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Expansion of 1/((1-8x8*x)*(1-10x10*x)).

1, 18, 244, 2952, 33616, 368928, 3951424, 41611392, 432891136, 4463129088, 45705032704, 465640261632, 4725122093056, 47800976744448, 482407813955584, 4859262511644672, 48874100093157376, 490992800745259008, 4927942405962072064, 49423539247696576512, 495388313981572612096, 4963106511852580896768

G. C. Greubel, <a href="/A016186/b016186_1.txt">Table of n, a(n) for n = 0..990</a>

a(n) = 5*10^n - 4*8^n = A081203(n+1). Binomial transform of A081035. - _From _R. J. Mathar_, Sep 18 2008: (Start)

a(n) = 5*10^n - 4*8^n = A081203(n+1).

Binomial transform of A081035. (End)

a(n) = 8*a(n-1) + 10^(n-1). - _From _Geoffrey Critzer_, Jan 24 2011: (Start)

a(n) = 8*a(n-1) + 10^(n-1).

E.g.f.: exp(9*x)*sinh(x) (with offset 1). - _Geoffrey Critzer_, Jan 24 2011(End)

a(n) = 10*a(n-1) + 8^n, a(0)=1. - _From _Vincenzo Librandi_, Feb 09 2011: (Start)

a(n) = 1810*a(n-1) - 80*a(+ 8^n-2), , a(0)=1, a(1)=18. - _Vincenzo Librandi_, Feb 09 2011

a(n) = 18*a(n-1) - 80*a(n-2), a(0)=1, a(1)=18. (End)

E.g.f.: exp(9*x)*( cosh(x) + 9*sinh(x) ). - G. C. Greubel, Nov 14 2024

RangeRest@With[0, 20]! {m=30}, CoefficientList[Series[Exp[9 x] Sinh[x], {x, 0, 20m}], x]*Range[0, m]!]

Join[{a=1, b=18}, Table[c=182^n*b(5^(n+1)-80*a; a=b; b=c, 4^(n+1)), {n, 0, 40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)

(Magma) [2^n*(5^(n+1)-4^(n+1)): n in [0..40]]; // G. C. Greubel, Nov 14 2024

(SageMath)

A016186=BinaryRecurrenceSequence(18, -80, 1, 18)

print([A016186(n) for n in range(41)]) # G. C. Greubel, Nov 14 2024

Cf. A016140, A060531, A081035, A081203.

More terms added by G. C. Greubel, Nov 14 2024

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Expansion of 1/((1-8x8*x)*(1-11x11*x)).

1, 19, 273, 3515, 42761, 503139, 5796673, 65860555, 741243321, 8287894259, 92240578673, 1023236299995, 11324318776681, 125117262357379, 1380687932442273, 15222751628953835, 167731742895202841, 1847300971660916499, 20338325086779563473, 223865691142651054075, 2463675524073768441801, 27109654136848307635619

G. C. Greubel, <a href="/A016187/b016187_1.txt">Table of n, a(n) for n = 0..950</a>

E.g.f.: (1/3)*(11*exp(11*x) - 8*exp(8*x)). - G. C. Greubel, Nov 14 2024

Join[{a=1, b=19}, Table[c=19*b(11^(n+1)-88*a; a=b; b=c, 8^(n+1))/3, {n, 0, 40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011 *)

LinearRecurrence[{19, -88}, {1, 19}, 40] (* G. C. Greubel, Nov 14 2024 *)

(Magma) [(11^(n+1)-8^(n+1))/3: n in [0..40]]; // G. C. Greubel, Nov 14 2024

(SageMath)

A016187=BinaryRecurrenceSequence(19, -88, 1, 19)

print([A016187(n) for n in range(41)]) # G. C. Greubel, Nov 14 2024

Cf. A016140.

More terms added by G. C. Greubel, Nov 14 2024

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E.g.f.: 3*exp(12*x) - 2*exp(8*x). - G. C. Greubel, Nov 14 2024

(Magma) [4^n*(3^(n+1)-2^(n+1)): n in [0..40]]; // G. C. Greubel, Nov 14 2024

(SageMath)

A016188=BinaryRecurrenceSequence(20, -96, 1, 20)

print([A016188(n) for n in range(41)]) # G. C. Greubel, Nov 14 2024

Cf. A016140.

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Expansion of 1/((1-3x3*x)*(1-7x7*x)).

1, 10, 79, 580, 4141, 29230, 205339, 1439560, 10083481, 70604050, 494287399, 3460188940, 24221854021, 169554572470, 1186886790259, 8308221880720, 58157596211761, 407103302622490, 2849723505777919, 19948065702706900, 139636463405732701, 977455254300482110, 6842186811484434379, 47895307774534219480

a(n) = ((5+sqrt4)^n - (5-sqrt4)^n)/4 in Fibonacci form. Offset 1. a(3)=79. - Al Hakanson (hawkuu(AT)gmail.com), Dec 31 2008

Join[{a=1, b=10}, Table[c=10*b(7^(n+1) -21*a; a=b; b=c, 3^(n+1))/4, {n, 600, 40}]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2011 *)

(Sage) [lucas_number1(n, 10, 21) for n in range(1, 2030)] # Zerinvary Lajos, Apr 26 2009

(Magma) [(7^(n+1)-3^(n+1))/4: n in [0..2030]]; // Vincenzo Librandi, Oct 09 2011

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Expansion of 1/((1-3x3*x)*(1-8x8*x)).

1, 11, 97, 803, 6505, 52283, 418993, 3354131, 26839609, 214736555, 1717951489, 13743789059, 109950843913, 879608345627, 7036871547985, 56294986732787, 450359936909017, 3602879624412299, 28823037382718881, 230584300224012515, 1844674405278884521, 14757395252691429371, 118059162052912494577, 944473296517443135443

In general, for expansion of 1/((1-bxb*x)*(1-cxc*x)): a(n) = (c^(n+1) - b^(n+1))/(c-b) = (b+c)*a(n-1) - bcb*c*a(n-2) = b*a(n-1) + c^n = c*a(n-1) + b^n = Sum_{i=0..n} b^i*c^(n-i). - Henry Bottomley, Jul 20 2000

a(n) = 11a11*a(n-1) - 24a24*a(n-2).

a(n) = 3a3*a(n-1) + 8^n.

a(n) = 8a8*a(n-1) + 3^n.

E.g.f.: (1/5)*(8*exp(8*x) - 3*exp(3*x)). - G. C. Greubel, Nov 14 2024

Join[{a = 1, b = 11}, Table[c = 11b (8^(n+1)- 24a; a = b; b = c, 3^(n+1))/5, {n, 600, 40}]] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011 *)

CoefficientList[Series[1 / ((1 - 3 x) (1 - 8 x)), {x, 0, 2030}], x] (* Vincenzo Librandi, Jun 24 2013 *)

(Sage) [lucas_number1(n, 11, 24) for n in range(1, 2030)] # Zerinvary Lajos, Apr 27 2009

(Magma) m:=2030; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1-8*x)))); // Vincenzo Librandi, Jun 24 2013

Sequences with g.f. 1/((1-n*x)*(1-8*x)): A001018 (n=0), A023001 (n=1), A016131 (n=2), this sequence (n=3), A016152 (n=4), A016162 (n=5), A016170 (n=6), A016177 (n=7), A053539 (n=8), A016185 (n=9), A016186 (n=10), A016187 (n=11), A016188 (n=12), A060195 (n=16).

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