A213590 Rectangular array: (row n) = b**c, where b(h) = h^2, c(h) = F(n-1+h), F = A000045 (Fibonacci numbers), n>=1, h>=1, and ** = convolution.

1, 5, 1, 15, 6, 2, 36, 20, 11, 3, 76, 51, 35, 17, 5, 148, 112, 87, 55, 28, 8, 273, 224, 188, 138, 90, 45, 13, 485, 421, 372, 300, 225, 145, 73, 21, 839, 758, 694, 596, 488, 363, 235, 118, 34, 1424, 1324, 1243, 1115, 968, 788, 588, 380, 191, 55, 2384, 2263, 2163, 2001, 1809, 1564, 1276, 951, 615, 309, 89

Offset

1,2

Comments

Principal diagonal: A213504.

Antidiagonal sums: A213557.

Row 1, (1,4,9,16,...)**(1,1,2,3,5,...): A053808.

Row 2, (1,4,9,16,...)**(1,2,3,5,8,...): A213586.

Row 3, (1,4,9,16,...)**(2,3,5,8,13,...).

For a guide to related arrays, see A213500.

Links

Clark Kimberling, Antidiagonals n = 1..60, flattened

Formula

Rows: T(n,k) = 4*T(n,k-1) -5*T(n,k-2) +*T(n,k-3) +2*T(n,k-4) -T(n,k-5).

Columns: T(n,k) = T(n-1,k) + T(n-2,k).

G.f. for row n: f(x)/g(x), where f(x) = F(n) + F(n+1)*x + F(n-1)*x^2 and g(x) = (1 - x - x^2)*(1 - x )^3.

T(n, k) = Fibonacci(n+k+6) - Fibonacci(n+6) - 2*k*Fibonacci(n+3) - k^2*Fibonacci(n+1). - G. C. Greubel, Jul 05 2019

Example

Northwest corner (the array is read by falling antidiagonals):
1....5....15....36....76.....148
1....6....20....51....112....224
2....11...35....87....188....372
3....17...55....138...300....596
5....28...90....225...488....868
8....45...145...363...788....1564
13...73...235...588...1276...2532

Mathematica

(* First program *)
b[n_]:= n^2; c[n_]:= Fibonacci[n];
T[n_, k_]:= Sum[b[k-i] c[n+i], {i, 0, k-1}]
TableForm[Table[T[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[T[n-k+1, k], {n, 12}, {k, n, 1, -1}]] (* A213590 *)
r[n_]:= Table[T[n, k], {k, 40}]  (* columns of antidiagonal triangle *)
Table[T[n, n], {n, 1, 40}] (* A213504 *)
s[n_]:= Sum[T[i, n+1-i], {i, 1, n}]
Table[s[n], {n, 1, 50}] (* A213557 *)
(* Second program *)
t[n_, k_]:= Fibonacci[n+7] - Fibonacci[k+6] - 2*(n-k+1)*Fibonacci[k+3] - (n-k+1)^2*Fibonacci[k+1]; Table[t[n, k], {n, 1, 12}, {k, 1, n}]//Flatten (* G. C. Greubel, Jul 05 2019 *)

Prog

(PARI) f=fibonacci; t(n, k) = f(n+7) -f(k+6) -2*(n-k+1)*f(k+3) -(n-k+1)^2 *f(k+1);
for(n=1, 12, for(k=1, n, print1(t(n, k), ", "))) \\ G. C. Greubel, Jul 05 2019
(Magma) F:=Fibonacci; [[F(n+7) -F(k+6) -2*(n-k+1)*F(k+3) -(n-k+1)^2 *F(k+1): k in [1..n]]: n in [1..12]]; // G. C. Greubel, Jul 05 2019
(Sage) f=fibonacci; [[f(n+7) -f(k+6) -2*(n-k+1)*f(k+3) - (n-k+1)^2* f(k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Jul 05 2019
(GAP) F:=Fibonacci;; Flat(List([1..12], n-> List([1..n], k-> F(n+7)-F(k+6) -2*(n-k+1)*F(k+3)-(n-k+1)^2*F(k+1) ))) # G. C. Greubel, Jul 05 2019

Crossrefs

Cf. A213500, A213587.

Sequence in context: A295574 A087727 A039807 * A185263 A264616 A157395

Adjacent sequences: A213587 A213588 A213589 * A213591 A213592 A213593

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Jun 19 2012

Status

approved