A047853 -id:A047853 - cp's OEIS frontend

A047848 Array A read by diagonals; n-th difference of (A(k,n), A(k,n-1),..., A(k,0)) is (k+2)^(n-1), for n=1,2,3,...; k=0,1,2,...

1, 2, 1, 5, 2, 1, 14, 6, 2, 1, 41, 22, 7, 2, 1, 122, 86, 32, 8, 2, 1, 365, 342, 157, 44, 9, 2, 1, 1094, 1366, 782, 260, 58, 10, 2, 1, 3281, 5462, 3907, 1556, 401, 74, 11, 2, 1, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1, 29525, 87382, 97657, 55988, 19609, 4682, 821, 112, 13, 2, 1

Offset: 0

Clark Kimberling

			Array, A(n, k), begins as:
  1, 2,  5,  14,   41, ... = A007051.
  1, 2,  6,  22,   86, ... = A047849.
  1, 2,  7,  32,  157, ... = A047850.
  1, 2,  8,  44,  260, ... = A047851.
  1, 2,  9,  58,  401, ... = A047852.
  1, 2, 10,  74,  586, ... = A047853.
  1, 2, 11,  92,  821, ... = A047854.
  1, 2, 12, 112, 1112, ... = A047855.
  1, 2, 13, 134, 1465, ... = A047856.
  1, 2, 14, 158, 1886, ... = A196791.
  1, 2, 15, 184, 2381, ... = A196792.
Downward antidiagonals, T(n, k), begins as:
      1;
      2,     1;
      5,     2,     1;
     14,     6,     2,     1;
     41,    22,     7,     2,     1;
    122,    86,    32,     8,     2,    1;
    365,   342,   157,    44,     9,    2,   1;
   1094,  1366,   782,   260,    58,   10,   2,   1;
   3281,  5462,  3907,  1556,   401,   74,  11,   2,  1;
   9842, 21846, 19532,  9332,  2802,  586,  92,  12,  2, 1;
  29525, 87382, 97657, 55988, 19609, 4682, 821, 112, 13, 2, 1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Cf. A047857 (row sums), A196793 (main diagonal).

A:= func< n,k | ((n+3)^k +n+1)/(n+2) >; // array A047848
[A(k,n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2025

A[n_, k_]:= ((n+3)^k +n+1)/(n+2);
A047848[n_, k_]:= A[k,n-k];
Table[A047848[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 11 2025 *)

def A(n,k): return (pow(n+3,k) +n+1)//(n+2) # array A047848
print(flatten([[A(k,n-k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 11 2025

A(n, k) = ((n+3)^k + n + 1)/(n+2). - Ralf Stephan, Feb 14 2004
From G. C. Greubel, Jan 11 2025: (Start)
T(n, k) = ((k+3)^(n-k) + k + 1)/(k+2) (antidiagonal triangle).
T(n, n) = A196793(n).
Sum_{k=0..n} T(n, k) = A047857(n). (End)

A226308 a(n) = a(n-1) + a(n-2) + 2*a(n-3) with a(0)=2, a(1)=1, a(2)=5.

Original entry on oeis.org

2, 1, 5, 10, 17, 37, 74, 145, 293, 586, 1169, 2341, 4682, 9361, 18725, 37450, 74897, 149797, 299594, 599185, 1198373, 2396746, 4793489, 9586981, 19173962, 38347921, 76695845, 153391690, 306783377, 613566757, 1227133514, 2454267025, 4908534053, 9817068106, 19634136209

Offset: 0

N. J. A. Sloane, Jun 07 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
G. Cerda-Morales, A note on modified third-order jacobsthal numbers, arXiv:1905.00725 [math.CO], 2019.
Charles K. Cook and Michael R. Bacon, Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41 (2013) pp. 27-39.
Evren Eyican Polatlı and Yüksel Soykan, On generalized third-order Jacobsthal numbers, Asian Res. J. of Math. (2021) Vol. 17, No. 2, 1-19, Article No. ARJOM.66022.
Index entries for linear recurrences with constant coefficients, signature (1,1,2).

Cf. A046636, A047853, A233328.

A226308 := n -> 1/7*(2^(n+3) + 6*cos(2*Pi*n/3) - 4*sqrt(3)*sin(2*Pi*n/3)):
seq(A226308(n), n = 0 .. 34); # Mélika Tebni, Mar 09 2024

CoefficientList[Series[-(2 x^2 - x + 2) / ((2 x - 1) (x^2 + x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 18 2013 *)
LinearRecurrence[{1,1,2},{2,1,5},40] (* Harvey P. Dale, Nov 03 2024 *)

a(n)=([0,1,0; 0,0,1; 2,1,1]^n*[2;1;5])[1,1] \\ Charles R Greathouse IV, Jul 19 2016

a0, a1, a2 = 2, 1, 5
for n in range(77):
  a = a2 + a1 + 2*a0
  print(a0, end=', ')
  a0, a1, a2 = a1, a2, a # Alex Ratushnyak, Jun 08 2013

G.f.: -(2*x^2-x+2) / ((2*x-1)*(x^2 + x + 1)). - Colin Barker, Jun 08 2013
a(3*n) = A047853(n+1), a(3*n+1) = A233328(n), a(3*n+2) = A046636(n+1). - Philippe Deléham, Feb 24 2014
From Mélika Tebni, Mar 09 2024: (Start)
E.g.f.: (1/7)*(8*exp(2*x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 4*sqrt(3)*sin(sqrt(3)*x/2))) (Charles K. Cook and Michael R. Bacon, 2013).
a(n) = (1/7)*(2^(n+3) + 6*cos(2*Pi*n/3) - 4*sqrt(3)*sin(2*Pi*n/3)). (End)

Deleted certain dangerous or potentially dangerous links. - N. J. A. Sloane, Jan 30 2021

A046636 Number of cubic residues mod 8^n.

Original entry on oeis.org

1, 5, 37, 293, 2341, 18725, 149797, 1198373, 9586981, 76695845, 613566757, 4908534053, 39268272421, 314146179365, 2513169434917, 20105355479333, 160842843834661, 1286742750677285, 10293942005418277, 82351536043346213, 658812288346769701, 5270498306774157605, 42163986454193260837

Offset: 0

David W. Wilson

Ralf Stephan, Prove or disprove: 100 conjectures from the OEIS, arXiv:math/0409509 [math.CO], 2004.
E. Wilmer and O. Schirokauer, A note on Stephan's conjecture 25, 2004. [broken link]
E. Wilmer and O. Schirokauer, A note on Stephan's conjecture 25, 2004. [cached copy]
Index entries for linear recurrences with constant coefficients, signature (9,-8).

Cf. A007583, A046530, A046630, A047853, A226308.

LinearRecurrence[{9, -8}, {1, 5}, 20] (* Jean-François Alcover, Jan 19 2019 *)

a(n) = (4*8^n + 3)/7.
a(n) = 8*a(n-1) - 3 (with a(0)=1). - Vincenzo Librandi, Nov 18 2010
From R. J. Mathar, Feb 28 2011: (Start)
a(n) = A046530(8^n) = A046630(3*n).
G.f.: (1-4*x)/((1-8*x)*(1-x)). (End)
a(n+1) = A226308(3*n+2). - Philippe Deléham, Feb 24 2014
From Elmo R. Oliveira, Apr 03 2025: (Start)
E.g.f.: exp(x)*(4*exp(7*x) + 3)/7.
a(n) = 9*a(n-1) - 8*a(n-2).
a(n) = A047853(n+1)/2. (End)

More terms from Elmo R. Oliveira, Apr 03 2025

A196791 a(n) = A047848(9, n).

Original entry on oeis.org

1, 2, 14, 158, 1886, 22622, 271454, 3257438, 39089246, 469070942, 5628851294, 67546215518, 810554586206, 9726655034462, 116719860413534, 1400638324962398, 16807659899548766, 201691918794585182, 2420303025535022174, 29043636306420266078, 348523635677043192926

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (13,-12).

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856.

Cf. A001021 (first differences).

Magma
```
[(12^n+10)/11: n in [0..20]];
```

LinearRecurrence[{13,-12},{1,2},30] (* Harvey P. Dale, Sep 07 2015 *)
(12^Range[0,40] +10)/11 (* G. C. Greubel, Jan 17 2025 *)

def A196791(n): return (pow(12, n) + 10)//11
print([A196791(n) for n in range(41)]) # G. C. Greubel, Jan 17 2025

a(n) = (12^n + 10)/11.
a(n) = 12*a(n-1) - 10, with a(0) = 1.
G.f.: (1-11*x)/((1-x)*(1-12*x)). - Bruno Berselli, Oct 11 2011
From Elmo R. Oliveira, Aug 30 2024: (Start)
E.g.f.: exp(x)*(exp(11*x) + 10)/11.
a(n) = 13*a(n-1) - 12*a(n-2) for n > 1. (End)

A196792 a(n) = A047848(10, n).

Original entry on oeis.org

1, 2, 15, 184, 2381, 30942, 402235, 5229044, 67977561, 883708282, 11488207655, 149346699504, 1941507093541, 25239592216022, 328114698808275, 4265491084507564, 55451384098598321, 720867993281778162, 9371283912663116095, 121826690864620509224, 1583746981240066619901

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900
Index entries for linear recurrences with constant coefficients, signature (14,-13).

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856.

Cf. A001022 (first differences).

Magma
```
[(13^n+11)/12: n in [0..20]];
```

(13^Range[0,40] +11)/12 (* G. C. Greubel, Jan 17 2025 *)

def A196792(n): return (pow(13, n) + 11)//12
print([A196792(n) for n in range(41)]) # G. C. Greubel, Jan 17 2025

a(n) = (13^n + 11)/12.
a(n) = 13*a(n-1) - 11, with a(0) = 1.
G.f.: (1-12*x)/((1-x)*(1-13*x)). - Bruno Berselli, Oct 11 2011
From Elmo R. Oliveira, Aug 30 2024: (Start)
E.g.f.: exp(x)*(exp(12*x) + 11)/12.
a(n) = 14*a(n-1) - 13*a(n-2) for n > 1. (End)

A196793 a(n) = A047848(n, n).

Original entry on oeis.org

1, 2, 7, 44, 401, 4682, 66431, 1111112, 21435889, 469070942, 11488207655, 311505013052, 9267595563617, 300239975158034, 10523614159962559, 396861212733968144, 16024522975978953761, 689852631578947368422, 31544039619835776489479

Offset: 0

Vincenzo Librandi, Oct 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A047848, A047849, A047850, A047851, A047852, A047853, A047854, A047855, A047856

Magma
```
[((n+3)^n+n+1)/(n+2): n in [0..20]]
```

A196793:=n->((n+3)^n+n+1)/(n+2); seq(A196793(k), k=0..50); # Wesley Ivan Hurt, Nov 08 2013

Table[((n+3)^n+n+1)/(n+2), {n,0,50}] (* Wesley Ivan Hurt, Nov 08 2013 *)

def A196793(n): return (pow(n+3, n) +n+1)//(n+2)
print([A196793(n) for n in range(51)]) # G. C. Greubel, Jan 17 2025

a(n) = ((n+3)^n + n + 1)/(n+2).

A166065 Triangle, read by rows, given by [0,1,1,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 0, 4, 2, 2, 0, 8, 4, 2, 2, 0, 16, 8, 4, 2, 2, 0, 32, 16, 8, 4, 2, 2, 0, 64, 32, 16, 8, 4, 2, 2, 0, 128, 64, 32, 16, 8, 4, 2, 2, 0, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 2, 0, 2048, 1024

Offset: 0

Philippe Deléham, Oct 05 2009

			Triangle begins :
1,
0,2,
0,2,2,
0,4,2,2,
0,8,4,2,2,
0,16,8,4,2,2,
0,32,16,8,4,2,2,
0,64,32,16,8,4,2,2,
0,128,64,32,16,8,4,2,2,
0,256,128,64,32,16,8,4,2,2,
0,512,256,128,64,32,16,8,4,2,2,

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A084247(n), A000007(n), A000079(n), A001787(n+1), A166060(n), A165665(n), A083585(n) for x= -1, 0, 1, 2, 3, 4, 5 respectively. Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A040000(n), A000079(n), A095121(n), A047851(n), A047853(n), A047855(n) for x = 0, 1, 2, 3, 4, 5 respectively.

G.f.: (1-2*x+x*y)/((-1+2*x)*(x*y-1)). - R. J. Mathar, Aug 11 2015

A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 2, 0, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Philippe Deléham, Oct 07 2009

			Triangle begins :
1 ;
0,2 ;
0,1,2 ;
0,1,1,2 ;
0,1,1,1,2 ;
0,1,1,1,1,2 ;
0,1,1,1,1,1,2 ; ...

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k)= A166122(n), A166114(n), A084222(n), A084247(n), A000034(n), A040000(n), A000027(n+1), A000079(n), A007051(n), A047849(n), A047850(n), A047851(n), A047852(n), A047853(n), A047854(n), A047855(n), A047856(n) for x= -5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^k= A000007(n), A000027(n+1), A033484(n), A134931(n), A083597(n) for x= 0,1,2,3,4 respectively.
T(n,k)= A166065(n,k)/2^(n-k).
G.f.: (1-x+x*y)/(1-x-x*y+x^2*y). - Philippe Deléham, Nov 09 2013
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 09 2013

cp's OEIS Frontend

A047848 Array A read by diagonals; n-th difference of (A(k,n), A(k,n-1),..., A(k,0)) is (k+2)^(n-1), for n=1,2,3,...; k=0,1,2,...

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A226308 a(n) = a(n-1) + a(n-2) + 2*a(n-3) with a(0)=2, a(1)=1, a(2)=5.

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A046636 Number of cubic residues mod 8^n.

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A196791 a(n) = A047848(9, n).

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A196792 a(n) = A047848(10, n).

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A196793 a(n) = A047848(n, n).

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A166065 Triangle, read by rows, given by [0,1,1,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A166124 Triangle, read by rows, given by [0,1/2,1/2,0,0,0,0,0,0,0,...] DELTA [2,-1,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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