A010883 -id:A010883 - cp's OEIS frontend

A203163 (n-1)-st elementary symmetric function of the first n terms of (1,2,3,4,1,2,3,4,1,2,3,4,...) = A010883.

1, 3, 11, 50, 74, 172, 564, 2400, 2976, 6528, 20736, 86400, 100224, 214272, 670464, 2764800, 3096576, 6524928, 20238336, 82944000, 90906624, 189775872, 585252864, 2388787200, 2579890176, 5350883328, 16434855936, 66886041600

Offset: 1

Clark Kimberling, Dec 30 2011

			Let esf abbreviate "elementary symmetric function". Then
0th esf of {1}:  1;
1st esf of {1,2}:  1+2 = 3;
2nd esf of {1,2,3} is 1*2 + 1*3 + 2*3 = 11.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,48,0,0,0,-576).

Cf. A010883, A203162, A203164, A203165.

f[k_] := 1 + Mod[k + 3, 4]; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 33}]  (* A203163 *)
LinearRecurrence[{0,0,0,48,0,0,0,-576},{1,3,11,50,74,172,564,2400},50] (* Harvey P. Dale, Aug 18 2020 *)

Vec(x*(36*x^6+28*x^5+26*x^4+50*x^3+11*x^2+3*x+1)/(24*x^4-1)^2 + O(x^100)) \\ Colin Barker, Aug 15 2014

G.f.: x*(36*x^6 + 28*x^5 + 26*x^4 + 50*x^3 + 11*x^2 + 3*x + 1) / (24*x^4 - 1)^2. - Colin Barker, Aug 15 2014

A239660 Triangle read by rows in which row n lists two copies of the n-th row of triangle A237593.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 3, 2, 2, 3, 3, 2, 2, 3, 4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4, 4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4, 5, 2, 1, 1, 2, 5, 5, 2, 1, 1, 2, 5, 5, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 5, 6, 2, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 2, 6, 6, 3, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1, 3, 6

Offset: 1

Omar E. Pol, Mar 24 2014

For the construction of this sequence also we can start from A235791.
This sequence can be interpreted as an infinite Dyck path: UDUDUUDD...
Also we use this sequence for the construction of a spiral in which the arms in the quadrants give the symmetric representation of sigma, see example.
We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016
The spiral has the property that the sum of the parts in the quadrants 1 and 3, divided by the sum of the parts in the quadrants 2 and 4, converges to 3/5. - Omar E. Pol, Jun 10 2019

			Triangle begins (first 15.5 rows):
1, 1, 1, 1;
2, 2, 2, 2;
2, 1, 1, 2, 2, 1, 1, 2;
3, 1, 1, 3, 3, 1, 1, 3;
3, 2, 2, 3, 3, 2, 2, 3;
4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4;
4, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 4;
5, 2, 1, 1, 2, 5, 5, 2, 1, 1, 2, 5;
5, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 5;
6, 2, 1, 1, 1, 1, 2, 6, 6, 2, 1, 1, 1, 1, 2, 6;
6, 3, 1, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 1, 3, 6;
7, 2, 2, 1, 1, 2, 2, 7, 7, 2, 2, 1, 1, 2, 2, 7;
7, 3, 2, 1, 1, 2, 3, 7, 7, 3, 2, 1, 1, 2, 3, 7;
8, 3, 1, 2, 2, 1, 3, 8, 8, 3, 1, 2, 2, 1, 3, 8;
8, 3, 2, 1, 1, 1, 1, 2, 3, 8, 8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
9, 3, 2, 1, 1, 1, 1, 2, 3, 9, ...
.
Illustration of initial terms as an infinite Dyck path (row n = 1..4):
.
.                            /\/\    /\/\
.       /\  /\  /\/\  /\/\  /    \  /    \
.  /\/\/  \/  \/    \/    \/      \/      \
.
.
Illustration of initial terms for the construction of a spiral related to sigma:
.
.  row 1     row 2          row 3           row 4
.                                          _ _ _
.                                               |_
.             _ _                                 |
.   _ _      |                                    |
.  |   |     |                                    |
.            |         |           |              |
.            |_ _      |_         _|              |
.                        |_ _ _ _|               _|
.                                          _ _ _|
.
.[1,1,1,1] [2,2,2,2] [2,1,1,2,2,1,1,2] [3,1,1,3,3,1,1,3]
.
The first 2*A003056(n) terms of the n-th row are represented in the A010883(n-1) quadrant and the last 2*A003056(n) terms of the n-th row are represented in the A010883(n) quadrant.
.
Illustration of the spiral constructed with the first 15.5 rows of triangle:
.
.               12 _ _ _ _ _ _ _ _
.                 |  _ _ _ _ _ _ _|_ _ _ _ _ _ _ 7
.                 | |             |_ _ _ _ _ _ _|
.                _| |                           |
.               |_ _|9 _ _ _ _ _ _              |_ _
.         12 _ _|     |  _ _ _ _ _|_ _ _ _ _ 5      |_
.      _ _ _| |      _| |         |_ _ _ _ _|         |
.     |  _ _ _|  9 _|_ _|                   |_ _ 3    |_ _ _ 7
.     | |      _ _| |   12 _ _ _ _          |_  |         | |
.     | |     |  _ _|    _|  _ _ _|_ _ _ 3    |_|_ _ 5    | |
.     | |     | |      _|   |     |_ _ _|         | |     | |
.     | |     | |     |  _ _|           |_ _ 3    | |     | |
.     | |     | |     | |    3 _ _        | |     | |     | |
.     | |     | |     | |     |  _|_ 1    | |     | |     | |
.    _|_|    _|_|    _|_|    _|_| |_|    _|_|    _|_|    _|_|    _
.   | |     | |     | |     | |         | |     | |     | |     | |
.   | |     | |     | |     |_|_ _     _| |     | |     | |     | |
.   | |     | |     | |    2  |_ _|_ _|  _|     | |     | |     | |
.   | |     | |     |_|_     2    |_ _ _|    _ _| |     | |     | |
.   | |     | |    4    |_               7 _|  _ _|     | |     | |
.   | |     |_|_ _        |_ _ _ _        |  _|    _ _ _| |     | |
.   | |    6      |_      |_ _ _ _|_ _ _ _| |    _|    _ _|     | |
.   |_|_ _ _        |_   4        |_ _ _ _ _|  _|     |    _ _ _| |
.  8      | |_ _      |                     15|      _|   |  _ _ _|
.         |_    |     |_ _ _ _ _ _            |  _ _|    _| |
.        8  |_  |_    |_ _ _ _ _ _|_ _ _ _ _ _| |      _|  _|
.             |_ _|  6            |_ _ _ _ _ _ _|  _ _|  _|
.                 |                             28|  _ _|
.                 |_ _ _ _ _ _ _ _                | |
.                 |_ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _| |
.                8                |_ _ _ _ _ _ _ _ _|
.                                                    31
.
The diagram contains A237590(16) = 27 parts.
The total area (also the total number of cells) in the n-th arm of the spiral is equal to sigma(n) = A000203(n), considering every quadrant and the axes x and y. (checked by hand up to row n = 128). The parts of the spiral are in A237270: 1, 3, 2, 2, 7...
Diagram extended by _Omar E. Pol_, Aug 23 2018

Robert Price, Table of n, a(n) for n = 1..30016 (rows n = 1..412, flattened)

Row n has length 4*A003056(n).
The sum of row n is equal to 4*n = A008586(n).
Row n is a palindromic composition of 4*n = A008586(n).
Both column 1 and right border are A008619, n >= 1.
The connection between A196020 and A237270 is as follows: A196020 --> A236104 --> A235791 --> A237591 --> A237593 --> this sequence --> A237270.
Cf. A000203, A000217, A003056, A008619, A010883, A112610, A193553, A237048, A237271, A237590, A239052, A239053, A239663, A239665, A239931, A239932, A239933, A239934, A240020, A240062, A244050, A245092, A262626, A296508, A299778.

A116081 Final nonzero digit of n^n.

Original entry on oeis.org

1, 4, 7, 6, 5, 6, 3, 6, 9, 1, 1, 6, 3, 6, 5, 6, 7, 4, 9, 6, 1, 4, 7, 6, 5, 6, 3, 6, 9, 9, 1, 6, 3, 6, 5, 6, 7, 4, 9, 6, 1, 4, 7, 6, 5, 6, 3, 6, 9, 5, 1, 6, 3, 6, 5, 6, 7, 4, 9, 6, 1, 4, 7, 6, 5, 6, 3, 6, 9, 9, 1, 6, 3, 6, 5, 6, 7, 4, 9, 6, 1, 4, 7, 6, 5, 6, 3, 6, 9, 1, 1, 6, 3, 6, 5, 6, 7, 4, 9, 1, 1, 4, 7, 6, 5

Offset: 1

Greg Dresden, Mar 12 2006

The decimal number .147656369116... formed from these digits is a transcendental number; see Dresden's second article. These digits are never eventually periodic.

Digits appear with predictable frequencies: 1/10 for 3, 4, and 7; 1/9 for 5; 3/25 for 9; 28/225 for 1; and 307/900 for 6. - Charles R Greathouse IV, Oct 03 2022

			a(4) = 6 because 4^4 (which is 256) ends in 6.

T. D. Noe, Table of n, a(n) for n = 1..1000
Articles can be found on Dresden's Home Page.
Gregory P. Dresden, Two Irrational Numbers From the Last Non-Zero Digits of n! and n^n, Math. Mag. 74 (October 2001), 316-320.
Gregory P. Dresden, Three transcendental numbers from the last non-zero digits of n^n, F_n and n!, Mathematics Magazine, pp. 96-105, vol. 81, 2008.
Jose María Grau and A. M. Oller-Marcen, On the last digit and the last non-zero digit of n^n in base b., arXiv:1203.4066 [math.NT], 2012.
Jose María Grau and A. M. Oller-Marcen, On the last digit and the last non-zero digit of n^n in base b, Bull. Korean Math. Soc. 51 (2014), No. 5, pp. 1325-1337.
S. Ikeda and K. Matsuoka, On transcendental numbers generated by certain integer sequences, Siauliai Math. Semin. 8 (2013), 63-69. Mentions this sequence.
Index entries for transcendental numbers

Cf. A008904, A010879, A065881.
Cf. A056849.
Cf. A204815, A204816, A204817, A204818, A204819, A230024, A204697.
Cf. A322489, A322490.

f:= proc(n) local d, m, p; d:= min(padic:-ordp(n,2), padic:-ordp(n,5));
   m:= n/10^d;
   p:= n - 1 mod 4 + 1;
   m &^ p mod 10;
end proc:
seq(f(n), n=1..1000); # Robert Israel, Oct 19 2014

f[n_] := Block[{m = n}, While[ Mod[m, 10] == 0, m /= 10]; PowerMod[m, n, 10]]; Array[f, 105] (* Robert G. Wilson v, Mar 13 2006 and modified Oct 12 2014 *)

f(n) = while(!(n % 10), n/=10); n % 10; \\ A065881
a(n) = lift(Mod(f(n), 10)^n); \\ Michel Marcus, Sep 13 2022

a(n)=my(k=n/10^valuation(n,10)); lift(Mod(k,10)^(n%4+4)) \\ Charles R Greathouse IV, Sep 13 2022

def a(n):
    k = n
    while k%10 == 0: k //= 10
    return pow(k, n, 10)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Sep 13 2022

def A116081(n): return pow(int(str(n).rstrip('0')[-1]),n,10) # Chai Wah Wu, Dec 07 2023

a(n) = A065881(n)^n mod 10 = A010879(A065881(n)^(A010883(n-1))). - Robert Israel, Oct 19 2014

More terms from Robert G. Wilson v, Mar 13 2006

A203164 (n-1)-st elementary symmetric function of the first n terms of the periodic sequence (4,1,2,3,4,1,2,3,...).

Original entry on oeis.org

1, 5, 14, 50, 224, 320, 736, 2400, 10176, 12480, 27264, 86400, 359424, 414720, 884736, 2764800, 11390976, 12718080, 26763264, 82944000, 339738624, 371589120, 775028736, 2388787200, 9746251776, 10510663680, 21785739264, 66886041600

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Cf. A010883, A203162, A203163, A203165.

f[k_] := 1 + Mod[k + 2, 4];
t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 33}] (* A203164 *)

Conjecture: a(n)=48*a(n-4)-576*a(n-8) with G.f. x*(1+5*x+14*x^2+50*x^3+176*x^4+80*x^5+64*x^6) / (-1+24*x^4)^2. - R. J. Mathar, Oct 15 2013

A203165 (n-1)-st elementary symmetric function of the first n terms of the periodic sequence (3,4,1,2,3,4,1,2,...).

Original entry on oeis.org

1, 7, 19, 50, 174, 768, 1056, 2400, 7776, 32832, 39744, 86400, 273024, 1133568, 1299456, 2764800, 8626176, 35500032, 39481344, 82944000, 256794624, 1051066368, 1146617856, 2388787200, 7357464576, 30003167232, 32296402944, 66886041600

Offset: 1

Clark Kimberling, Dec 30 2011

nonn

Cf. A010883, A203162, A203163, A203164.

f[k_] := 1 + Mod[k + 1, 4];
t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 33}]  (* A203165 *)

Conjecture: G.f. x*(1+7*x+19*x^2+50*x^3+126*x^4+432*x^5+144*x^6) / (-1+24*x^4)^2 with a(n) = 48*a(n-4)-576*a(n-8). - R. J. Mathar, Jul 02 2013

A252922 a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3), with a(1)=0, a(2)=1, a(3)=4.

Original entry on oeis.org

0, 1, 4, 8, 14, 17, 25, 26, 35, 36, 46, 43, 58, 54, 66, 62, 79, 73, 88, 77, 101, 94, 110, 92, 120, 115, 133, 113, 138, 126, 158, 134, 167, 143, 165, 150, 193, 177, 189, 154, 206, 188, 228, 182, 224, 206, 234, 198, 244, 229, 274, 222, 263, 224, 272, 246, 312, 272, 290, 230, 318, 290, 326, 262, 327, 315, 355, 296

Offset: 1

Omar E. Pol, Dec 24 2014

This is also a rectangular array read by rows, with four columns, in which T(j,k) is the number of cells (also the area) of the j-th gap between the arms in the k-th quadrant of the spiral of the symmetric representation of sigma described in A239660, with j >= 1 and 1 <= k <= 4 and starting with T(1,1) = 0, see example.

We can find the spiral (mentioned above) on the terraces of the stepped pyramid described in A244050. - Omar E. Pol, Dec 07 2016

			a(5) = sigma(4) + sigma(3) + sigma(2) = 7 + 4 + 3 = 14. On the other hand a(5) = A024916(4) - A024916(1) = 15 - 1 = 14.
...
Also, if written as a rectangular array T(j,k) with four columns the sequence begins:
    0,   1,   4,   8;
   14,  17,  25,  26;
   35,  36,  46,  43;
   58,  54,  66,  62;
   79,  73,  88,  77;
  101,  94, 110,  92;
  120, 115, 133, 113;
  138, 126, 158, 134;
  167, 143, 165, 150;
  193, 177, 189, 154;
  206, 188, 228, 182;
  224, 206, 234, 198;
  244, 229, 274, 222;
  263, 224, 272, 246;
  312, 272, 290, 230;
  318, 290, 326, 262;
  ...
In this case T(2,1) = a(5) = 14.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A010883, A024916, A092403, A112610, A193553, A196020, A236104, A237270, A237271, A237593, A239052, A239053, A239931-A239934, A239660, A240020, A244050, A245092, A262626.

L:= [0,0,0,seq(numtheory:-sigma(n), n=1..100)]:
L[1..101]+L[2..102]+L[3..103]; # Robert Israel, Dec 07 2016

a252922[n_] := Block[{f}, f[1] = 0; f[2] = 1; f[3] = 4;
  f[x_] := DivisorSigma[1, x - 1] + DivisorSigma[1, x - 2] +
DivisorSigma[1, x - 3]; Table[f[i], {i, n}]]; a252922[68] (* Michael De Vlieger, Dec 27 2014 *)

v=concat([0,1,4],vector(100,n,sigma(n)+sigma(n+1)+sigma(n+2))) \\ Derek Orr, Dec 30 2014

a(1) = 0, a(2) = sigma(1) = 1, a(3) = sigma(2) + sigma(1) = 4; for n >= 4, a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3).

a(n) = A024916(n-1) - A024916(n-4) for n >= 5.

A328943 a(n) = 2 + (n mod 4).

Original entry on oeis.org

2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3, 4, 5, 2, 3

Offset: 0

David Nacin, Oct 31 2019

nonn

Terms of the simple continued fraction of (36+sqrt(1806))/34.
2345/9999=0.234523452345...
Partial sums are given by A130482(n) + 2*n + 2.
Example of a sequence where the largest of any four consecutive terms equals the sum of the two smallest.

Michael De Vlieger, Table of n, a(n) for n = 0..10000
David Nacin, Van der Laan Sequences and a Conjecture on Padovan Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.2.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 1).

Cf. A010883, A010873, A130482

Mathematica
```
PadRight[{}, 120, {2, 3, 4, 5}]
```
Python
```
def a(n):
   return n%4+2
```

a(n) = 2 + (n mod 4).
G.f.: (5x^3 + 4x^2 + 3x + 2)/(1 - x^4).
a(n) = A010873(n) + 2 = A010883(n) + 1.
a(n) = 14 - a(n-1) - a(n-2) - a(n-3) for n > 2.

A177037 Decimal expansion of (9 + 2*sqrt(39))/15.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 01 2010

Continued fraction expansion of (9 + 2*sqrt(39))/15 is A010883.

The positive solution to 15*x^2 - 18*x - 5 = 0. - Michal Paulovic, Feb 23 2023

			1.43266639978645309411...

Cf. A010493 (decimal expansion of sqrt(39)), A010883 (repeat 1, 2, 3, 4).

evalf(3/5 + sqrt(52/75), 100); # Michal Paulovic, Feb 24 2023

RealDigits[(9+2*Sqrt[39])/15,10,120][[1]] (* Harvey P. Dale, Feb 12 2013 *)

my(c=(9+2*quadgen(4*39))/15); a_vector(len) = digits(floor(c*10^(len-1)));
a_vector(100) \\ Kevin Ryde, Feb 24 2023

Equals sqrt(1/3 + (6/5) * sqrt(1/3 + (6/5) * sqrt(1/3 + (6/5) * ...))). - Michal Paulovic, Feb 23 2023

cp's OEIS Frontend

A203163 (n-1)-st elementary symmetric function of the first n terms of (1,2,3,4,1,2,3,4,1,2,3,4,...) = A010883.

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A239660 Triangle read by rows in which row n lists two copies of the n-th row of triangle A237593.

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A116081 Final nonzero digit of n^n.

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A203164 (n-1)-st elementary symmetric function of the first n terms of the periodic sequence (4,1,2,3,4,1,2,3,...).

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A203165 (n-1)-st elementary symmetric function of the first n terms of the periodic sequence (3,4,1,2,3,4,1,2,...).

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A252922 a(n) = sigma(n-1) + sigma(n-2) + sigma(n-3), with a(1)=0, a(2)=1, a(3)=4.

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A328943 a(n) = 2 + (n mod 4).

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A177037 Decimal expansion of (9 + 2*sqrt(39))/15.

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