A094825 -id:A094825 - cp's OEIS frontend

A191797 a(n) = binomial(F(n), 2) where F(n) = A000045(n).

0, 0, 0, 1, 3, 10, 28, 78, 210, 561, 1485, 3916, 10296, 27028, 70876, 185745, 486591, 1274406, 3337236, 8738290, 22879230, 59901985, 156830905, 410597496, 1074972528, 2814337800, 7368069528, 19289917153, 50501756955, 132215475106, 346144864780, 906219437046

Offset: 0

Emeric Deutsch, Jun 21 2011

			a(7) = binomial(13,2) = 78.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Cf. A000045, A000071, A000217, A001622, A056014, A094825 (binomial transform), A122931.

with(combinat): seq(binomial(fibonacci(n), 2), n = 0 .. 30);

Table[Binomial[Fibonacci[n], 2], {n, 0, 39}] (* Alonso del Arte, Apr 04 2013 *)

a(n) = binomial(fibonacci(n), 2); \\ Michel Marcus, Sep 07 2015

concat(vector(3), Vec(x^3 / ((1+x)*(1-x-x^2)*(1-3*x+x^2)) + O(x^40))) \\ Colin Barker, Mar 26 2017

from sympy import binomial, fibonacci
def a(n): return binomial(fibonacci(n), 2) # Indranil Ghosh, Mar 26 2017

a(n) = 3*a(n-1) + 1*a(n-2) - 5*a(n-3) - 1*a(n-4) + 1*a(n-5).
G.f.: x^3/(1-3*x-x^2+5*x^3+x^4-x^5) = x^3/((1+x)*(1-x-x^2)*(1-3*x+x^2)).
a(n) + a(n+1) = A056014(n+1). - R. J. Mathar, Jun 24 2011
a(n) = (2*F(n)^2 - F(n+4) + 3*F(n+1))/4, F(n) = A000045(n). - Gary Detlefs, Jan 05 2013
a(n) = Sum_{k=1..n-2} A122931(k). - J. M. Bergot, Apr 05 2013
a(n) = A000217(A000071(n)). - Peter M. Chema, Mar 26 2017
a(n) = (2^(-1-n)*(-(-1)^n*2^(1+n) + sqrt(5)*(1-sqrt(5))^n + (3-sqrt(5))^n - sqrt(5)*(1+sqrt(5))^n + (3+sqrt(5))^n)) / 5. - Colin Barker, Mar 26 2017
a(n) ~ phi^(2*n) / 10, where phi is the golden ratio (A001622). - Amiram Eldar, Aug 26 2025

A217593 Square array T, read by antidiagonals: T(n,k) = 0 if n-k >=1 or if k-n >= 9, T(0,k) = 1 for k = 0..8, T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 5, 0, 0, 0, 1, 5, 9, 5, 0, 0, 0, 1, 6, 14, 14, 0, 0, 0, 0, 1, 7, 20, 28, 14, 0, 0, 0, 0, 0, 8, 27, 48, 42, 0, 0, 0, 0, 0, 0, 8, 35, 75, 90, 42, 0, 0, 0, 0, 0, 0, 0, 43, 110, 165, 132, 0, 0, 0, 0, 0, 0, 0, 0, 43, 153, 275, 297, 132, 0, 0, 0, 0, 0, 0, 0, 0, 0, 196, 428, 572, 429, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Mar 18 2013

			Square array begins :
1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 0, 0, ...
0, 0, 2, 5, 9, 14, 20, 27, 35, 43, 43, 0, 0, ...
0, 0, 0, 5, 14, 28, 75, 110, 153, 196, 196, 0, 0, ....
0, 0, 0, 0, 14, 42, 90, 165, 275, 428, 624, 820, 820, 0, 0, ...
...
Square array, read by rows, with 0 omitted:
1, 1, 1, 1, 1, 1, 1, 1, 1
1, 2, 3, 4, 5, 6, 7, 8, 8
2, 5, 9, 14, 20, 27, 35, 43, 43
5, 14, 28, 48, 75, 110, 153, 196, 196
14, 42, 90, 165, 275, 428, 624, 820, 820
42, 132, 297, 572, 1000, 1624, 2444, 3264, 3264
132, 429, 1001, 2001, 3625, 6069, 9333, 12597, 12597
429, 1430, 3431, 7056, 13125, 22458, 35055, 47652, 47652
...

A hexagon arithmetic of E. Lucas.

T(n,n)   = A033191(n).
T(n,n+1) = A033191(n+1).
T(n,n+2) = A033190(n+1).
T(n,n+3) = A094667(n+1).
T(n,n+4) = A093131(n+1) = A030191(n).
T(n,n+5) = A094788(n+2).
T(n,n+6) = A094825(n+3).
T(n,n+7) = T(n,n+8) = A094865(n+3).
Sum_{k, 0<=k<=n} T(n-k,k) = A178381(n).

cp's OEIS Frontend

A191797 a(n) = binomial(F(n), 2) where F(n) = A000045(n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI