A100216 -id:A100216 - cp's OEIS frontend

A100213 Expansion of g.f.: x(4-7x+2x^2-8x^4+16x^5-16x^6)/((1-2x) (1-2x^2) (1-2x+2x^2) * (1+2*x^2)).

4, 9, 14, 18, 32, 64, 128, 256, 544, 1104, 2144, 4128, 8192, 16384, 32768, 65536, 131584, 263424, 525824, 1049088, 2097152, 4194304, 8388608, 16777216, 33562624, 67129344, 134242304, 268443648, 536870912, 1073741824, 2147483648, 4294967296, 8590065664

Offset: 1

Creighton Dement, Nov 11 2004

The sequence can be created applying the pos operator (which sums over the positive coefficients) to the n-th power of the Floretion element (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e).

			a(5) = 32 because (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)^5 = - 2 'j - 2 'k - 2 j' - 2 k' + 6 'ii' + 10 'jj' + 10 'kk' + 6 e,
and the sum of all positive coefficients is 6+10+10+6 = 32.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Creighton Dement, Floretion Online Multiplier.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,4,-16,24,-16).

Cf. A000079, A009116, A009545, A038503, A077957, A077966, A100212, A100215, A100216.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-4*x+6*x^2-4*x^3)*(1-4*x^4)) )); // G. C. Greubel, Mar 29 2024

Rest[CoefficientList[Series[x(4-7x+2x^2-8x^4+16x^5-16x^6)/((1-2x)(1-2x^2)(1-2x+2x^2)(1+2x^2)),{x,0,40}],x]] (* or *) LinearRecurrence[{4,-6,4,4,-16,24,-16},{4,9,14,18,32,64,128},40] (* Harvey P. Dale, Aug 23 2015 *)

def A100213_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-4*x+6*x^2-4*x^3)*(1-4*x^4)) ).list()
a=A100213_list(51); a[1:] # G. C. Greubel, Mar 29 2024

a(n) = A100215(n) - A100212(n).
a(n) = (-1)^n*A009116(n+3) + A100216 + A038503(n+1).
Equation above in Floretian Algebra operator speak: (pos) + (neg) = (ves) = (jes) + (les) + (tes)
a(n-1) = A000079(n+1) + (5*A077957(n) + 6*A077957(n-1))/4 + A009545(n)/2 + A009545(n+1) + A077966(n-1) - A077966(n)/4. - R. J. Mathar, May 07 2008
From G. C. Greubel, Mar 29 2024: (Start)
a(n) = (1/16)*( 2^(n+4) - 2*((1+5*i)*(1+i)^n + (1-5*i)*(1-i)^n) + (1 - (-1)^n)*2^((n+1)/2)*(5+i^(n+1)) + (1+(-1)^n)*2^(1+n/2)*(3-2*i^n) ).
a(2*n-1) = 2^(n-3)*( 2^(n+2) + 5 + (-1)^n - 6*cos(n*Pi/2) + 4*sin(n*Pi/2) ), for n >= 1.
a(2*n) = 2^(n-2)*( 2^(n+2) + 3 - 2*(-1)^n - cos(n*Pi/2) + 5*sin(n*Pi/2) ), n >= 1.
E.g.f.: -1 + exp(2*x) + (1/8)*(6*cosh(sqrt(2)*x) + 5*sqrt(2)* sinh(sqrt(2)*x) - (4*cos(sqrt(2)*x) + sqrt(2)*sin(sqrt(2)*x)) - 2*exp(x)*(cos(x) - 5*sin(x)) ). (End)

Replaced definition with generating function, changed offset to 1. - R. J. Mathar, Mar 12 2010

A100215 Expansion of (4 - 7x + 2x^2)/((1-2x)(1 - 2x + 2x^2)).

Original entry on oeis.org

4, 9, 14, 18, 24, 44, 104, 248, 544, 1104, 2144, 4128, 8064, 16064, 32384, 65408, 131584, 263424, 525824, 1049088, 2095104, 4189184, 8382464, 16775168, 33562624, 67129344, 134242304, 268443648, 536838144

Offset: 0

Creighton Dement, Nov 11 2004

a(n) = (-1)^n*A009116(n+3) + A100216(n) + A038503(n+1), where A009116, A100216 and A038503 can be generated by the operators jes, les and tes of the Floretion algebra, which is a product factor space Q x Q /{(1,1), (-1,-1)}.

Binomial transform of the sequence 4,5,0,-1 (repeated with period length 4). - R. J. Mathar, Apr 18 2009

			a(2) = 14 because (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)^3 = 1'j + 1'k + 1j' + 1k' + 3'ii' + 2'jj' + 2'kk' + 1'jk' + 1'kj' + 1e and the sum of these coefficients is 1 + 1 + 1 + 1 + 3 + 2 + 2 + 1 + 1 + 1 = 14 (see comment).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Creighton Dement, Floretion Online Multiplier.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4).

Cf. A009116, A038503, A099087, A100213, A100214, A100216.

I:=[4, 9, 14]; [n le 3 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Jun 25 2012

LinearRecurrence[{4,-6,4},{4,9,14},40] (* Vincenzo Librandi, Jun 25 2012 *)

A099087=BinaryRecurrenceSequence(2,-2,1,2)
def A100215(n): return 2^(n+1) + 2*A099087(n) + A099087(n-1)
[A100215(n) for n in range(41)] # G. C. Greubel, Mar 29 2024

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3).
a(n) = (-1)^n*A009116(n+3) + A100216(n) + A038503(n+1).
a(n) = vesseq(.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e), where ves sums over all floretion basis vector coefficients.
a(n) = 2^(n+1) + 2*A099087(n) + A099087(n-1). - R. J. Mathar, Apr 18 2009

Definition replaced with the more precise g.f. by R. J. Mathar, Nov 17 2010

cp's OEIS Frontend

A100213 Expansion of g.f.: x(4-7x+2x^2-8x^4+16x^5-16x^6)/((1-2x) (1-2x^2) (1-2x+2x^2) * (1+2*x^2)).

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A100215 Expansion of (4 - 7x + 2x^2)/((1-2x)(1 - 2x + 2x^2)).

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A100213 Expansion of g.f.: x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-2*x) * (1-2*x^2) * (1-2*x+2*x^2) * (1+2*x^2)).

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A100215 Expansion of (4 - 7*x + 2*x^2)/((1-2*x)*(1 - 2*x + 2*x^2)).

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A100213 Expansion of g.f.: x(4-7x+2x^2-8x^4+16x^5-16x^6)/((1-2x) (1-2x^2) (1-2x+2x^2) * (1+2*x^2)).

A100215 Expansion of (4 - 7x + 2x^2)/((1-2x)(1 - 2x + 2x^2)).