A053142 -id:A053142 - cp's OEIS frontend

A016278 Expansion of g.f. 1/((1-2x)(1-3x)(1-9x)).

1, 14, 145, 1370, 12541, 113534, 1023865, 9221090, 83008981, 747138854, 6724424785, 60520350410, 544684739821, 4902167424974, 44119521140905, 397075733249330, 3573681728253061, 32163135941435894, 289468224634660225

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Counting families of generalized balancing numbers, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.
Index entries for linear recurrences with constant coefficients, signature (14,-51,54).

Cf. A053141, A053142.

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-2*x)*(1-3*x)*(1-9*x)))); // Vincenzo Librandi, Jun 24 2013

CoefficientList[Series[1 / ((1 - 2 x) (1 - 3 x) (1 - 9 x)), {x, 0, 20}], x] (* Vincenzo Librandi, Jun 24 2013 *)

Vec(1/((1-2*x)*(1-3*x)*(1-9*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 14*a(n-1) - 51*a(n-2) + 54*a(n-3); a(n) = (4/7)*2^(n-1) + (-3/2)*3^(n-1) + (27/14)*9^(n-1). - Antonio Alberto Olivares, Apr 21 2008, Apr 22 2008

E.g.f.: exp(2*x)*(8 - 21*exp(x) + 27*exp(7*x))/14. - Stefano Spezia, Feb 12 2025

A055141 Matrix inverse of triangle A055140.

Original entry on oeis.org

1, 0, 1, -2, 0, 1, -8, -6, 0, 1, -36, -32, -12, 0, 1, -224, -180, -80, -20, 0, 1, -1880, -1344, -540, -160, -30, 0, 1, -19872, -13160, -4704, -1260, -280, -42, 0, 1, -251888, -158976, -52640, -12544, -2520, -448, -56, 0, 1, -3712256, -2266992

Offset: 0

Christian G. Bower, May 09 2000

T is an example of the group of matrices outlined in the table in A132382--the associated matrix for aC(1,1). The e.g.f. for the row polynomials is exp(x*t) * exp(x) * (1-2*x)^(1/2). T(n,k) = Binomial(n,k)* s(n-k) where s = A055142 with an e.g.f. of exp(x) * (1-2*x)^(1/2) which is the reciprocal of the e.g.f. of A053871. The row polynomials form an Appell sequence. [From Tom Copeland, Sep 11 2008]

			1; 0,1; -2,0,1; -8,-6,0,1; -36,-32,-12,0,1; ...

a(n, k) = A053142(n-k)*C(n, k).

A077784 Numbers k such that (10^k - 1)/3 + 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

Original entry on oeis.org

3, 5, 35, 159, 237, 325, 355, 371, 481, 1649, 3641, 4709, 269623

Offset: 1

Patrick De Geest, Nov 16 2002

Prime versus probable prime status and proofs are given in the author's table.

a(13) > 2*10^5. - Robert Price, Apr 03 2016

			5 is a term because (10^5 - 1)/3 + 2*10^2 = 33533.

C. Caldwell and H. Dubner, "Journal of Recreational Mathematics", Volume 28, No. 1, 1996-97, pp. 1-9.

Patrick De Geest, World!Of Numbers, Palindromic Wing Primes (PWP's)
Makoto Kamada, Prime numbers of the form 33...33533...33
Index entries for primes involving repunits.

Partial sums of S(n, x), for x=1...9: A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420.

Cf. A004023, A077775-A077798, A107123-A107127, A107648, A107649, A115073, A183174-A183187.

Do[ If[ PrimeQ[(10^n + 6*10^Floor[n/2] - 1)/3], Print[n]], {n, 3, 4800, 2}] (* Robert G. Wilson v, Dec 16 2005 *)

a(n) = 2*A183175(n) + 1.

Name corrected by Jon E. Schoenfield, Oct 31 2018

a(13) from Robert Price, Aug 03 2024

A077828 Expansion of 1/(1-3x-3x^2-3*x^3).

Original entry on oeis.org

1, 3, 12, 48, 189, 747, 2952, 11664, 46089, 182115, 719604, 2843424, 11235429, 44395371, 175422672, 693160416, 2738935377, 10822555395, 42763953564, 168976333008, 667688525901, 2638286437419, 10424853888984, 41192486556912, 162766880649945, 643152663287523

Offset: 0

N. J. A. Sloane, Nov 17 2002

Michael De Vlieger, Table of n, a(n) for n = 0..1675
Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 19.
Index entries for linear recurrences with constant coefficients, signature (3,3,3).

Partial sums of S(n, x), for x=1...12, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-7.

Cf. A071675.

CoefficientList[Series[1/(1-3x-3x^2-3x^3),{x,0,30}],x] (* or *) LinearRecurrence[ {3,3,3},{1,3,12},30] (* Harvey P. Dale, Dec 25 2018 *)

Vec(1/(1-3*x-3*x^2-3*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

a(n) = sum{k=0..n, T(n-k, k)3^(n-k)}, T(n, k) = trinomial coefficients (A027907). - Paul Barry, Feb 15 2005

a(n) = sum{k=0..n, sum{i=0..floor((n-k)/2), C(n-k-i, i)C(k, n-k-i)}*3^k}. - Paul Barry, Apr 26 2005

A077829 Expansion of 1/(1-3x-3x^2-2*x^3).

Original entry on oeis.org

1, 3, 12, 47, 183, 714, 2785, 10863, 42372, 165275, 644667, 2514570, 9808261, 38257827, 149227404, 582072215, 2270414511, 8855914986, 34543132921, 134737972743, 525555146964, 2049965624963, 7996038261267, 31189121952618, 121655411891581, 474525678055131

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (3,3,2).

Partial sums of S(n, x), for x=1...14, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139.

CoefficientList[Series[1/(1 - 3*x - 3*x^2 - 2*x^3), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 20 2024 *)
LinearRecurrence[{3,3,2},{1,3,12},30] (* Harvey P. Dale, Dec 20 2024 *)

Vec(1/(1-3*x-3*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

G.f.: 1/(1-3*x-3*x^2-2*x^3).

a(n) = 3*a(n-1) + 3*a(n-2) + 2*a(n-3). - Wesley Ivan Hurt, Jan 20 2024

A077831 Expansion of 1/(1-3x-2x^2-2*x^3).

Original entry on oeis.org

1, 3, 11, 41, 151, 557, 2055, 7581, 27967, 103173, 380615, 1404125, 5179951, 19109333, 70496151, 260067021, 959412031, 3539362437, 13057045415, 48168685181, 177698871247, 655548074933, 2418379337655, 8921631905325, 32912750541151, 121418274109413

Offset: 0

N. J. A. Sloane, Nov 17 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,2).

Partial sums of S(n, x), for x=1...16, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139, A097829, A097830.

CoefficientList[Series[1/(1-3x-2x^2-2x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,2,2},{1,3,11},30] (* Harvey P. Dale, Feb 28 2025 *)

Vec(1/(1-3*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

A077826 Expansion of (1-x)^(-1)/(1-2x-3x^2-2*x^3).

Original entry on oeis.org

1, 3, 10, 32, 101, 319, 1006, 3172, 10001, 31531, 99410, 313416, 988125, 3115319, 9821846, 30965900, 97627977, 307797347, 970410426, 3059468848, 9645763669, 30410754735, 95877738174, 302279267892, 953013259777, 3004619799579, 9472837914274, 29865561746840

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (3,1,-1,-2).

Partial sums of S(n, x), for x=1...10, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784.

Partial sums of A077833.

LinearRecurrence[{3,1,-1,-2},{1,3,10,32},30] (* Harvey P. Dale, May 12 2024 *)

Vec((1-x)^(-1)/(1-2*x-3*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

From Wesley Ivan Hurt, Jun 26 2022: (Start)
G.f.: (1-x)^(-1)/(1-2*x-3*x^2-2*x^3).
a(n) = 3*a(n-1) + a(n-2) - a(n-3) - 2*a(n-4). (End)

A077827 Expansion of (1-x)^(-1)/(1-2x-2x^2-2*x^3).

Original entry on oeis.org

1, 3, 9, 27, 79, 231, 675, 1971, 5755, 16803, 49059, 143235, 418195, 1220979, 3564819, 10407987, 30387571, 88720755, 259032627, 756281907, 2208070579, 6446770227, 18822245427, 54954172467, 160446376243, 468445588275, 1367692273971, 3993168476979, 11658612678451

Offset: 0

N. J. A. Sloane, Nov 17 2002, Jun 05 2007

Index entries for linear recurrences with constant coefficients, signature (3,0,0,-2).

Partial sums of S(n, x), for x=1...11, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826.

CoefficientList[Series[(1-x)^(-1)/(1-2x-2x^2-2x^3),{x,0,40}],x]  (* Harvey P. Dale, Mar 27 2011 *)

Vec((1-x)^(-1)/(1-2*x-2*x^2-2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

A077830 Expansion of 1/(1-3x-2x^2-3*x^3).

Original entry on oeis.org

1, 3, 11, 42, 157, 588, 2204, 8259, 30949, 115977, 434606, 1628619, 6103000, 22870056, 85702025, 321155187, 1203479779, 4509855786, 16899992477, 63330128340, 237319937332, 889320046107, 3332590398005, 12488371098225, 46798254229006, 175369276077483

Offset: 0

N. J. A. Sloane, Nov 17 2002

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,2,3).

Partial sums of S(n, x), for x=1...15, A021823, A000217, A027941, A061278, A089817, A053142, A092521, A076765, A092420, A097784, A097826-A097828, A076139, A097829.

CoefficientList[Series[1/(1-3x-2x^2-3x^3),{x,0,40}],x] (* or *) LinearRecurrence[{3,2,3},{1,3,11},40] (* Harvey P. Dale, Nov 05 2021 *)

A358682 Numbers k such that 8k^2 + 8k - 7 is a square.

Original entry on oeis.org

1, 7, 43, 253, 1477, 8611, 50191, 292537, 1705033, 9937663, 57920947, 337588021, 1967607181, 11468055067, 66840723223, 389576284273, 2270616982417, 13234125610231, 77134136678971, 449570694463597, 2620290030102613, 15272169486152083, 89012726886809887, 518804191834707241

Offset: 1

Stefano Spezia, Nov 26 2022

a(n) is the n-th almost cobalancing number of second type (see Tekcan and Erdem).

			a(2) = 7 is a term since 8*7^2 + 8*7 - 7 = 441 = 21^2.

Ahmet Tekcan and Alper Erdem, General Terms of All Almost Balancing Numbers of First and Second Type, arXiv:2211.08907 [math.NT], 2022.
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Cf. A002315, A006452, A077443, A077446, A106328, A106329, A216134.

Mathematica
```
LinearRecurrence[{7,-7,1},{1,7,43},24]
```

a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 3.
a(n) = (3*(3 - 2*sqrt(2))^n*(2 + sqrt(2)) + 3*(2 - sqrt(2))*(3 + 2*sqrt(2))^n - 4)/8.
O.g.f.: x*(1 + x^2)/((1 - x)*(1 - 6*x + x^2)).
E.g.f.: (3*(2 + sqrt(2))*(cosh(3*x - 2*sqrt(2)*x) + sinh(3*x - 2*sqrt(2)*x)) + 3*(2 - sqrt(2))*(cosh(3*x + 2*sqrt(2)*x) + sinh(3*x + 2*sqrt(2)*x)) - 4*(cosh(x) + sinh(x)) - 8)/8.
a(n) = 3*A011900(n) - 2 = 6*A053142(n) + 1. - Hugo Pfoertner, Nov 26 2022

cp's OEIS Frontend

A016278 Expansion of g.f. 1/((1-2x)(1-3x)(1-9x)).

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A055141 Matrix inverse of triangle A055140.

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A077784 Numbers k such that (10^k - 1)/3 + 2*10^floor(k/2) is a palindromic wing prime (a.k.a. near-repdigit palindromic prime).

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A077828 Expansion of 1/(1-3*x-3*x^2-3*x^3).

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A077829 Expansion of 1/(1-3*x-3*x^2-2*x^3).

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A077831 Expansion of 1/(1-3*x-2*x^2-2*x^3).

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A077826 Expansion of (1-x)^(-1)/(1-2*x-3*x^2-2*x^3).

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A077827 Expansion of (1-x)^(-1)/(1-2*x-2*x^2-2*x^3).

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A077828 Expansion of 1/(1-3x-3x^2-3*x^3).

A077829 Expansion of 1/(1-3x-3x^2-2*x^3).

A077831 Expansion of 1/(1-3x-2x^2-2*x^3).

A077826 Expansion of (1-x)^(-1)/(1-2x-3x^2-2*x^3).

A077827 Expansion of (1-x)^(-1)/(1-2x-2x^2-2*x^3).

A077830 Expansion of 1/(1-3x-2x^2-3*x^3).

A358682 Numbers k such that 8k^2 + 8k - 7 is a square.