A048473 -id:A048473 - cp's OEIS frontend

A198688 6*7^n-1.

5, 41, 293, 2057, 14405, 100841, 705893, 4941257, 34588805, 242121641, 1694851493, 11863960457, 83047723205, 581334062441, 4069338437093, 28485369059657, 199397583417605, 1395783083923241, 9770481587462693, 68393371112238857

Offset: 0

Vincenzo Librandi, Oct 29 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-7).

Cf. A048473, A081655, A198480, A198647.

Magma
```
[6*7^n-1: n in [0..30]]
```

a(n) = 7*a(n-1)+6. a(n) = 8*a(n-1)-7*a(n-2), n>1.

G.f. ( 5+x ) / ( (7*x-1)*(x-1) ). - R. J. Mathar, Oct 30 2011

A210202 Triangle of coefficients of polynomials v(n,x) jointly generated with A210201; see the Formula section.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 7, 17, 17, 12, 12, 35, 50, 40, 24, 20, 70, 120, 135, 92, 48, 33, 134, 275, 365, 346, 208, 96, 54, 251, 593, 930, 1033, 856, 464, 192, 88, 461, 1236, 2206, 2874, 2784, 2064, 1024, 384, 143, 835, 2500, 5015, 7389, 8355, 7240, 4880

Offset: 1

Clark Kimberling, Mar 18 2012

Column 1: F(n+2)-1, where F=A000045 (Fibonacci numbers)
Row sums: A048473
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
2....3
4....7....6
7....17...17...12
12...35...50...40...24
First three polynomials v(n,x): 1, 2 + 3x , 4 + 7x + 6x^2.

Cf. A210201, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210201 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210202 *)

u(n,x)=u(n-1,x)+v(n-1,x)+1,
v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A210800 Triangle of coefficients of polynomials v(n,x) jointly generated with A210799; see the Formula section.

Original entry on oeis.org

1, 1, 2, 5, 4, 3, 5, 14, 9, 5, 17, 28, 36, 19, 8, 17, 70, 88, 83, 38, 13, 53, 136, 251, 245, 181, 73, 21, 53, 298, 557, 746, 613, 379, 137, 34, 161, 568, 1376, 1930, 2030, 1439, 769, 252, 55, 161, 1162, 2888, 5026, 5818, 5139, 3221, 1524, 457, 89, 485

Offset: 1

Clark Kimberling, Mar 27 2012

Row n starts a term of A048473 and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Alternating row sums: 1,2,3,4,5,6,7,...
For a discussion and guide to related arrays, see A208510.

			First five rows:
1
1....2
5....4....3
5....14...9....5
17...28...36...19...8
First three polynomials v(n,x): 1, 1 + 2x, 5 + 4x + 3x^2

Cf. A210799, A208510.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 1; c = 1; h = 2; p = -1; f = 0;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210799 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210800 *)

u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2) + a(k) with a(0) = 2, a(1) = 1, a(k) = 0 if k>1, T(1,0) = T(2,0) = 1, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k >n. - Philippe Deléham, Mar 31 2012

A079362 Sequence of sums of alternating powers of 3.

Original entry on oeis.org

1, 4, 5, 14, 17, 44, 53, 134, 161, 404, 485, 1214, 1457, 3644, 4373, 10934, 13121, 32804, 39365, 98414, 118097, 295244, 354293, 885734, 1062881, 2657204, 3188645, 7971614, 9565937, 23914844, 28697813, 71744534, 86093441, 215233604

Offset: 1

Cino Hilliard, Feb 15 2003

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3).

Cf. A079360, A079363, A028242, A048473 (bisection).

a:=[1,4,5];; for n in [4..30] do a[n]:=a[n-1]+3*a[n-2]-3*a[n-3]; od; a; # G. C. Greubel, Aug 07 2019

I:=[1,4,5]; [n le 3 select I[n] else Self(n-1) +3*Self(n-2) -3*Self(n-3): n in [1..40]]; // G. C. Greubel, Aug 07 2019

a[1]:=1:a[2]:=4:for n from 3 to 100 do a[n]:=3*a[n-2]+2 od: seq(a[n], n=1..33); # Zerinvary Lajos, Mar 17 2008

LinearRecurrence[{1,3,-3},{1,4,5},40] (* Harvey P. Dale, Oct 18 2016 *)

a(n)=if(n<1,0,1+sum(k=2,n,3^((k\2)-(k%2))))

a(n)=if(n<0,0,(5/3-3*n%2)*2^ceil(n/2)-1)

@CachedFunction
def a(n):
    if (n==0): return 1
    elif (1<=n<=2): return n+3
    else: return a(n-1) + 3*a(n-2) - 3*a(n-3)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 07 2019

G.f.: x*(1+3*x-2*x^2)/((1-x)*(1-3*x^2)). - Michael Somos, Feb 18 2003

For n >= 1, a(2n-1) = (2/3)*3^n - 1, a(2n) = (5/3)*3^n - 1. - Benoit Cloitre, Feb 16 2003

A198646 a(n) = 11*3^n-1.

Original entry on oeis.org

10, 32, 98, 296, 890, 2672, 8018, 24056, 72170, 216512, 649538, 1948616, 5845850, 17537552, 52612658, 157837976, 473513930, 1420541792, 4261625378, 12784876136, 38354628410, 115063885232, 345191655698, 1035574967096, 3106724901290

Offset: 0

Vincenzo Librandi, Oct 28 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A027107, A048473, A171498.

Magma
```
[11*3^n-1: n in [0..30]]
```

11*3^Range[0, 30] - 1 (* Wesley Ivan Hurt, Oct 02 2021 *)

a(n) = 3*a(n-1)+2, a(0)=10.
G.f.: (10-8*x) / ((3*x-1)*(x-1)). - R. J. Mathar, Oct 30 2011
a(n) = 4*a(n-1)-3*a(n-2). - Wesley Ivan Hurt, Oct 02 2021
a(n) = 2*A052909(n+1). - R. J. Mathar, Apr 07 2022

A338486 Numbers n whose symmetric representation of sigma(n) consists of 3 regions with maximum width 2.

Original entry on oeis.org

15, 35, 45, 70, 77, 91, 110, 130, 135, 143, 154, 170, 182, 187, 190, 209, 221, 225, 238, 247, 266, 286, 299, 322, 323, 350, 374, 391, 405, 418, 437, 442, 493, 494, 506, 527, 550, 551, 572, 589, 598, 638, 646, 650, 667, 682, 703, 713, 748, 754, 782, 806, 814, 836, 850

Offset: 1

Hartmut F. W. Hoft, Oct 30 2020

nonn

This sequence is a subsequence of A279102. The definition of the sequence excludes squares of primes, A001248, since the 3 regions of their symmetric representation of sigma have width 1 (first column in the irregular triangle of A247687).
Table of numbers in this sequence arranged by the number of prime factors, counting multiplicities:
     2     3     4      5      6      7  ...
  ------------------------------------------
    15    45   135    405   1215   3645
    35    70   225   1125   5625    ...
    77   110   350   1750   8750 744795
    91   130   550   2584    ...    ...
   143   154   572   2750  85455
   187   170   650   3128    ...
   209   182   748   3250
   221   190   836   3496
   247   238   850   3944
   299   266   884   4216
   ...   ...   ...    ...
              1035   9585
               ...    ...
The numbers in the first row of the table above are b(k) = 5*3^k, k>=1, (see A005030) so that infinitely many odd numbers occur outside of the first column. The central region of the symmetric representation of sigma(b(k)) contains 2*k-1 separate contiguous sections consisting of sequences of entire legs of width 2, k>=1 (see Lemma 2 in the link).
Conjecture: The combined extent of these sections in sigma(b(k)) is 2*3^(k-1) - 1 = A048473(k-1), k>=1.
Since each number n in the first column and first row has a prime factor of odd exponent a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 2. For odd numbers n not in the first row or column in which all prime factors have even powers, such as 225 and 5625 in the second row, a contiguous section of the symmetric representation of sigma(n) centered on the diagonal has width 1 (see Lemma 1 in the link).
For each k>=3 and every prime p such that b(k-1) < 2*p < 4*b(k-2), the odd number p*b(k-1) is in the column of b(k). The two inequalities are equivalent to b(k-1) <= row(p*b(k-1)) < 2*b(k-1) ensuring that the symmetric representation of sigma(p*b(k-1)) consists of 3 regions.
45 is the only odd number in its column (see Lemma 3 in the link).
Since the factors of n = p*q satisfy 2 < p < q < 2*p the first column in the table above is a subsequence of A082663 and of A087718 (see Lemma 4 in the link). Each of the two outer regions consists of a single leg of width 1 and length (1 + p*q)/2. The center region of size p+q consists of two subparts (see A196020 & A280851) of width 1 of sizes 2*p-q and 2*q-p, respectively (see Lemma 5 in the link). The table below arranges the first column in the table above according to the length 2*p-q of their single contiguous extent of width 2 in the center region:
     1     3      5      7      9     11     13     15  ...
------------------------------------------------------
    15    35    187    247    143    391   2257    323
    91    77    493    589    221   1363   3139    437
   703   209    943   2479    551   2911   6649    713
  1891   299   1537   3397    851   3901    ...   1247
  2701   527   4183   8509   1643   6313          1457
  ...    ...    ...    ...    ...    ...          ....
A129521: p*q satisfies 2*p - q = 1 (excluding A129521(1)=6)
A226755: p*q satisfies 2*p - q = 3 (excluding A226755(1)=9)
Sequences with larger differences 2*p - q are not in OEIS.

			a(6) = 91 = 7*13 is in the sequence and in the 2-column of the first table since 1 < 2 < 7 < 13 = row(91) representing the 4 odd divisors 1 - 91 - 7 - 13 (see A237048) results in the following pattern for the widths of the legs (see A249223): 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2. It also is in the 1-column of the second table since it has a single area of width 2 which is 1 unit long.
a(29) = 405 = 5*3^4 is in the sequence and in the 5-column of the first table since 1 < 2 < 3 < 5 < 6 < 9 < 10 < 15 < 18 < 27 = row(405) representing the 10 odd divisors 1 - 405 - 3 - 5 - 135 - 9 - 81 - 15 - 45 - 27 results in the following pattern for the widths of the legs: 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2 for 3 regions with width not exceeding 2, and 7 = 2*4 - 1 sections of width 2 in the central region.
a(35) = 506 = 2*11*23 is in the sequence since positions 1 < 4 < 11 < 23 < row(506) = 31 representing the 4 odd divisors 1 - 253 - 11 - 23 results in the following pattern for the widths of the legs: 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2 for 3 regions with width not exceeding 2, with the two outer regions consisting of 3 legs of width 1, and a single area of width 2 in the central region.

Hartmut F. W. Hoft, proofs for quoted lemmas

Cf. A001248, A005030, A048473, A082663, A087718, A129521, A196020, A226755, A235791, A237048, A237270, A237271, A237591, A237593, A247687, A249223, A279102, A280107, A280851.

(* Functions path and a237270 are defined in A237270 *)
maxDiagonalLength[n_] := Max[Map[#[[1]]-#[[2]]&, Transpose[{Drop[Drop[path[n], 1], -1], path[n-1]}]]]
a338486[m_, n_] := Module[{r, list={}, k}, For[k=m, k<=n, k++, r=a237270[k]; If[Length[r]== 3 && maxDiagonalLength[k]==2,AppendTo[list, k]]]; list]
a338486[1, 850]

A090842 Square array of numbers read by antidiagonals where T(n,k) = ((k+3)*(k+2)^n-2)/(k+1).

Original entry on oeis.org

1, 1, 4, 1, 5, 10, 1, 6, 17, 22, 1, 7, 26, 53, 46, 1, 8, 37, 106, 161, 94, 1, 9, 50, 187, 426, 485, 190, 1, 10, 65, 302, 937, 1706, 1457, 382, 1, 11, 82, 457, 1814, 4687, 6826, 4373, 766, 1, 12, 101, 658, 3201, 10886, 23437, 27306, 13121, 1534, 1, 13, 122, 911, 5266

Offset: 0

Paul Barry, Dec 09 2003

Nodes on a tree with degree k interior nodes and degree 1 boundary nodes.

			Rows begin:
  1 4 10 22 ...
  1 5 17 53 ...
  1 6 26 106 ...
  1 7 37 187 ...

L. He, X. Liu and G. Strang, Trees with Cantor Eigenvalue Distribution, Studies in Applied Mathematics 110 (2), 123-138, 2003.

Rows include A033484, A048473, A020989, A057651, A061801, A090843.

The total number of nodes on a tree with degree k interior nodes and degree 1 boundary nodes is given by N(k, r) = (k*(k-1)^r-2)/(k-2).

G.f.: Sum_{k>=0} (1+x*y)/(1-x*y)/(1-(k+2)*x*y)*y^k. - Vladeta Jovovic, Dec 12 2003

A102105 a(n) = (195^n - 163^n + 1) / 4.

Original entry on oeis.org

1, 12, 83, 486, 2645, 13872, 71303, 362346, 1829225, 9198612, 46150523, 231225006, 1157542205, 5791962552, 28972567343, 144901100466, 724620293585, 3623445841692, 18118262329763, 90594411012726, 452981353155365, 2264934660052032, 11324756983085783

Offset: 0

Gary W. Adamson, Dec 30 2004

Sum of the entries in the last row of the 3 X 3 matrix M^n, where M = {{1, 0, 0}, {2, 3, 0}, {3, 4, 5}}.

Sum of the entries in the second row of M^n = A048473(n).

			a(4) = 2645 = 9*486 - 23*83 + 15*12 = 9*a(3) - 23*a(2) + 15*a(1).
a(4) = 2645 since M^4 * {1, 1, 1} = {1, 161, 2645}, where 161 = A048473(4).

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-23,15).

Cf. A000326, A048473, A094727.

List([0..30], n-> (19*5^n -16*3^n +1)/4); # G. C. Greubel, Oct 27 2019

[(19*5^n -16*3^n +1)/4: n in [0..30]]; // G. C. Greubel, Oct 27 2019

with(linalg): M[1]:=matrix(3,3,[1,0,0,2,3,0,3,4,5]): for n from 2 to 23 do M[n]:=multiply(M[1],M[n-1]) od: 1,seq(multiply(M[n],matrix(3,1,[1,1,1]))[3,1],n=1..23);
seq((19*5^n -16*3^n +1)/4, n=0..30); # G. C. Greubel, Oct 27 2019

Table[(19*5^n -16*3^n +1)/4, {n,0,30}] (* G. C. Greubel, Oct 27 2019 *)
LinearRecurrence[{9,-23,15},{1,12,83},30] (* Harvey P. Dale, Sep 19 2021 *)

Vec((1 + 3*x - 2*x^2) / ((1 - x)*(1 - 3*x)*(1 - 5*x)) + O(x^30)) \\ Colin Barker, Mar 03 2017

[(19*5^n -16*3^n +1)/4 for n in (0..30)] # G. C. Greubel, Oct 27 2019

a(n) = 9*a(n-1) - 23*a(n-2) + 15*a(n-3), a(0)=1,a(1)=12,a(2)=83 (derived from the minimal polynomial of the matrix M).
G.f.: (1 + 3*x - 2*x^2) / ((1 - x)*(1 - 3*x)*(1 - 5*x)). - Colin Barker, Mar 03 2017
E.g.f.: (exp(x) - 16*exp(3*x) + 19*exp(5*x))/4. - G. C. Greubel, Oct 27 2019

Corrected by T. D. Noe, Nov 07 2006
Edited by N. J. A. Sloane, Dec 02 2006
New definition from Ralf Stephan, May 17 2007

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

Original entry on oeis.org

3, 32, 355, 4110, 48887, 588886, 7111107, 85555550, 1022222215, 12111111102, 142222222211, 1655555555542, 19111111111095, 218888888888870, 2488888888888867, 28111111111111086, 315555555555555527, 3522222222222222190, 39111111111111111075, 432222222222222222182

Offset: 0

Ctibor O. Zizka, Mar 06 2008

Sequence generalized: a(n) = a(0)*(B^n) + F(n)* ((B^n)-1)/(B-1); a(0), B integers, F(n) arithmetic function.
Examples:
a(0) = 1, B = 10, F(n) = 1 gives A002275, F(n) = 2 gives A090843, F(n) = 3 gives A097166, F(n) = 4 gives A099914, F(n) = 5 gives A099915.
a(0) = 1, B = 2, F(n) = 1 gives A000225, F(n) = 2 gives A033484, F(n) = 3 gives A036563, F(n) = 4 gives A048487, F(n) = 5 gives A048488, F(n) = 6 gives A048489.
a(0) = 1, B = 3, F(n) = 1 gives A003462, F(n) = 2 gives A048473, F(n) = 3 gives A134931, F(n) = 4 gives A058481, F(n) = 5 gives A116952.
a(0) = 1, B = 4, F(n) = 1 gives A002450, F(n) = 2 gives A020989, F(n) = 3 gives A083420, F(n) = 4 gives A083597, F(n) = 5 gives A083584.
a(0) = 1, B = 5, F(n) = 1 gives A003463, F(n) = 2 gives A057651, F(n) = 3 gives A117617, F(n) = 4 gives A081655.
a(0) = 2, B = 10, F(n) = 1 gives A037559, F(n) = 2 gives A002276.

			a(3) = 3*10^3 + (3*3 + 1)*(10^3 - 1)/9 = 4110.

G. C. Greubel, Table of n, a(n) for n = 0..985
Index entries for linear recurrences with constant coefficients, signature (33,-393,1991,-3930,3300,-1000).

Cf. A002275, A011557.

Table[3*10^n +(n^2 +1)*(10^n -1)/9, {n,0,30}] (* G. C. Greubel, Jan 05 2022 *)

a(n) = 3*(10^n) + (n*n+1)*((10^n)-1)/9; \\ Jinyuan Wang, Feb 27 2020

[3*10^n +(1+n^2)*(10^n -1)/9 for n in (0..30)] # G. C. Greubel, Jan 05 2022

a(n) = 3*(10^n) + (n^2 + 1)*(10^n - 1)/9.

O.g.f.: (3 - 67*x + 478*x^2 - 1002*x^3 + 850*x^4 - 100*x^5)/((1-x)^3 * (1-10*x)^3). - R. J. Mathar, Mar 16 2008

More terms from R. J. Mathar, Mar 16 2008

More terms from Jinyuan Wang, Feb 27 2020

A155155 a(n) = 2(103^n - 1).

Original entry on oeis.org

18, 58, 178, 538, 1618, 4858, 14578, 43738, 131218, 393658, 1180978, 3542938, 10628818, 31886458, 95659378, 286978138, 860934418, 2582803258, 7748409778, 23245229338, 69735688018, 209207064058, 627621192178, 1882863576538

Offset: 0

Paul Curtz, Jan 21 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A003462, A048473.

[2*(10*3^n-1): n in [0..30]]; // Vincenzo Librandi, Aug 07 2011

seq(2*(10*3^n -1), n=0..30); # G. C. Greubel, Mar 20 2021

2*(10*3^Range[0,30] -1) (* G. C. Greubel, Mar 20 2021 *)

[2*(10*3^n - 1) for n in (0..30)] # G. C. Greubel, Mar 20 2021

a(n) = 4*a(n-1) - 3*a(n-2).
G.f.: ( 18 - 14*x ) / ( (1-x)*(1-3*x) ).
a(n) = 10*A048473(n) + 8 = A048473(n) + A048473(n+2).
a(n) = 3*a(n-1) - a(n-2) + 3*a(n-3) + 8.
From G. C. Greubel, Mar 20 2021: (Start)
a(n) = 18*A003462(n+1) - 14*A003462(n).
E.g.f.: 2*( 10*exp(3*x) - exp(x) ). (End)
a(n) = 2 * A198645(n). - Joerg Arndt, Mar 21 2021

cp's OEIS Frontend

A198688 6*7^n-1.

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Magma

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A210202 Triangle of coefficients of polynomials v(n,x) jointly generated with A210201; see the Formula section.

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A210800 Triangle of coefficients of polynomials v(n,x) jointly generated with A210799; see the Formula section.

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A079362 Sequence of sums of alternating powers of 3.

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A198646 a(n) = 11*3^n-1.

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Magma

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A338486 Numbers n whose symmetric representation of sigma(n) consists of 3 regions with maximum width 2.

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A090842 Square array of numbers read by antidiagonals where T(n,k) = ((k+3)*(k+2)^n-2)/(k+1).

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A102105 a(n) = (19*5^n - 16*3^n + 1) / 4.

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A102105 a(n) = (195^n - 163^n + 1) / 4.

A137215 a(n) = 3(10^n) + (n^2 + 1)(10^n - 1)/9.

A155155 a(n) = 2(103^n - 1).