A285551 - OEIS

A285551

Volume of each square prism building the next 3-dimensional box in A100538 where side lengths form the Padovan spiral number sequence (A134816), starting with 1 X 1 X 1, 1 X 1 X 2, 2 X 2 X 2, 2 X 2 X 3, 4 X 4 X 5, ...

1, 2, 8, 12, 36, 80, 175, 441, 972, 2304, 5376, 12348, 29008, 67081, 156065, 363350, 843144, 1962396, 4560200, 10600000, 24648975, 57288465, 133194600, 309636096, 719790336, 1673379352, 3890033728, 9043304417, 21023197601, 48872682810, 113615800200, 264124052396

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Robert Dickau, Padovan's Spiral Numbers, Wolfram Demonstrations Project Published: August 19, 2010.

Index entries for linear recurrences with constant coefficients, signature (1,2,3,-2,4,-4,-1,-1,0,-1).

FORMULA

a(n) = A000931(n+5)^2*A000931(n+6).

a(n) = A100538(n+1) - A100538(n).

From Elmo R. Oliveira, May 12 2026: (Start)

a(n) = a(n-1) + 2*a(n-2) + 3*a(n-3) - 2*a(n-4) + 4*a(n-5) - 4*a(n-6) - a(n-7) - a(n-8) - a(n-10).

G.f.: x*(1-x)*(1+2*x+6*x^2+3*x^3+7*x^4+3*x^5+2*x^6+x^7+x^8) / ((-1+x) * (-1+3*x-2*x^2+x^3) * (1+3*x+5*x^2+5*x^3+5*x^4+3*x^5+x^6)). (End)

MATHEMATICA

A[n_]:=Sum[Binomial[k, n - 2k], {k, 0, Floor[n/2]}]; a000931[n_]:=If[n==0, 1, If[n<3, 0, A[n - 3]]]; a[n_]:=a000931[n + 5]^2*a000931[n + 6]; Table[a[n], {n, 0, 50}] (* Indranil Ghosh, Apr 26 2017 *)

(* Alternative: *)

LinearRecurrence[{1, 2, 3, -2, 4, -4, -1, -1, 0, -1}, {1, 2, 8, 12, 36, 80, 175, 441, 972, 2304}, 40] (* Vincenzo Librandi, Jul 19 2017 *)

PROG

(PARI) A(n) = sum(k=0, n\2, binomial(k, n - 2*k));

a000931(n) = if(n==0, 1, if(n<3, 0, A(n - 3)));

a(n) = a000931(n + 5)^2*a000931(n + 6); \\ Indranil Ghosh, Apr 26 2017

(Python)

from sympy import binomial

def A(n): return sum([binomial(k, n - 2*k) for k in range(int(n/2) + 1)])

def a000931(n): return 1 if n==0 else 0 if n<3 else A(n - 3)

def a(n): return a000931(n + 5)**2*a000931(n + 6) # Indranil Ghosh, Apr 26 2017

CROSSREFS