A047850 -id:A047850 - 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)

A227076 A triangle formed like Pascal's triangle, but with 5^n on the borders instead of 1.

Original entry on oeis.org

1, 5, 5, 25, 10, 25, 125, 35, 35, 125, 625, 160, 70, 160, 625, 3125, 785, 230, 230, 785, 3125, 15625, 3910, 1015, 460, 1015, 3910, 15625, 78125, 19535, 4925, 1475, 1475, 4925, 19535, 78125, 390625, 97660, 24460, 6400, 2950, 6400, 24460, 97660, 390625

Offset: 0

T. D. Noe, Aug 06 2013

All rows except the zeroth are divisible by 5. Is there a closed-form formula for these numbers, as there is for binomial coefficients?

			Triangle begins as:
       1;
       5,     5;
      25,    10,    25;
     125,    35,    35,  125;
     625,   160,    70,  160,  625;
    3125,   785,   230,  230,  785, 3125;
   15625,  3910,  1015,  460, 1015, 3910, 15625;
   78125, 19535,  4925, 1475, 1475, 4925, 19535, 78125;
  390625, 97660, 24460, 6400, 2950, 6400, 24460, 97660, 390625;

T. D. Noe, Rows n = 0..50 of triangle, flattened

Cf. A007318 (Pascal's triangle), A228053 ((-1)^n on the borders).
Cf. A051601 (n on the borders), A137688 (2^n on borders).
Cf. A083585 (row sums: (8*5^n - 5*2^n)/3), A227074 (4^n edges), A227075 (3^n edges).
Cf. A000351.

function T(n,k) // T = A227076
  if k eq 0 or k eq n then return 5^n;
  else return T(n-1,k) + T(n-1,k-1);
  end if;
end function;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 10 2025

A227076 := proc(n,k)
    if k = 0 or k = n then
        5^n ;
    elif k < 0 or k > n then
        0;
    else
        procname(n-1,k)+procname(n-1,k-1) ;
    end if;
end proc: # R. J. Mathar, Aug 09 2013

t = {}; Do[r = {}; Do[If[k == 0 || k == n, m = 5^n, m = t[[n, k]] + t[[n, k + 1]]]; r = AppendTo[r, m], {k, 0, n}]; AppendTo[t, r], {n, 0, 10}]; t = Flatten[t]

from sage.all import *
@CachedFunction
def T(n,k): # T = A227076
    if k==0 or k==n: return pow(5,n)
    else: return T(n-1,k) + T(n-1,k-1)
print(flatten([[T(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 10 2025

From R. J. Mathar, Aug 09 2013: (Start)
T(n,0) = 5^n.
T(n,1) = 5*A047850(n-1).
T(n,2) = 5*(5^n/80 + 3*n/4 + 51/16).
T(n,3) = 5*(5^n/320 + 45*n/16 + 3*n^2/8 + 819/64). (End)
Sum_{k=0..n} (-1)^k*T(n, k) = 20*(1+(-1)^n)*A009969(floor((n-1)/2)) - (3/5)*[n = 0]. - G. C. Greubel, Jan 10 2025

A132079 a(n) = (5^n + 3)/2.

Original entry on oeis.org

2, 4, 14, 64, 314, 1564, 7814, 39064, 195314, 976564, 4882814, 24414064, 122070314, 610351564, 3051757814, 15258789064, 76293945314, 381469726564, 1907348632814, 9536743164064, 47683715820314

Offset: 0

Mohit Maheshwari (mohitmahe1989(AT)gmail.com), Oct 30 2007

(5^n + 3)/2 is always divisible by 2 and the next value can be generated by appending 4 at the ones place and adding the rest of the number with the respective power of 5. For example, 314 can be generated after 64 by just writing 4 at the ones place and adding 5^2 to 6 to get 31 and appending 31 to 4 to get 314.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A047850.

[(5^n+3)/2: n in [0..30]]; // Vincenzo Librandi, May 02 2011

Table[(5^n + 3)/2, {n, 0, 24}] (* Alonso del Arte, May 02 2011 *)

a(n)=5^n\2+2 \\ Charles R Greathouse IV, May 02 2011

From Elmo R. Oliveira, Dec 21 2023: (Start)
a(n) = 2*A047850(n).
a(n) = 5*a(n-1) - 6 for n>0.
a(n) = 6*a(n-1) - 5*a(n-2) for n>1.
G.f.: (2-8*x)/((1-x)*(1-5*x)).
E.g.f.: exp(x)*(exp(4*x) + 3)/2. (End)

Offset changed from 4 to 0; corrected and extended by Vincenzo Librandi, May 02 2011

A123490 Triangle whose k-th column satisfies a(n) = (k+3)a(n-1)-(k+2)a(n-2).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 5, 2, 1, 16, 14, 6, 2, 1, 32, 41, 22, 7, 2, 1, 64, 122, 86, 32, 8, 2, 1, 128, 365, 342, 157, 44, 9, 2, 1, 256, 1094, 1366, 782, 260, 58, 10, 2, 1, 512, 3281, 5462, 3907, 1556, 401, 74, 11, 2, 1, 1024, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1

Offset: 0

Paul Barry, Oct 01 2006

			Triangle begins
     1;
     2,    1;
     4,    2,     1;
     8,    5,     2,     1;
    16,   14,     6,     2,    1;
    32,   41,    22,     7,    2,    1;
    64,  122,    86,    32,    8,    2,   1;
   128,  365,   342,   157,   44,    9,   2,  1;
   256, 1094,  1366,   782,  260,   58,  10,  2,  1;
   512, 3281,  5462,  3907, 1556,  401,  74, 11,  2, 1;
  1024, 9842, 21846, 19532, 9332, 2802, 586, 92, 12, 2, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Columns include A000079, A007051, A047849, A047850, A047851.

Cf. A047848, A103439 (row sums), A123491 (diagonal sums).

[((k+2)^(n-k) +k)/(k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 15 2021

Table[((k+2)^(n-k) +k)/(k+1), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 14 2017 *)

for(n=0, 10, for(k=0,n, print1(((k+2)^(n-k)+k)/(k+1), ", "))) \\ G. C. Greubel, Oct 14 2017

flatten([[((k+2)^(n-k) +k)/(k+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 15 2021

Column k has g.f.: x^k*(1-x(1+k))/((1-x)*(1-x(2+k))).
T(n,k) = ((k+2)^(n-k) + k)/(k+1), for 0 <= k <= n.
Sum_{k=0..n} T(n, k) = A103439(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A123491(n).

A153776 Sequence S such that 1 is in S and if x is in S, then 5x-3 and 5x-1 are in S.

Original entry on oeis.org

1, 2, 4, 7, 9, 17, 19, 32, 34, 42, 44, 82, 84, 92, 94, 157, 159, 167, 169, 207, 209, 217, 219, 407, 409, 417, 419, 457, 459, 467, 469, 782, 784, 792, 794, 832, 834, 842, 844, 1032, 1034, 1042, 1044, 1082, 1084, 1092, 1094, 2032, 2034, 2042, 2044, 2082, 2084

Offset: 1

Clark Kimberling, Jan 02 2009

nonn

Subsequences include A047850, A083065.
First generation: 1
2nd generation: 2, 4
3rd generation: 7, 9, 17, 19
4th generation: 32, 34, 42, 44, 82, 84, 92, 94
Does every generation contain p or 2p for some prime p?

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A047850, A083065, A153773, A153775.

nxt[n_] := Flatten[5 # + {-3, -1} & /@ n]; Union[Flatten[NestList[nxt, {1}, 5]]] (* G. C. Greubel, Aug 28 2016 *)

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).

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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A227076 A triangle formed like Pascal's triangle, but with 5^n on the borders instead of 1.

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A132079 a(n) = (5^n + 3)/2.

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A123490 Triangle whose k-th column satisfies a(n) = (k+3)*a(n-1)-(k+2)*a(n-2).

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A153776 Sequence S such that 1 is in S and if x is in S, then 5x-3 and 5x-1 are in S.

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

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

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A123490 Triangle whose k-th column satisfies a(n) = (k+3)a(n-1)-(k+2)a(n-2).