A126646 -id:A126646 - cp's OEIS frontend

A232602 a(n) = Sum_{k=0..n} k^p*q^k, where p=3, q=-2.

0, -2, 30, -186, 838, -3162, 10662, -33242, 97830, -275418, 748582, -1977306, 5100582, -12897242, 32060454, -78531546, 189903910, -454052826, 1074770982, -2521320410, 5867287590, -13554437082

Offset: 0

Stanislav Sykora, Nov 27 2013

			a(3) = 0^3*2^0 - 1^3*2^1 + 2^3*2^2 - 3^3*2^3 = -186.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-7,-16,-8,16,16).

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

[2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27: n in [0..35]]; // G. C. Greubel, Mar 31 2021

A232602:= n-> 2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27; seq(A232602(n), n=0..35); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-7,-16,-8,16,16}, {0,-2,30,-186,838}, 40] (* G. C. Greubel, Mar 31 2021 *)

a(n)=((-1)^n*2^(n+1)*(27*n^3+27*n^2-9*n-3)+6)/81;

[2*(1 -(-2)^n*(1 +3*n -9*n^2 -9*n^3))/27 for n in (0..35)] # G. C. Greubel, Mar 31 2021

a(n) = 2*(1 - (-2)^n*(1 +3*n -9*n^2 -9*n^3))/27.
G.f.: -2*x*(1-8*x+4*x^2) / ( (1-x)*(1+2*x)^4 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (2/27)*(exp(x) - (1 +30*x -144*x^2 +72*x^3)*exp(-2*x)). - G. C. Greubel, Mar 31 2021
a(n) = - 7*a(n-1) - 16*a(n-2) - 8*a(n-3) + 16*a(n-4) + 16*a(n-5). - Wesley Ivan Hurt, Mar 31 2021

A255047 1 together with the positive terms of A000225.

Original entry on oeis.org

1, 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607, 16777215, 33554431, 67108863, 134217727, 268435455, 536870911, 1073741823, 2147483647, 4294967295

Offset: 0

Omar E. Pol, Feb 15 2015

Also, right border of A246674 arranged as an irregular triangle.
Essentially the same as A168604, A126646 and A000225.
Total number of lambda-parking functions induced by all partitions of n. a(0)=1: [], a(1)=1: [1], a(2)=3: [1], [2], [1,1], a(4)=7: [1], [2], [3], [1,1], [1,2], [2,1], [1,1,1]. - Alois P. Heinz, Dec 04 2015
Also, the decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 19 2017
Also number of multiset partitions of {1,1} U [n] into exactly 2 nonempty parts.  a(2) = 3: 111|2, 11|12, 1|112. - Alois P. Heinz, Aug 18 2017
Also, the number of unlabeled connected P-series (equivalently, connected P-graphs) with n+1 elements. - Salah Uddin Mohammad, Nov 19 2021

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
R. Stanley, Parking Functions, 2011.
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A011782, A028310, A246674, A253909, A265007, A265202, A349276.
Row n=1 of A263159.
Column k=2 of A291117.
Cf. A078485.

[1] cat [2^n -1: n in [1..40]]; // G. C. Greubel, Feb 07 2021

CoefficientList[Series[(1 -2*x +2*x^2)/((1-x)*(1-2*x)), {x, 0, 33}], x] (* or *) LinearRecurrence[{3, -2}, {1,1,3}, 40] (* Vincenzo Librandi, Jul 20 2017 *)
Table[2^n -1 +Boole[n==0], {n, 0, 40}] (* G. C. Greubel, Feb 07 2021 *)

def A255047(n): return -1^(-1<Chai Wah Wu, Dec 21 2022

[1]+[2^n -1 for n in (1..40)] # G. C. Greubel, Feb 07 2021

From Alois P. Heinz, Feb 19 2015: (Start)
O.g.f.: (1 -2*x +2*x^2)/((1-x)*(1-2*x)).
E.g.f.: exp(2*x) - exp(x) + 1. (End)
a(n) = A078485(n+1) for n > 2. - Georg Fischer, Oct 22 2018

A036827 a(n) = 26 + 2^(n+1)(-13 +9n -3*n^2 +n^3).

Original entry on oeis.org

0, 2, 34, 250, 1274, 5274, 19098, 63002, 194074, 567322, 1591322, 4317210, 11395098, 29392922, 74350618, 184942618, 453378074, 1097334810, 2626158618, 6222250010, 14610858010, 34032582682, 78693531674, 180757725210, 412685959194

Offset: 0

N. J. A. Sloane

			a(3) = 2^0*0^3 + 2^1*1^3 + 2^2*2^3 + 2^3*3^3 = 250.

M. Petkovsek et al., A=B, Peters, 1996, p. 97.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (9,-32,56,-48,16).

Cf. A059841 (p=0,q=-1), A130472 (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), this sequence (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

a036827 n = 2^(n+1) * (n^3 - 3*n^2 + 9*n - 13) + 26
-- Reinhard Zumkeller, May 24 2012

[2*(13 + 2^n*(-13 +9*n -3*n^2 +n^3)): n in [0..35]]; // G. C. Greubel, Mar 31 2021

A036827:= n-> 2*(13 + 2^n*(-13 +9*n -3*n^2 +n^3)); seq(A026827(n), n=0..30); # G. C. Greubel, Mar 31 2021

Table[26 +2^(n+1)(-13 +9n -3n^2 +n^3), {n, 0, 30}] (* or *) LinearRecurrence[ {9, -32, 56, -48, 16}, {0, 2, 34, 250, 1274}, 31] (* Harvey P. Dale, Dec 15 2011 *)

a(n)=26+2^(n+1)*(-13+9*n-3*n^2+n^3) \\ Charles R Greathouse IV, Oct 07 2015

[2*(13 + 2^n*(-13 +9*n -3*n^2 +n^3)) for n in (0..35)] # G. C. Greubel, Mar 31 2021

a(n) = Sum_{k=0..n} 2^k*k^3. - Benoit Cloitre, Jun 11 2003
G.f.: 2*x*(1 +8*x +4*x^2)/((1-x)*(1-2*x)^4). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009
a(n) = 9*a(n-1) -32*a(n-2) +56*a(n-3) -48*a(n-4) +16*a(n-5) for n>4 with a(0)=0, a(1)=2, a(2)=34, a(3)=250, a(4)=1274. - Harvey P. Dale, Dec 15 2011
a(n) = Sum_{k=0..n} Sum_{i=0..n} k^3 * C(k,i). - Wesley Ivan Hurt, Sep 21 2017
E.g.f.: 2 (13*exp(x) + (-13 +14*x +8*x^3)*exp(2*x)). - G. C. Greubel, Mar 31 2021

A261922 a(n) = smallest nonnegative number that is not a substring of n in its binary representation.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Sep 16 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Similar to A261461.

Cf. A007088, A030308, A062289, A126646, A262281.

import Data.List (isInfixOf)
a261922 x = f a030308_tabf where
   f (cs:css) = if isInfixOf cs (a030308_row x)
                   then f css else foldr (\d v -> 2 * v + d) 0 cs
-- Reinhard Zumkeller, Sep 17 2015

bstr(n) = if (n==0, "0", my(s="", b=binary(n)); for (i=1, #b, s=concat(s, b[i])); s);
a(n) = my(sn=btostr(n), k=0); while (#strsplit(sn, bstr(k)) != 1, k++); k; \\ Michel Marcus, Sep 20 2023

def a(n): b=bin(n)[2:]; return next(k for k in range(2**len(b)) if bin(k)[2:] not in b)
print([a(n) for n in range(99)]) # Michael S. Branicky, Sep 21 2023

From Reinhard Zumkeller, Sep 17 2015: (Start)
a(A062289(n)) = A261461(A062289(n)).
a(A126646(n)) != A261461(A126646(n)). (End)

A293574 a(n) = Sum_{k=0..n} n^(n-k)*binomial(n+k-1,k).

Original entry on oeis.org

1, 2, 11, 82, 787, 9476, 139134, 2422218, 48824675, 1118286172, 28679699578, 814027423892, 25330145185646, 857375286365768, 31360145331198428, 1232586016712594010, 51805909208539809315, 2318588202311267591852, 110085368092924083334626, 5526615354023679440754396, 292501304641192746350100410

Offset: 0

Ilya Gutkovskiy, Oct 12 2017

nonn

a(n) is the n-th term of the main diagonal of iterated partial sums array of powers of n (see example).

			For n = 2 we have:
----------------------------
0   1   [2]   3    4     5
----------------------------
1,  2,   4,   8,  16,   32, ... A000079 (powers of 2)
1,  3,   7,  15,  31,   63, ... A126646 (partial sums of A000079)
1,  4, [11], 26,  57,  120, ... A000295 (partial sums of A126646)
----------------------------
therefore a(2) = 11.

Cf. A000312, A001700, A031973, A032443, A100192, A100193.

Join[{1}, Table[Sum[n^(n - k) Binomial[n + k - 1, k], {k, 0, n}], {n, 1, 20}]]
Table[SeriesCoefficient[1/((1 - x)^n (1 - n x)), {x, 0, n}], {n, 0, 20}]
Join[{1, 2}, Table[n^(2 n)/(n - 1)^n - Binomial[2 n, n + 1] Hypergeometric2F1[1, 2 n + 1, n + 2, 1/n]/n, {n, 2, 20}]]

a(n) = sum(k=0, n, n^(n-k)*binomial(n+k-1,k)); \\ Michel Marcus, Oct 12 2017

a(n) = [x^n] 1/((1 - x)^n*(1 - n*x)).

a(n) ~ exp(1) * n^n. - Vaclav Kotesovec, Oct 16 2017

A306714 Permanent of the circulant matrix whose first row is given by the binary expansion of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 6, 1, 2, 4, 9, 2, 9, 9, 24, 1, 2, 2, 13, 2, 13, 13, 44, 2, 13, 13, 44, 13, 44, 44, 120, 1, 2, 4, 20, 8, 17, 17, 80, 4, 17, 36, 82, 17, 80, 82, 265, 2, 20, 17, 80, 17, 82, 80, 265, 20, 80, 82, 265, 80, 265, 265, 720, 1, 2, 2, 31, 2, 24, 24

Offset: 0

Alois P. Heinz, Mar 05 2019

			The circulant matrix for n = 23 = 10111_2 is
  [1 0 1 1 1]
  [1 1 0 1 1]
  [1 1 1 0 1]
  [1 1 1 1 0]
  [0 1 1 1 1] and has permanent 44, thus a(23) = 44.
a(10) = 4 != a(12) = 2 although 10 = 1010_2 and 12 = 1100_2 have the same number of 0's and 1's.

Alois P. Heinz, Table of n, a(n) for n = 0..8191
Wikipedia, Circulant matrix
Wikipedia, Permanent (mathematics)
Index entries for sequences related to binary expansion of n

Cf. A000051, A000079, A000142, A000225, A007283, A007395, A008305, A040000, A113473, A126646, A306595.

a:= n-> (l-> LinearAlgebra[Permanent](Matrix(nops(l),
         shape=Circulant[l])))(convert(n, base, 2)):
seq(a(n), n=0..100);

a(n) = 1 <=> n in { A000079 }.
a(n) = floor(log_2(2n))! for n in { A126646 }.
a(A000225(n)) = A000142(n) for n >= 1.
a(A000051(n)) = A040000(n).
a(A007283(n)) = A007395(n+1).

A337277 Stern's triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1

Offset: 0

N. J. A. Sloane, Sep 09 2020

The first two rows are 1, then 1,1,1. To get row n, copy row n-1, and insert c+d between every pair of adjacent terms c,d, and finally insert a 1 at the beginning and end of the row.

The maximum value in row n is A000045(n+1). - Alois P. Heinz, Sep 09 2020

			Triangle begins:
  1;
  1, 1, 1;
  1, 1, 2, 1, 2, 1, 1;
  1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1;
  1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1;
  ...

Stanley, Richard P. "Some Linear Recurrences Motivated by Stern’s Diatomic Array." The American Mathematical Monthly 127.2 (2020): 99-111.

Alois P. Heinz, Rows n = 0..14, flattened
Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019.

Cf. A000045, A002487, A049456, A334627.
Row sums give A000244.
Row lengths give A126646.

T:= proc(n) option remember; `if`(n=0, 1, (L-> [1, L[1], seq(
      [L[i-1]+L[i], L[i]][], i=2..nops(L)), 1][])([T(n-1)]))
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Sep 09 2020

Nest[Append[#, Flatten@ Join[{1}, If[Length@ # > 1, Map[{#1, #1 + #2} & @@ # &, Partition[#[[-1]], 2, 1] ], {}], {#[[-1, -1]]}, {1}]] &, {{1}}, 5] // Flatten (* Michael De Vlieger, Sep 09 2020 *)

T(n,n) = A002487(n+1). - Alois P. Heinz, Sep 09 2020

A351894 Numbers that contain only odd digits in their factorial-base representation.

Original entry on oeis.org

1, 3, 9, 21, 33, 45, 81, 93, 153, 165, 201, 213, 393, 405, 441, 453, 633, 645, 681, 693, 873, 885, 921, 933, 1113, 1125, 1161, 1173, 1353, 1365, 1401, 1413, 2313, 2325, 2361, 2373, 2553, 2565, 2601, 2613, 2793, 2805, 2841, 2853, 3753, 3765, 3801, 3813, 3993, 4005

Offset: 1

Amiram Eldar, Feb 24 2022

All the terms above 1 are odd multiples of 3.

			3 is a term since its factorial-base presentation, 11, has only odd digits.
21 is a term since its factorial-base presentation, 311, has only odd digits.

Amiram Eldar, Table of n, a(n) for n = 1..18056 (terms below 11!)
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Cf. A007623, A016945, A351893.
Subsequence: A007489
Similar sequences: A003462 \ {0} (ternary), A014261 (decimal), A032911 (base 4), A032912 (base 5), A033032 (base 6), A033033 (base 7), A033034 (base 8), A033035 (base 9), A033036 (base 11), A033037 (base 12), A033038 (base 13), A033039 (base 14), A033040 (base 15), A033041 (base 16), A126646 (binary).

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; Select[Range[1, max!, 2], AllTrue[fctBaseDigits[#], OddQ] &]

A248687 Sum of the numbers in row n of the triangular array at A248686.

Original entry on oeis.org

1, 3, 10, 43, 221, 1371, 9696, 78751, 712447, 7173853, 79106413, 952587175, 12397677007, 173864946685, 2609479384942, 41786786069887, 710577455524223, 12795789975272877, 243154034699436147, 4864103085730989101, 102153340062463300261, 2247608818115460466681

Offset: 1

Clark Kimberling, Oct 11 2014

			First seven rows of the array at A248686:
1
1   2
1   3    6
1   6    12    24
1   10   30    60    120
1   20   90    180   360    720
1   35   210   630   1260   2520   5040
The row sums are 1, 3, 10, ...

Alois P. Heinz, Table of n, a(n) for n = 1..450 (first 100 terms from Clark Kimberling)

Cf. A248686.

b:= proc(n, k) option remember; `if`(k<1,
     `if`(n=k, 1, 0), n!/mul(iquo(n+i, k)!, i=0..k-1))
    end:
a:= n-> add(b(n,k), k=0..n):
seq(a(n), n=1..22);  # Alois P. Heinz, Feb 20 2024

f[n_, k_] := f[n, k] = n!/Product[Floor[(n + i)/k]!, {i, 0, k - 1}]
t = Table[f[n, k], {n, 0, 10}, {k, 1, n}];
u = Flatten[t]  (* A248686 sequence *)
TableForm[t]    (* A248686 array *)
Table[Sum[f[n, k], {k, 1, n}], {n, 1, 22}] (* A248687 *)

a(n) = Sum_{k=1..n} n!/(n(1)!*n(2)!* ... *n(k)!), where n(i) = floor((n + i - 1)/k) for i = 1..k.
a(n) ~ 2 * n!. - Vaclav Kotesovec, Oct 21 2014
a(n) mod 2 = 0 <=> n in { A126646 } \ { 1 }. - Alois P. Heinz, Feb 20 2024

A307078 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k-x^k).

Original entry on oeis.org

1, 1, 3, 1, 2, 7, 1, 2, 4, 15, 1, 2, 3, 8, 31, 1, 2, 3, 5, 16, 63, 1, 2, 3, 4, 10, 32, 127, 1, 2, 3, 4, 6, 21, 64, 255, 1, 2, 3, 4, 5, 12, 43, 128, 511, 1, 2, 3, 4, 5, 7, 28, 86, 256, 1023, 1, 2, 3, 4, 5, 6, 14, 64, 171, 512, 2047, 1, 2, 3, 4, 5, 6, 8, 36, 136, 341, 1024, 4095

Offset: 0

Seiichi Manyama, Mar 22 2019

			Square array begins:
     1,   1,   1,   1,  1,  1,  1,  1, 1, ...
     3,   2,   2,   2,  2,  2,  2,  2, 2, ...
     7,   4,   3,   3,  3,  3,  3,  3, 3, ...
    15,   8,   5,   4,  4,  4,  4,  4, 4, ...
    31,  16,  10,   6,  5,  5,  5,  5, 5, ...
    63,  32,  21,  12,  7,  6,  6,  6, 6, ...
   127,  64,  43,  28, 14,  8,  7,  7, 7, ...
   255, 128,  86,  64, 36, 16,  9,  8, 8, ...
   511, 256, 171, 136, 93, 45, 18, 10, 9, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-6 give A126646, A000079, A024494(n+1), A038504(n+1), A133476(n+1), A119336.

Cf. A306846, A306915, A307079.

T[n_, k_] := Sum[Binomial[n+1, k*j+1], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+1,k*j+1).

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i,k*j) * binomial(n-i,k*j).

cp's OEIS Frontend

A232602 a(n) = Sum_{k=0..n} k^p*q^k, where p=3, q=-2.

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A255047 1 together with the positive terms of A000225.

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A036827 a(n) = 26 + 2^(n+1)*(-13 +9*n -3*n^2 +n^3).

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A261922 a(n) = smallest nonnegative number that is not a substring of n in its binary representation.

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A293574 a(n) = Sum_{k=0..n} n^(n-k)*binomial(n+k-1,k).

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A306714 Permanent of the circulant matrix whose first row is given by the binary expansion of n.

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A337277 Stern's triangle read by rows.

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A036827 a(n) = 26 + 2^(n+1)(-13 +9n -3*n^2 +n^3).