A053428 -id:A053428 - cp's OEIS frontend

A053404 Expansion of 1/((1+3x)(1-4*x)).

1, 1, 13, 25, 181, 481, 2653, 8425, 40261, 141361, 624493, 2320825, 9814741, 37664641, 155441533, 607417225, 2472715621, 9761722321, 39434309773, 156574977625, 629786694901, 2508686426401, 10066126765213, 40170363882025

Offset: 0

Barry E. Williams, Jan 07 2000

Hankel transform is := 1,12,0,0,0,... - Philippe Deléham, Nov 02 2008

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=2, 13*a(n-2) equals the number of 13-colored compositions of n with all parts >=2, such that no adjacent parts have the same color. - Milan Janjic, Nov 26 2011

A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
F. P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, March 2014; Preprint on ResearchGate.
A. K. Whitford, Binet's formula generalized, Fib. Quart., 15 (1977), pp. 21, 24, 29.
Index entries for linear recurrences with constant coefficients, signature (1,12).

Cf. A000045, A001045, A006130, A006131, A015440 - A015443, A015445 - A015447, A053404, A053428, A168579, A350468, A350469.

[((4^(n+1)) - (-3)^(n+1))/7: n in [0..30]]; // G. C. Greubel, Jan 16 2018

seq(simplify(hypergeom([1/2 - (1/2)*n, -(1/2)*n], [-n], -48)), n = 1..40); # Peter Bala, Jul 05 2025

CoefficientList[Series[1/((1 + 3 x) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 06 2014 *)

a(n)=([0,1; 12,1]^n*[1;1])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[lucas_number1(n,1,-12) for n in range(1, 25)] # Zerinvary Lajos, Apr 22 2009

a(n) = ((4^(n+1))-(-3)^(n+1))/7.
a(n) = a(n-1) + 12*a(n-2), n > 1; a(0)=1, a(1)=1.
From Paul Barry, Jul 30 2004: (Start)
Convolution of 4^n and (-3)^n.
G.f.: 1/((1+3x)(1-4x)); a(n) = Sum_{k=0..n, 4^k*(-3)^(n-k)} = Sum_{k=0..n, (-3)^k*4^(n-k)}. (End)
a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*(-12)^(n-k). - Philippe Deléham, Oct 26 2008
a(n) = (sum_{1<=k<=n+1, k odd} C(n+1,k)*7^(k-1))/2^n. - Vladimir Shevelev, Feb 05 2014
From Peter Bala, Jun 27 2025: (Start)
a(n) = hypergeom([1/2 - (1/2)*n, -(1/2)*n], [-n], -48) for n >= 1.
The following products telescope:
Product_{k >= 0} (1 + 12^k/a(2*k+1)) = 8.
Product_{k >= 1} (1 - 12^k/a(2*k+1)) = 4/25.
Product_{k >= 0} (1 + (-12)^k/a(2*k+1)) = 8/7.
Product_{k >= 1} (1 - (-12)^k/a(2*k+1)) = 28/25. (End)

More terms from James Sellers, Feb 02 2000

A081297 Array T(k,n), read by antidiagonals: T(k,n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 5, 1, 1, 1, 13, 13, 11, 1, 1, 1, 21, 25, 55, 21, 1, 1, 1, 31, 41, 181, 133, 43, 1, 1, 1, 43, 61, 461, 481, 463, 85, 1, 1, 1, 57, 85, 991, 1281, 2653, 1261, 171, 1, 1, 1, 73, 113, 1891, 2821, 10501, 8425, 4039, 341, 1, 1, 1, 91, 145, 3305

Offset: 0

Paul Barry, Mar 17 2003

Square array of solutions of a family of recurrences.
Rows of the array give solutions to the recurrences a(n)=a(n-1)+k(k-1)a(n-2), a(0)=a(1)=1.
Subarray of array in A072024. - Philippe Deléham, Nov 24 2013

			Rows begin
  1, 1,  1,  1,   1,    1, ...
  1, 1,  3,  5,  11,   21, ...
  1, 1,  7, 13,  55,  133, ...
  1, 1, 13, 25, 181,  481, ...
  1, 1, 21, 41, 461, 1281, ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Cf. A059259, A072024.
Rows include A001045, A015441, A053404, A053428, A053430.
Columns include A002061, A001844, A072025.
Diagonals include A081298, A081299, A081300, A081301, A081302.

T[n_, k_]:=((n + 1)^(k + 1) - (-n)^(k + 1)) / (2n + 1); Flatten[Table[T[n - k, k], {n, 0, 10}, {k, 0, n}]] (* Indranil Ghosh, Mar 27 2017 *)

for(k=0, 10, for(n=0, 9, print1(((k+1)^(n+1)-(-k)^(n+1))/(2*k+1), ", "); ); print(); ) \\ Andrew Howroyd, Mar 26 2017

def T(n, k): return ((n + 1)**(k + 1) - (-n)**(k + 1)) // (2*n + 1)
for n in range(11):
    print([T(n - k, k) for k in range(n + 1)]) # Indranil Ghosh, Mar 27 2017

T(k, n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1).

Rows of the array have g.f. 1/((1+kx)(1-(k+1)x)).

Name clarified by Andrew Howroyd, Mar 27 2017

A072024 Table by antidiagonals of T(n,k) = ((n+1)^k - (-n)^k)/(2*n+1).

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 5, 7, 1, 1, 0, 1, 11, 13, 13, 1, 1, 0, 1, 21, 55, 25, 21, 1, 1, 0, 1, 43, 133, 181, 41, 31, 1, 1, 0, 1, 85, 463, 481, 461, 61, 43, 1, 1, 0, 1, 171, 1261, 2653, 1281, 991, 85, 57, 1, 1, 0, 1, 341, 4039, 8425, 10501, 2821, 1891, 113, 73, 1, 1, 0

Offset: 0

Henry Bottomley, Jun 06 2002

Rows of the array have g.f. x/((1+k*x)*(1-(k+1)*x)). - Philippe Deléham, Nov 24 2013

			Rows start:
0 1 1  1   1    1     1      1       1        1 ...
0 1 1  3   5   11    21     43      85      171 ...
0 1 1  7  13   55   133    463    1261     4039 ...
0 1 1 13  25  181   481   2653    8425    40261 ...
0 1 1 21  41  461  1281  10501   36121   246141 ...
0 1 1 31  61  991  2821  32551  117181  1093711 ...
0 1 1 43  85 1891  5461  84883  314245  3879331 ...
0 1 1 57 113 3305  9633 194713  734161 11638089 ...
...

Andrew Howroyd, Table of n, a(n) for n = 0..1275

Rows include A057427, A001045, A015441, A053404, A053428, A053430, A065874, etc. Columns include A000004, A000012, A000012, A002061, A001844, A072025, etc.

Cf. A081297.

[((k+1)^(n-k) - (-k)^(n-k))/(2*k+1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 27 2020

seq(seq( ((k+1)^(n-k) - (-k)^(n-k))/(2*k+1), k=0..n), n=0..12); # G. C. Greubel, Jan 27 2020

T[n_, k_]:= ((n + 1)^k - (-n)^k)/(2n + 1); Flatten[Join[{0}, Table[T[k, n- k], {n, 1, 15}, {k, 0, n}]]] (* Indranil Ghosh, Mar 27 2017 *)

for(n=0, 10, for(k=0, 9, print1(((n+1)^k-(-n)^k)/(2*n+1), ", "); ); print(); ) \\ Andrew Howroyd, Mar 26 2017

def T(n, k): return ((n+1)^k - (-n)^k)/(2*n+1)
[[T(k,n-k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 27 2020

T(n, k) = T(n, k-1) + n*(n+1)*T(n, k-2) = A060959(A002378(n), k).

T(k, 2n) = (2n+1)*A047969(n, k+1).

A321133 a(n) = 3a(n-1) + 10a(n-2), n >= 2; a(0)=-1, a(1)=23.

Original entry on oeis.org

-1, 23, 59, 407, 1811, 9503, 46619, 234887, 1170851, 5861423, 29292779, 146492567, 732405491, 3662142143, 18310481339, 91552865447, 457763409731, 2288818883663, 11444090748299, 57220461081527, 286102290727571, 1430511482997983, 7152557356269659, 35762786898788807, 178813934259063011

Offset: 0

Seiichi Manyama, Aug 27 2019

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,10).

Cf. A053428, A224473 (trimorphic number), A320468, A320469.

LinearRecurrence[{3,10},{-1,23},30] (* Harvey P. Dale, Mar 11 2023 *)

PARI
```
{a(n) = 3*5^n-4*(-2)^n}
```

N=40; x='x+O('x^N); Vec((-1+26*x)/((1-5*x)*(1+2*x)))

a(n) = 3*5^n - 4*(-2)^n.
G.f.: (-1+26*x)/((1-5*x)*(1+2*x)).
a(n) == 7*A320469(n)*A224473(n) mod 10^n.
a(n)*A224473(n) == 7*A320469(n) mod 10^n.

A065874 a(n) = (7^(n+1) - (-6)^(n+1))/13.

Original entry on oeis.org

1, 1, 43, 85, 1891, 5461, 84883, 314245, 3879331, 17077621, 180009523, 897269605, 8457669571, 46142992981, 401365114963, 2339370820165, 19196705648611, 117450280095541, 923711917337203, 5856623681349925, 44652524209512451, 290630718826209301, 2166036735625732243

Offset: 0

Len Smiley, Dec 07 2001

A second-order recurrence of promic type (integer roots).
If the number j = A002378(m) is promic (= i(i+1)), then a(n) = a(n-1) + j*a(n-2), a(0) = a(1) = 1 has a closed-form solution involving only powers of integers. The binomial coefficient sum solves the recurrence regardless of promicity (cf. GKP reference).
Hankel transform is := 1,42,0,0,0,0,0,0,0,0,0,0,... - Philippe Deléham, Nov 02 2008

R. L. Graham, D. E. Knuth, O. Patashnik, "Concrete Mathematics", Addison-Wesley, 1994, p. 204.

Harry J. Smith, Table of n, a(n) for n = 0..150
Index entries for linear recurrences with constant coefficients, signature (1,42).

Cf. A001045 (j=2), A015441 (j=6), A053404 (j=12), A053428 (j=20), A053430 (j=30).

Maple
```
n->sum(binomial(n-k, k)*(42)^k, k=0..n)
```

LinearRecurrence[{1,42},{1,1},30] (* Harvey P. Dale, Apr 30 2017 *)

a(n) = { (7^(n+1) - (-6)^(n+1))/13 } \\ Harry J. Smith, Nov 02 2009

a(n) = a(n-1) + 42a(n-2); a(0) = a(1) = 1.

G.f.: -1/((6*x+1)*(7*x-1)). - R. J. Mathar, Nov 16 2007

A320468 a(n) = a(n-1) + 20*a(n-2), n >= 2; a(0)=1, a(1)=41.

Original entry on oeis.org

1, 41, 61, 881, 2101, 19721, 61741, 456161, 1690981, 10814201, 44633821, 260917841, 1153594261, 6371951081, 29443836301, 156882857921, 745759583941, 3883416742361, 18798608421181, 96466943268401, 472439111692021, 2401777977060041, 11850560210900461, 59886119752101281, 296897323970110501

Offset: 0

Seiichi Manyama, Aug 27 2019

Index entries for linear recurrences with constant coefficients, signature (1,20).

Cf. A053428, A224473 (trimorphic number), A320469, A321133.

PARI
```
{a(n) = (-4)^(n+1)+5^(n+1)}
```

N=40; x='x+O('x^N); Vec((1+40*x)/((1-5*x)*(1+4*x)))

a(n) = (-4)^(n+1) + 5^(n+1).
G.f.: (1+40*x)/((1-5*x)*(1+4*x)).
a(n) == 9*A053428(n)*A224473(n) mod 10^n.
a(n)*A224473(n) == 9*A053428(n) mod 10^n.

A320469 a(n) = 3a(n-1) + 10a(n-2), n >= 2; a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 13, 49, 277, 1321, 6733, 33409, 167557, 836761, 4185853, 20925169, 104634037, 523153801, 2615801773, 13078943329, 65394847717, 326973976441, 1634870406493, 8174350983889, 40871757016597, 204358780888681, 1021793912832013, 5108969547382849, 25544847770468677

Offset: 0

Seiichi Manyama, Aug 27 2019

Index entries for linear recurrences with constant coefficients, signature (3,10).

Cf. A053428, A224473 (trimorphic number), A320468, A321133.

PARI
```
{a(n) = (4*(-2)^n+3*5^n)/7}
```

N=40; x='x+O('x^N); Vec((1-2*x)/((1-5*x)*(1+2*x)))

a(n) = (4*(-2)^n + 3*5^n)/7.

G.f.: (1-2*x)/((1-5*x)*(1+2*x)).

cp's OEIS Frontend

A053404 Expansion of 1/((1+3*x)*(1-4*x)).

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A081297 Array T(k,n), read by antidiagonals: T(k,n) = ((k+1)^(n+1)-(-k)^(n+1))/(2k+1).

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A072024 Table by antidiagonals of T(n,k) = ((n+1)^k - (-n)^k)/(2*n+1).

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A321133 a(n) = 3*a(n-1) + 10*a(n-2), n >= 2; a(0)=-1, a(1)=23.

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A065874 a(n) = (7^(n+1) - (-6)^(n+1))/13.

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Programs

Maple

Mathematica

PARI

Formula

A320468 a(n) = a(n-1) + 20*a(n-2), n >= 2; a(0)=1, a(1)=41.

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PARI

PARI

Formula

A053404 Expansion of 1/((1+3x)(1-4*x)).

A321133 a(n) = 3a(n-1) + 10a(n-2), n >= 2; a(0)=-1, a(1)=23.

A320469 a(n) = 3a(n-1) + 10a(n-2), n >= 2; a(0)=1, a(1)=1.