A122803 -id:A122803 - cp's OEIS frontend

A137447 a(n) = 4*a(n-4) for n>3, a(0)=-1, a(1)=-4, a(2)=2, a(3)=12.

-1, -4, 2, 12, -4, -16, 8, 48, -16, -64, 32, 192, -64, -256, 128, 768, -256, -1024, 512, 3072, -1024, -4096, 2048, 12288, -4096, -16384, 8192, 49152, -16384, -65536, 32768, 196608, -65536, -262144, 131072, 786432, -262144, -1048576, 524288, 3145728, -1048576

Offset: 0

Paul Curtz, Apr 18 2008

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,4).

Cf. A084221, A102561, A122803.

&cat[[-(-2)^n,2^n-5*(-2)^n]: n in [0..20]];  // Bruno Berselli, Nov 02 2011

LinearRecurrence[{0,0,0,4},{-1,-4,2,12},50] (* or *) CoefficientList[ Series[(1+4x-2x^2-12x^3)/(4x^4-1),{x,0,50}],x] (* Harvey P. Dale, Jun 27 2011 *)

def A137447(n): return 2^(n//2)*(-1)^(n//2+1) if n%2==0 else 2^((n-1)//2)*(1 - 5*(-1)^((n-1)//2))
[A137447(n) for n in range(51)] # G. C. Greubel, Sep 15 2023

G.f.: (1+4*x-2*x^2-12*x^3)/(4*x^4-1). - Harvey P. Dale, Jun 27 2011
From Bruno Berselli, Nov 02 2011: (Start)
a(n) = (1-(-1)^n-2*(3-2*(-1)^n)*(-1)^floor(n/2))*2^(floor(n/2)-1).
a(2n) = -A122803(n).
a(2n+1) = (-1)^(n+1)*A084221(n+2). (End)
E.g.f.: (1/sqrt(2))*( sinh(sqrt(2)*x) - 5*sin(sqrt(2)*x) - sqrt(2)*cos(sqrt(2)*x) ). - G. C. Greubel, Sep 15 2023

More terms from Harvey P. Dale, Jun 27 2011

A140589 Triangle A(k,n) = (-2)^k+2^n read by rows.

Original entry on oeis.org

2, -1, 0, 5, 6, 8, -7, -6, -4, 0, 17, 18, 20, 24, 32, -31, -30, -28, -24, -16, 0, 65, 66, 68, 72, 80, 96, 128, -127, -126, -124, -120, -112, -96, -64, 0, 257, 258, 260, 264, 272, 288, 320, 384, 512, -511, -510, -508, -504, -496, -480, -448, -384, -256, 0, 1025, 1026, 1028, 1032

Offset: 0

Paul Curtz, Jul 06 2008

The flattened sequence a(A000217(k)+j)=A(k,j) obeys a(n+1)-2a(n)= -5, 2, 5, -4, -4, -23, 8, 8, 8, 17, -16, -16, -16, -16, -95, ..., which is a dispersion of 2, -4, -4, 8, 8, 8, ... (a signed version of A140513) with -5, 5, -23, 17, -95, 65,... The latter sequence is A(k,0)-2*A(k-1,k-1), an alternation of the negative of A140529 with each second element of A000051.

			Rows starting at k=0: (2), (-1,0); (5, 6, 8); (-7,-6,-4,0); (17,18,20,24,32);...

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020

A(k,n) = A000079(n)+A122803(k).

Edited by R. J. Mathar, Jul 08 2008

A239285 a(n) = (15^n - (-2)^n)/17.

Original entry on oeis.org

0, 1, 13, 199, 2977, 44671, 670033, 10050559, 150758257, 2261374111, 33920611153, 508809168319, 7632137522737, 114482062845151, 1717230942669073, 25758464140052479, 386376962100754417, 5795654431511381791, 86934816472670595793, 1304022247090059199039

Offset: 0

Felix P. Muga II, Mar 14 2014

Let k and t be positive integers and consider a(n) = k*a(n-1)+t*a(n-2) for n>=2, with a(0)=0, a(1)=1.
The roots of its characteristic equation are r1 = (k+sqrt(k^2+4t))/2 and r2 = (k-sqrt(k^2+4t))/2. Hence, the solution to the recurrence relation is the sequence {a(n)} where a(n) = alpha1*r1^n + alpha2*r2^n. It can be shown that alpha1 = 1/sqrt(k^2+4t) and alpha2 = -alpha1. It can be shown also that |r2/r1| < 1. Thus, a(n+1)/a(n) converges to r1 as n approaches infinity.
Note that limit a(n+1)/a(n) = 15 with k=13 and t=30.
If n > 20, then |a(n+1)/a(n) - 15| < 10^(-16).
Let b(n) be the number of strings of length n containing the 13-ary digits: 0,...,9,A,B,C or the 30 two-consecutive digits D0,D1,...,D9,DA,...,DT where A corresponds to 10, ..., T corresponds to 29. Then b(0)=1=a(2) and b(1)=13=a(3). The strings q_1q_2...q_n of length n can be partitioned into 2 groups A and B where A contains the strings where q_1=0,1,...,9,A,B,C and B contains the strings where q_1=D. Thus, |A|=13*b(n-1) and |B|=30*b(n-2). Hence, b(n) = 13*b(n-1) + 30*b(n-2) for n>1. Since b(0)=a(2) and b(1)=a(3), we can show that b(n) = a(n+2).

G. C. Greubel, Table of n, a(n) for n = 0..849
Index entries for linear recurrences with constant coefficients, signature (13,30).

Cf. A001024, A122803.

[(15^n -(-2)^n)/17: n in [0..30]]; // G. C. Greubel, May 26 2018

CoefficientList[Series[x/(1-13*x-30*x^2), {x,0,50}], x] (* or *) Table[
(15^n - (-2)^n)/17, {n,0,30}] (* or *) LinearRecurrence[{13,30}, {0,1}, 30] (* G. C. Greubel, May 26 2018 *)

a(n) = (15^n-(-2)^n)/17; \\ Michel Marcus, Mar 16 2014

x='x+O('x^30); concat([0], Vec()) \\ G. C. Greubel, May 26 2018

G.f.: x/(1 - 13*x - 30*x^2).
a(n) = 13*a(n-1) + 30*a(n-2) for n >= 2, a(0)=0, a(1)=1.
a(n) = (1/17)*(A001024(n) - A122803(n)), n >= 0.
a(0)=0, a(n) = Sum_{k=0..n-1} A001024(k)*A122803(n-1-k) for n > 0.
E.g.f.: (exp(15*x) - exp(-2*x))/17. - G. C. Greubel, May 26 2018

A351401 Decimal expansion of erfi(1)/e, where erfi is the imaginary error function.

Original entry on oeis.org

6, 0, 7, 1, 5, 7, 7, 0, 5, 8, 4, 1, 3, 9, 3, 7, 2, 9, 1, 1, 5, 0, 3, 8, 2, 3, 5, 8, 0, 0, 7, 4, 4, 9, 2, 1, 1, 6, 1, 2, 2, 0, 9, 2, 8, 6, 6, 5, 1, 5, 6, 9, 1, 5, 9, 1, 6, 9, 4, 4, 1, 9, 1, 9, 2, 7, 2, 0, 8, 7, 6, 9, 4, 9, 2, 0, 2, 8, 1, 1, 8, 2, 0, 1, 6, 3, 9, 1, 3, 1, 6, 5, 2, 6, 3, 3, 2, 6, 8, 5, 4, 8, 1, 0, 4

Offset: 0

Amiram Eldar, Feb 10 2022

The alternating sum of reciprocals of the factorials of the positive half-integers.

			0.60715770584139372911503823580074492116122092866515...

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, and Sergei Rogosin, Mittag-Leffler Functions, Related Topics and Applications, New York, NY: Springer, 2020. See p. 94, eq. (4.12.9.6).
Constantin Milici, Gheorghe Drăgănescu, and J. Tenreiro Machado, Fractional Differential Equations, Introduction to Fractional Differential Equations, Springer, Cham, 2019. See p. 12, eq. (1.9).

Eric Weisstein's World of Mathematics, Erfi.
Eric Weisstein's World of Mathematics, Mittag-Leffler Function.
Wikipedia, Mittag-Leffler function.

Cf. A001113, A001147, A068985, A087197, A099288, A122803, A351400.

evalf(exp(-1)*erfi(1), 120);  # Alois P. Heinz, Feb 10 2022

Mathematica
```
RealDigits[Erfi[1]/E, 10, 100][[1]]
```

real(-I*(1.0-erfc(I)))/exp(1) \\ Michel Marcus, Feb 10 2022

Equals Sum_{k>=0} (-1)^k/(k + 1/2)! = Sum_{k>=1} (-1)^(k+1)/Gamma(k + 1/2).
Equals E_{1, 3/2}(-1), where E_{a,b}(z) is the two-parameter Mittag-Leffler function.
Equals (-1/sqrt(Pi)) * Sum_{k>=1} (-2)^k/(2*k-1)!!.
Equals A068985 * A099288.

A210239 Triangle, read by rows, given by (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 2, 2, 5, 3, 2, 9, 12, 5, 2, 13, 28, 25, 8, 2, 17, 52, 74, 50, 13, 2, 21, 84, 167, 177, 96, 21, 2, 25, 124, 320, 470, 397, 180, 34, 2, 29, 172, 549, 1041, 1211, 850, 331, 55, 2, 33, 228, 870, 2034, 3042, 2928, 1758, 600, 89

Offset: 0

Philippe Deléham, Mar 19 2012

			Triangle begins :
1
2, 2
2, 5, 3
2, 9, 12, 5
2, 13, 28, 25, 8
2, 17, 52, 74, 50, 13
2, 21, 84, 167, 177, 96, 21
2, 25, 124, 320, 470, 397, 180, 34

Cf. A000045, A026150, A112087 (3rd column, n>2).

G.f.: (1+x+y*x)/(1-x-y*x-y*x^2-y^2*x^2).
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A122803(n), A000007(n), A040000(n), A026150(n+1) for x = -2, -1, 0, 1 respectively.
T(n,n) = Fibonacci(n+2) = A000045(n+2), T(n+1,n) = A067331(n).

A352692 a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.

Original entry on oeis.org

4, -3, 5, -1, 9, 7, 25, 39, 89, 167, 345, 679, 1369, 2727, 5465, 10919, 21849, 43687, 87385, 174759, 349529, 699047, 1398105, 2796199, 5592409, 11184807, 22369625, 44739239, 89478489, 178956967, 357913945, 715827879, 1431655769, 2863311527, 5726623065, 11453246119, 22906492249

Offset: 0

Paul Curtz, Mar 29 2022

Difference table D(n,k) = D(n-1,k+1) - D(n-1,k), D(0,k) = a(k):
    4,  -3,   5,  -1,   9,   7, 25, ...
   -7,   8,  -6,  10,  -2,  18, 14,  50, ...
   15, -14,  16, -12,  20,  -4, 36,  28, 100, ...
  -29,  30, -28,  32, -24,  40, -8,  72,  56, 200, ...
   59, -58,  60, -56,  64, -48, 80, -16, 144, 112, 400, ...
  ...
The diagonals are given by D(n,n+k) = a(k)*2^n.
D(n,1) = -(-1)^n* A340627(n).
a(n)   - a(n) =  0,  0,  0,  0,   0, ...       (trivially)
a(n+1) + a(n) =  1,  2,  4,  8,  16, ... = 2^n (by definition)
a(n+2) - a(n) =  1,  2,  4,  8,  16, ... = 2^n
a(n+3) + a(n) =  3,  6, 12, 24,  48, ... = 2^n*3
a(n+4) - a(n) =  5, 10, 20, 40,  80, ... = 2^n*5
a(n+5) + a(n) = 11, 22, 44, 88, 176, ... = 2^n*11
     (...)
This table is given by T(r,n) = A001045(r)*2^n with r, n >= 0.
Sums of antidiagonals are A045883(n).
Main diagonal: A192382(n).
First upper diagonal: A054881(n+1).
First subdiagonal: A003683(n+1).
Second subdiagonal: A246036(n).
Now consider the array from c(n) = (-1)^n*a(n) with its difference table:
   4,   3,   5,    1,    9,   -7,   25,   -39, ...  = c(n)
  -1,   2,  -4,    8,  -16,   32,  -64,   128, ...  = -A122803(n)
   3,  -6,  12,  -24,   48,  -96,  192,  -384, ...  =
  -9,  18, -36,   72, -144,  288, -576,  1152, ...
  27, -54, 108, -216,  432, -864, 1728, -3456, ...
...
The first subdiagonal is -A000400(n). The second is A169604(n).

Index entries for linear recurrences with constant coefficients, signature (1,2).

If a(0) = k then A001045 (k=0), A078008 (k=1), A140966 (k=2), A154879 (k=3), this sequence (k=4).
Essentially the same as A115335.
Cf. A000079, A002450, A340627.
Cf. A020714, A175805.
Cf. A045883, A192382, A003683, A246036.
Cf. A054881, A000400, A169604, A122803.
Cf. A132805, A082311, A082365.
Cf. A024495, A132804, A163868.

a := proc(n) option remember; ifelse(n = 0, 4, 2^(n-1) - a(n-1)) end: # Peter Luschny, Mar 29 2022
A352691 := proc(n)
    (11*(-1)^n + 2^n)/3
end proc: # R. J. Mathar, Apr 26 2022

LinearRecurrence[{1, 2}, {4, -3}, 40] (* Amiram Eldar, Mar 29 2022 *)

a(n) = (11*(-1)^n + 2^n)/3; \\ Thomas Scheuerle, Mar 29 2022

abs(a(n)) = A115335(n-1) for n >= 1.
a(3*n) - (-1)^n*4 = A132805(n).
a(3*n+1) + (-1)^n*4 = A082311(n).
a(3*n+2) - (-1)^n*4 = A082365(n).
From Thomas Scheuerle, Mar 29 2022: (Start)
G.f.: (-4 + 7*x)/(-1 + x + 2*x^2).
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(m + 2*n-k) = a(m)*2^n.
Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(1 + n-k) = -(-1)^n*A340627(n).
a(n) = (11*(-1)^n + 2^n)/3.
a(n + 2*m) = a(n) + A002450(m)*2^n.
a(2*n) = A192382(n+1) + (-1)^n*a(n).
a(n) = ( A045883(n) - Sum_{k=0..n-1}(-1)^k*a(k) )/n, for n > 0. (End)
a(n) = A001045(n) + 4*(-1)^n.
a(n+1) = 2*a(n) -11*(-1)^n.
a(n+2) = a(n) + 2^n.
a(n+4) = a(n) + A020714(n).
a(n+6) = a(n) + A175805(n).
a(2*n) = A163868(n).
a(2*n+1) = (2^(2*n+1) - 11)/3.

Warning: The DATA is correct, but there may be errors in the COMMENTS, which should be rechecked. - Editors of OEIS, Apr 26 2022

Edited by M. F. Hasler, Apr 26 2022.

A370627 a(n) = 2^(n - 1)((-1)^(n + 1) + 72^n)/3 = 2^(n - 1)*A062092(n).

Original entry on oeis.org

1, 5, 18, 76, 296, 1200, 4768, 19136, 76416, 305920, 1223168, 4893696, 19572736, 78295040, 313171968, 1252704256, 5010784256, 20043202560, 80172679168, 320690978816, 1282763390976, 5131054612480, 20524216352768, 82096869605376, 328387470032896, 1313549896908800, 5254199554080768

Offset: 0

Paul Curtz, Jul 03 2024

Index entries for linear recurrences with constant coefficients, signature (2,8).

Cf. A003683, A122803, A133125, A255470, A255471.

Cf. A108982, A062092.

LinearRecurrence[{2, 8}, {1, 5}, 27] (* Amiram Eldar, Jul 03 2024 *)

a(n) = 2^(n-1)*((-1)^(n+1) + 7*2^n)/3 \\ Thomas Scheuerle, Jul 03 2024

Binomial transform of A133125.
G.f.: (1 + 3*x)/(1 - 2*x - 8*x^2).
E.g.f.: (1/3)*exp(x)*(3*exp(3*x) + sinh(3*x)).
a(n) = 2*a(n-1) + 8*a(n-2), for n > 1.
a(n) = 4*a(n-1) + (-2)^n, for n > 0.
a(n) = (a(n+2) - 2*a(n+1))/8.
From Thomas Scheuerle, Jul 03 2024: (Start)
a(n) = 2^(n - 1)*((-1)^(n + 1) + 7*2^n)/3.
a(n) = A003683(n) + 4^n.
a(n) = A255470(2^n - 1) - A255470(2^(n-1) - 1) = A255471(n) - A255471(n-1), for n > 0. (End)
Binomial transform: A108982.

cp's OEIS Frontend

A137447 a(n) = 4*a(n-4) for n>3, a(0)=-1, a(1)=-4, a(2)=2, a(3)=12.

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A140589 Triangle A(k,n) = (-2)^k+2^n read by rows.

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A239285 a(n) = (15^n - (-2)^n)/17.

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A351401 Decimal expansion of erfi(1)/e, where erfi is the imaginary error function.

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A210239 Triangle, read by rows, given by (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A352692 a(n) + a(n+1) = 2^n for n >= 0 with a(0) = 4.

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A370627 a(n) = 2^(n - 1)*((-1)^(n + 1) + 7*2^n)/3 = 2^(n - 1)*A062092(n).

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A370627 a(n) = 2^(n - 1)((-1)^(n + 1) + 72^n)/3 = 2^(n - 1)*A062092(n).