A047854 -id:A047854 - 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)

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).

A062508 a(n) = 3^(2n)+7.

Original entry on oeis.org

8, 16, 88, 736, 6568, 59056, 531448, 4782976, 43046728, 387420496, 3486784408, 31381059616, 282429536488, 2541865828336, 22876792454968, 205891132094656, 1853020188851848, 16677181699666576, 150094635296999128

Offset: 0

Jason Earls, Jun 24 2001

D. M. Burton, Elementary Number Theory, Allyn and Bacon, Inc. Boston, MA, 1976, pp. 29.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10, -9).

Cf. A047854.

A062508:=n->3^(2*n)+7: seq(A062508(n), n=0..30); # Wesley Ivan Hurt, Feb 11 2017

3^(2Range[0,20])+7 (* or *) LinearRecurrence[{10,-9},{8,16},20] (* Harvey P. Dale, May 13 2012 *)

a(0)=8, a(1)=16, a(n)=10*a(n-1)-9*a(n-2). - Harvey P. Dale, May 13 2012

A377735 Cogrowth sequence of the 16-element group Q8 X C2 = .

Original entry on oeis.org

1, 1, 7, 103, 829, 7261, 66595, 598627, 5377849, 48426745, 435876607, 3922582687, 35303534581, 317733991381, 2859598948507, 25736384863003, 231627537879409, 2084647743751921, 18761829221081335, 168856464809568727, 1519708183900618669, 13677373637498037325

Offset: 0

Sean A. Irvine, Nov 10 2024

Gives the even terms, all the odd terms are 0.

Index entries for linear recurrences with constant coefficients, signature (8, 2, 72, -81).

Cf. A047854 (D4 X C2), A377840 (C8 X C2), A071930 (Q8).

G.f.: (27*x^3+3*x^2+7*x-1) / ((1-x) * (9*x-1) * (9*x^2+2*x+1)).

A377943 Cogrowth sequence of the 16-element Pauli group C4 o D4 = .

Original entry on oeis.org

1, 1, 11, 91, 821, 7381, 66431, 597871, 5380841, 48427561, 435848051, 3922632451, 35303692061, 317733228541, 2859599056871, 25736391511831, 231627523606481, 2084647712458321, 18761829412124891, 168856464709124011, 1519708182382116101, 13677373641439044901

Offset: 0

Sean A. Irvine, Nov 11 2024

Gives the even terms, all the odd terms are 0.

Index entries for linear recurrences with constant coefficients, signature (9, 1, -9).

Cf. A070775 (C4 X C4), A377855 (C4:C4), A047854 (D4 X C2).

G.f.: (x^2-8*x+1) / ((x-1) * (9*x-1) * (x+1)).

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

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

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Magma

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

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A062508 a(n) = 3^(2n)+7.

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A377735 Cogrowth sequence of the 16-element group Q8 X C2 = .

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A377943 Cogrowth sequence of the 16-element Pauli group C4 o D4 = .

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Formula

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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