A187093 a(0)=0, a(1)=a(2)=1; thereafter, a(n+1) = n^2 - a(n-1).
0, 1, 1, 3, 8, 13, 17, 23, 32, 41, 49, 59, 72, 85, 97, 111, 128, 145, 161, 179, 200, 221, 241, 263, 288, 313, 337, 363, 392, 421, 449, 479, 512, 545, 577, 611, 648, 685, 721, 759, 800, 841, 881, 923, 968, 1013, 1057, 1103, 1152, 1201, 1249, 1299, 1352, 1405, 1457
Offset: 0
Keywords
Links
- Harvey P. Dale, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (3,-4,4,-3,1).
Programs
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Maple
A000034 := proc(n) op(1+(n mod 2),[1,2]) ; end proc: A187093 := proc(n) (n^2-1+(-1)^floor(n/2)*A000034(n))/2 ;end proc: # R. J. Mathar
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Mathematica
LinearRecurrence[{3, -4, 4, -3, 1}, {0, 1, 1, 3, 8}, 60] (* Jean-François Alcover, Mar 30 2020 *) Join[{0},RecurrenceTable[{a[1]==a[2]==1,a[n+1]==n^2-a[n-1]},a,{n,60}]] (* Harvey P. Dale, Jan 05 2023 *) -
PARI
a(n) = (n^2-1+(-1)^(n\2)*(1 + (n % 2)))/2; \\ Michel Marcus, Sep 11 2016
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Python
print(0, end=',') # a(-1)=0 prpr = prev = 1 # a(0)=a(1)=1 for n in range(2, 77): print(prpr, end=',') curr = n*n - prpr # a(n) = n^2 - a(n-2) prpr = prev prev = curr # from Alex Ratushnyak, Aug 05 2012
Formula
a(n) = (n^2 - 1 + (-1)^floor(n/2) * A000034(n))/2.
G.f.: x*(-1+2*x+x^3-4*x^2) / ( (x^2+1)*(x-1)^3 ).
a(2^(n+1)) = A081654(n). - Anton Zakharov, Sep 13 2016
Extensions
Edited by N. J. A. Sloane, Mar 09 2011
More terms from Alex Ratushnyak, Aug 05 2012
Comments