A248216 -id:A248216 - cp's OEIS frontend

A155597 a(n) = 6^n - 2^n + 1.

1, 5, 33, 209, 1281, 7745, 46593, 279809, 1679361, 10077185, 60465153, 362795009, 2176778241, 13060685825, 78364147713, 470184951809, 2821109841921, 16926659313665, 101559956406273, 609359739486209, 3656158439014401, 21936950638280705, 131621703838072833

Offset: 0

Mohammad K. Azarian, Jan 25 2009

Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Cf. A074501, A020515, A155588, A155590, A155592, A155593, A155594, A155596.

Table[6^n-2^n+1,{n,0,20}] (* or *) LinearRecurrence[{9,-20,12},{1,5,33},20] (* Harvey P. Dale, Jul 13 2011 *)

a(n) = 6^n-2^n+1 \\ Charles R Greathouse IV, Jul 25 2011

G.f.: 1/(1-6*x)-1/(1-2*x)+1/(1-x).
E.g.f.: exp(6*x)-exp(2*x)+exp(x).
a(n) = 8*a(n-1)-12*a(n-2)+5 with a(0)=1, a(1)=5. - Vincenzo Librandi, Jul 21 2010
a(0)=1, a(1)=5, a(2)=33, a(n) = 9*a(n-1)-20*a(n-2)+12*a(n-3). - Harvey P. Dale, Jul 13 2011
a(n) = A248216(n)+1. - R. J. Mathar, Mar 10 2022

A248217 a(n) = 8^n - 2^n.

Original entry on oeis.org

0, 6, 60, 504, 4080, 32736, 262080, 2097024, 16776960, 134217216, 1073740800, 8589932544, 68719472640, 549755805696, 4398046494720, 35184372056064, 281474976645120, 2251799813554176, 18014398509219840, 144115188075331584, 1152921504605798400

Offset: 0

Vincenzo Librandi, Oct 04 2014

If 2^(n+1) is the length of the even leg of a primitive Pythagorean triangle (PPT) then it constrains the odd leg to have a length of 4^n-1 and the hypotenuse to have a length of 4^n+1. The resulting triangle has a semiperimeter of 4^n+2^n, an area of 8^n-2^n and an inradius of 2^n-1. For n > 0, a(n) is the area of such triangles. - Frank M Jackson, Sep 07 2018

Maximum anomalous cancellation multiplicity of (2n+1)-digit integers: number of (2n+1)-digit integers which can be anomalously canceled with a fixed (2n+1)-digit integer. The maximum is obtained at 88...88911...11 containing n 8's and n 1's (see Example below). Anomalous cancellation is a "canceling" of digits of a and b in the numerator and denominator of a fraction a/b which results in a fraction equal to the original, and no 0 or digits that appear different times in a and b are canceled. For example, 49/98 = 4/8, 138/184 = 3/4, 1985/5955 = 185/555, 88911/43956 = 8811/4356, but 120/340 is not because canceling the 0's is not an anomalous cancellation. - Xiaohan Zhang, Nov 21 2019

			For n=1, there are 6 numbers with 3 digits that can be anomalously canceled with 891: 297, 396, 495, 594, 693, 792. For n=2 there are 60 numbers with 88911: 12987, 13986, 14985, 15984, 16983, 17982, 21978, 22977, 23976, 24975, 25974, 26973, 27972, 28971, 31968, 32967, 33966, 34965, 35964, 36963, 37962, 38961, 41958, 42957, 43956, 44955, 45954, 46953, 47952, 48951, 51948, 52947, 53946, 54945, 55944, 56943, 57942, 58941, 61938, 62937, 63936, 64935, 65934, 66933, 67932, 68931, 71928, 72927, 73926, 74925, 75924, 76923, 77922, 78921, 82917, 83916, 84915, 85914, 86913, 87912. For n=3 504 numbers with 8889111, and no other (2n+1)-digit number has greater multiplicity. There seems to be a pattern of integer partitions in these examples, because the sum of the digits of numbers above are all multiples of 9. - _Xiaohan Zhang_, Nov 21 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Anomalous Cancellation
Index entries for linear recurrences with constant coefficients, signature (10,-16).

Cf. similar sequences listed in A248216.

Cf. A000079, A024036.

Magma
```
[8^n-2^n: n in [0..25]];
```

Table[8^n - 2^n, {n, 0, 25}] (* or *) CoefficientList[Series[6 x /((1 - 2 x) (1 - 8 x)), {x, 0, 30}], x]
LinearRecurrence[{10,-16},{0,6},30] (* Harvey P. Dale, Mar 29 2015 *)

a(n) = 8^n-2^n; \\ Altug Alkan, Sep 07 2018

def A248217(n): return 6*binomial(pow(2,n) +1, 3)
print([A248217(n) for n in range(41)]) # G. C. Greubel, Dec 26 2024

G.f.: 6*x/((1-2*x)*(1-8*x)).
a(n) = 10*a(n-1) - 16*a(n-2).
a(n) = 2^n*(4^n-1) = A000079(n) * A024036(n) = A001018(n) - A000079(n).
E.g.f.: exp(2*x)*(-1 + exp(6*x)). - Stefano Spezia, Sep 07 2018
a(n) = 6*A016131(n-1). - R. J. Mathar, Mar 10 2022

A174719 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 3, read by rows.

Original entry on oeis.org

1, 1, 1, 1, -7, 1, 1, -51, -51, 1, 1, -239, -399, -239, 1, 1, -967, -2177, -2177, -967, 1, 1, -3639, -10191, -13831, -10191, -3639, 1, 1, -13115, -43719, -74323, -74323, -43719, -13115, 1, 1, -45919, -177119, -360799, -452639, -360799, -177119, -45919, 1

Offset: 0

Roger L. Bagula, Mar 28 2010

The row sums of this class of sequences, for varying q, is given by Sum_{k=0..n} T(n, k, q) = q^n * (n+1) + 2^n * (1 - q^n). - G. C. Greubel, Feb 09 2021

			Triangle begins as:
  1;
  1,      1;
  1,     -7,       1;
  1,    -51,     -51,       1;
  1,   -239,    -399,    -239,       1;
  1,   -967,   -2177,   -2177,    -967,       1;
  1,  -3639,  -10191,  -13831,  -10191,   -3639,       1;
  1, -13115,  -43719,  -74323,  -74323,  -43719,  -13115,      1;
  1, -45919, -177119, -360799, -452639, -360799, -177119, -45919, 1;

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Cf. A000012 (q=1), A174718 (q=2), this sequence (q=3), A174720 (q=4).

Cf. A027471, A248216.

T:= func< n,k,q | 1 + (1-q^n)*(Binomial(n,k) -1) >;
[T(n,k,3): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021

T[n_, k_, q_]:= 1 +(1-q^n)*(Binomial[n, k] -1);
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten

def T(n,k,q): return 1 + (1-q^n)*(binomial(n,k) - 1)
flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 09 2021

T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q=3.

Sum_{k=0..n} T(n, k, 3) = 3^n*(n+1) + 2^n*(1 - 3^n) = A027471(n+2) - A248216(n). - G. C. Greubel, Feb 09 2021

Edited by G. C. Greubel, Feb 09 2021

A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 3, 4, 5, 7, 10, 14, 19, 15, 22, 32, 46, 65, 31, 46, 68, 100, 146, 211, 63, 94, 140, 208, 308, 454, 665, 127, 190, 284, 424, 632, 940, 1394, 2059, 255, 382, 572, 856, 1280, 1912, 2852, 4246, 6305, 511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171

Offset: 0

Paul Curtz, Jun 20 2025

			Triangle begins:
    0;
    1,   1;
    3,   4,    5;
    7,  10,   14,   19;
   15,  22,   32,   46,   65;
   31,  46,   68,  100,  146,  211;
   63,  94,  140,  208,  308,  454,  665;
  127, 190,  284,  424,  632,  940, 1394, 2059;
  255, 382,  572,  856, 1280, 1912, 2852, 4246,  6305;
  511, 766, 1148, 1720, 2576, 3856, 5768, 8620, 12866, 19171;
  ...

Columns k=0..2: A000225, A033484, A053209 (sans 1).
Diagonals: A001047, A027649, A053581 (sans 1), A291012 (sans 2).
Cf. A028243 (row sums), A036561, A059268, A081656, A248216, A321003.

/* As triangle */ [[2^(n-k)*3^k - 2^k : k in [0..n]]: n in [0..9]]; // Vincenzo Librandi, Jun 27 2025

T:= proc(n,k) option remember;
     `if`(n=k, 3^n-2^n, T(n, k+1)-T(n-1, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jun 24 2025

t[n_, 0] := 3^n - 2^n; t[n_, k_] := t[n, k] = t[n + 1, k - 1] - t[n, k - 1]; Table[t[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 20 2025 *)

T(n,n) = 3^n - 2^n = A001047(n).
T(n,k) = T(n,k+1) - T(n-1,k) for 0 <= k < n.
T(n,k) = 2^(n-k)*3^k - 2^k = A036561(n,k) - A059268(n,k).
T(2n,n) = A248216(n+1).

cp's OEIS Frontend

A155597 a(n) = 6^n - 2^n + 1.

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A248217 a(n) = 8^n - 2^n.

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A174719 Triangle T(n, k, q) = (1-q^n)*( binomial(n, k) - 1 ) + 1, with q = 3, read by rows.

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A385178 Triangle T(n,k) read by rows in which the n-th diagonal lists the n-th differences of A001047, 0 <= k <= n.

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