 2013

			a(5) = 15 = sum of row 5 in A128619: (5 + 0 + 5 + 0 + 5).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,4,-3,-4,1,1).

Cf. A000045, A128619, A199512.

[Floor((n+2)/2)*Fibonacci(n+1): n in [0..40]]; // G. C. Greubel, Mar 15 2024

LinearRecurrence[{1,4,-3,-4,1,1}, {1,1,4,6,15,24}, 40] (* or *)
Table[Floor[(n+2)/2] Fibonacci[n+1], {n, 0, 40}] (* Bruno Berselli, Dec 02 2013 *)

a(n)= ((n+2)\2) * fibonacci(n+1); \\ Michel Marcus, Dec 02 2013

[int((n+2)/2)*fibonacci(n+1) for n in range(41)] # G. C. Greubel, Mar 15 2024

a(n) = floor((n+2)/2)*Fibonacci(n+1). - Philippe Deléham, Dec 01 2013

G.f.: (1 - x^2 + x^3)/((1 + x - x^2)*(1 - x - x^2)^2). - Bruno Berselli, Dec 02 2013

More terms from Philippe Deléham, Dec 01 2013

a(31) corrected from Bruno Berselli, Dec 02 2013

A128618 Triangle read by rows: A128174 * A127647 as infinite lower triangular matrices.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 2, 0, 5, 0, 1, 0, 3, 0, 8, 1, 0, 2, 0, 5, 0, 13, 0, 1, 0, 3, 0, 8, 0, 21, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 1, 0, 3, 0, 8, 0, 21, 0, 55, 0, 144, 1, 0, 2, 0, 5, 0, 13, 0, 34, 0, 89, 0, 233

Offset: 1

Gary W. Adamson, Mar 14 2007

This triangle is different from A128619, which is A128619 = A127647 * A128174.

			First few rows of the triangle are:
  1;
  0, 1;
  1, 0, 2;
  0, 1, 0, 3;
  1, 0, 2, 0, 5;
  0, 1, 0, 3, 0, 8;
  1, 0, 2, 0, 5, 0, 13;
  0, 1, 0, 3, 0, 8,  0, 21;
  1, 0, 2, 0, 5, 0, 13,  0, 34;
  0, 1, 0, 3, 0, 8,  0, 21,  0, 55;
  1, 0, 2, 0, 5, 0, 13,  0, 34,  0, 89;
  ...

G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Cf. A000045, A052952, A127647, A128174, A128619, A355020.

[((n+k+1) mod 2)*Fibonacci(k): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 17 2024

Table[Fibonacci[k]*Mod[n-k+1,2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Mar 17 2024 *)

flatten([[((n-k+1)%2)*fibonacci(k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Mar 17 2024

By columns, Fibonacci(k) interspersed with alternate zeros in every column, k=1,2,3,...
Sum_{k=1..n} T(n, k) = A052952(n-1) (row sums).
From G. C. Greubel, Mar 17 2024: (Start)
T(n, k) = (1/2)*(1 + (-1)^(n+k))*Fibonacci(k).
T(n, n) = A000045(n).
T(2*n-1, n) = (1/2)*(1-(-1)^n)*A000045(n).
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A052952(n-1).
Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1 - (-1)^n)*(Fibonacci((n+ 5)/2) - 1).
Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n) *  A355020(floor((n-1)/2)). (End)

a(6) corrected and more terms from Georg Fischer, May 30 2023

cp's OEIS Frontend

A128620 Row sums of A128619.

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