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

0,4

Comments

The original definition was equivalent to: Let S(n) = sum_{i=0..n} a(i), then n^2+a(n)-S(n+1) = S(n-2). This in turn simplifies to the present definition.

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).

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

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

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 *)

Prog

(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
(PARI) a(n) = (n^2-1+(-1)^(n\2)*(1 + (n % 2)))/2; \\ Michel Marcus, Sep 11 2016

Crossrefs

Cf. A194274, A081654.

Sequence in context: A310303 A022807 A292655 * A081766 A152676 A197062

Adjacent sequences: A187090 A187091 A187092 * A187094 A187095 A187096

Keyword

nonn

Author

Benjamin Coinsin, Mar 04 2011

Extensions

Edited by N. J. A. Sloane, Mar 09 2011

More terms from Alex Ratushnyak, Aug 05 2012

Status

approved