A140504 -id:A140504 - cp's OEIS frontend

A254367 a(n) = 52^(n+2) + 2^(2n+2) + 103^n + 5^n + 35.

70, 126, 294, 846, 2814, 10326, 40614, 168126, 723534, 3208806, 14570934, 67417806, 316645854, 1505245686, 7225414854, 34956689886, 170199537774, 832952952966, 4093454620374, 20184631056366, 99800366967294, 494533989722646, 2454868429675494

Offset: 0

Luciano Ancora, Jan 30 2015

This is the sequence of fifth terms of "fourth partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas, page 2.
Luciano Ancora, Recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (15,-85,225,-274,120).

Cf. A140504, A254365, A254366.

[5*2^(n+2)+2^(2*n+2)+10*3^n+5^n+35: n in [0..30]]; // Vincenzo Librandi, Feb 02 2015

Table[5 2^(n + 2) + 2^(2 n + 2) + 10 3^n + 5^n + 35, {n, 0, 30}] (* Vincenzo Librandi, Feb 02 2015 *)

vector(30, n, n--; 5*2^(n+2) + 2^(2*n+2) + 10*3^n + 5^n + 35) \\ Colin Barker, Jan 30 2015

a(n) = 15*a(n-1)-85*a(n-2)+225*a(n-3)-274*a(n-4)+120*a(n-5). - Colin Barker, Jan 30 2015

G.f.: -2*(2972*x^4-4302*x^3+2177*x^2-462*x+35) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)). - Colin Barker, Jan 30 2015

A242475 a(n) = 2^n + 8.

Original entry on oeis.org

9, 10, 12, 16, 24, 40, 72, 136, 264, 520, 1032, 2056, 4104, 8200, 16392, 32776, 65544, 131080, 262152, 524296, 1048584, 2097160, 4194312, 8388616, 16777224, 33554440, 67108872, 134217736, 268435464, 536870920, 1073741832

Offset: 0

Vincenzo Librandi, May 20 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000051, A052548, A062709, A140504, A168614, A153972, A168415, A188165.

Magma
```
[2^n+8: n in [0..40]];
```

Table[2^n + 8, {n, 0, 40}] (* or *) CoefficientList[Series[(9 - 17 x)/((1 - x) (1 - 2 x)),{x, 0, 30}], x]
LinearRecurrence[{3,-2},{9,10},40] (* Harvey P. Dale, May 21 2025 *)

G.f.: (9 - 17*x)/((1 - x)*(1 - 2*x)).
a(n) = 2*a(n-1) - 8 = 3*a(n-1) - 2*a(n-2).
a(n) = A052548(n)+6 = A140504(n)+4 = A153972(n)+2.
E.g.f.: exp(2*x) + 8*exp(x). - Elmo R. Oliveira, Nov 11 2023

A246139 a(n) = 2^n + 10.

Original entry on oeis.org

11, 12, 14, 18, 26, 42, 74, 138, 266, 522, 1034, 2058, 4106, 8202, 16394, 32778, 65546, 131082, 262154, 524298, 1048586, 2097162, 4194314, 8388618, 16777226, 33554442, 67108874, 134217738, 268435466, 536870922, 1073741834, 2147483658, 4294967306

Offset: 0

Vincenzo Librandi, Aug 18 2014

First trisection of A085688. [Bruno Berselli, Aug 19 2014]

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. Sequences of the form 2^n + k: A000079 (k=0), A000051 (k=1), A052548 (k=2), A062709 (k=3), A140504 (k=4), A168614 (k=5), A153972 (k=6), A168415 (k=7), A242475 (k=8), A188165 (k=9), this sequence (k=10).

Cf. A085688.

Magma
```
[2^n+10: n in [0..40]];
```
Mathematica
```
Table[2^n + 10, {n, 0, 40}]
```

vector(50, n, 2^(n-1)+10) \\ Derek Orr, Aug 18 2014

G.f.: (11 - 21*x)/(1 - 3*x + 2*x^2).
a(n) = A000079(n) + 10.
a(n) = 3*a(n-1) - 2*a(n-2) for n > 1.
E.g.f.: exp(2*x) + 10*exp(x). - Elmo R. Oliveira, Nov 11 2023

A140505 Second differences of Jacobsthal sequence A001045, pairs with even and odd indices swapped.

Original entry on oeis.org

2, -1, 4, 0, 12, 4, 44, 20, 172, 84, 684, 340, 2732, 1364, 10924, 5460, 43692, 21844, 174764, 87380, 699052, 349524, 2796204, 1398100, 11184812, 5592404, 44739244, 22369620, 178956972, 89478484, 715827884, 357913940, 2863311532, 1431655764, 11453246124

Offset: 0

Paul Curtz, Jun 30 2008

The second differences are -1, 2, 0, 4, 4, 12, 20, 44, ... (-1)^(n+1)*A084247(n), essentially A097073, which are listed here with -1 <=> 2, 0 <=> 4 etc. swapped in pairs.

a(n) = 4*A092808(n-2), n>1.
a(n+1) - 2a(n) = (-1)^n*A140504(n).
O.g.f.: (2+x-5x^2)/[(1+x)(1-2x)(1+2x)]. - R. J. Mathar, Jul 08 2008

Edited by R. J. Mathar, Jul 08 2008

A254465 a(n) = 352^n + 104^n + 203^n + 45^n + 6^n + 56.

Original entry on oeis.org

126, 252, 672, 2232, 8592, 36552, 166992, 804552, 4037712, 20923272, 111231312, 603667272, 3331889232, 18646768392, 105558814032, 603280840392, 3475274371152, 20152803339912, 117513698083152, 688425727971912, 4048693055291472, 23888489018765832, 141334996634766672, 838119509472869832

Offset: 0

Luciano Ancora, Jan 31 2015

This is the sequence of sixth terms of "fourth partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas, page 2.
Luciano Ancora, Recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (21,-175,735,-1624,1764,-720).

Cf. A140504, A254365, A254366, A254367, A254466.

Table[35 2^n + 10 4^n + 20 3^n + 4 5^n + 6^n + 56, {n, 0, 24}] (* Michael De Vlieger, Jan 31 2015 *)
LinearRecurrence[{21,-175,735,-1624,1764,-720},{126,252,672,2232,8592,36552},30] (* Harvey P. Dale, Aug 02 2024 *)

vector(30, n, n--; 35*2^n + 10*4^n + 20*3^n + 4*5^n + 6^n + 56) \\ Colin Barker, Jan 31 2015

G.f.: -6*(10036*x^5 -16454*x^4 +10065*x^3 -2905*x^2 +399*x -21) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)). - Colin Barker, Jan 31 2015

a(n) = 21*a(n-1)-175*a(n-2)+735*a(n-3)-1624*a(n-4)+1764*a(n-5)-720*a(n-6). - Colin Barker, Jan 31 2015

A254466 a(n) = 562^n + 204^n + 353^n + 46^n + 10*5^n + 7^n + 84.

Original entry on oeis.org

210, 462, 1386, 5214, 22770, 110022, 571626, 3136014, 17944290, 106156182, 645091866, 4006997214, 25344197010, 162737255142, 1058251916106, 6955456112814, 46130658756930, 308314670926902, 2074188361172346, 14032607275346814, 95392686703000050

Offset: 0

Luciano Ancora, Jan 31 2015

This is the sequence of seventh terms of "fourth partial sums of m-th powers".

Colin Barker, Table of n, a(n) for n = 0..1000
Luciano Ancora, Demonstration of formulas, page 2.
Luciano Ancora, Recurrence relations for partial sums of m-th powers
Index entries for linear recurrences with constant coefficients, signature (28,-322,1960,-6769,13132,-13068,5040).

Cf. A140504, A254365, A254366, A254367, A254465.

Table[56 2^n + 20 4^n + 35 3^n + 4 6^n + 10 5^n + 7^n + 84, {n, 0, 24}] (* Michael De Vlieger, Jan 31 2015 *)

vector(30, n, n--; 56*2^n + 20*4^n + 35*3^n + 4*6^n + 10*5^n + 7^n + 84) \\ Colin Barker, Jan 31 2015

G.f.: -6*(110440*x^6 -199272*x^5 +139840*x^4 -49405*x^3 +9345*x^2 -903*x +35) / ((x -1)*(2*x -1)*(3*x -1)*(4*x -1)*(5*x -1)*(6*x -1)*(7*x -1)). - Colin Barker, Jan 31 2015

a(n) = 28*a(n-1) -322*a(n-2) +1960*a(n-3) -6769*a(n-4) +13132*a(n-5) -13068*a(n-6) +5040*a(n-7). - Colin Barker, Jan 31 2015

A295077 a(n) = 2n(n-1) + 2^n - 1.

Original entry on oeis.org

0, 1, 7, 19, 39, 71, 123, 211, 367, 655, 1203, 2267, 4359, 8503, 16747, 33187, 66015, 131615, 262755, 524971, 1049335, 2097991, 4195227, 8389619, 16778319, 33555631, 67110163, 134219131, 268436967, 536872535, 1073743563, 2147485507

Offset: 0

Franck Maminirina Ramaharo, Nov 13 2017

We have a(0) = 0, and for n > 0, a(n) is a subsequence of A131098 where the indices are given by the partial sums of A288382.
For n > 0, a(n) gives the number of words of length n over the alphabet A = {a,b,c,d} such that: a word containing 'c' does not contain 'b' or 'd'; a word cannot be fully written with 'a'; a word contains letters from {b,d} if and only if it contains exactly a unique couple of letters from {b,d}. Thus a(1) = 1 where the corresponding word is "c" since 'c' is the only letter allowed to be written alone.
Primes in the sequence are 7, 19, 71, 211, 367, 2267, 16747, 524971, ... which are of the form 4*k + 3 (A002145).
The second difference of this sequence is A140504.

			a(4) = 39. The corresponding words are aabb, aabd, aadb, aadd, abab, abad, abba, abda, adab, adad, adba, adda, aaac, aaca, aacc, acaa, acac, acca, accc, baab, baad, baba, bada, bbaa, bdaa, caaa, caac, caca, cacc, ccaa, ccac, ccca, cccc, daab, daad, daba, dada, dbaa, ddaa.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..70 from Franck Maminirina Ramaharo)
Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.
Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2)

Cf. A000079, A002145, A008586, A046092, A126646, A131098, A131924, A140504, A288382.

[2*n*(n-1)+2^n-1 : n in [0..40]]; // Wesley Ivan Hurt, Nov 26 2017

A295077:=n->2*n*(n-1)+2^n-1; seq(A295077(n), n=0..70);

Table[2 n (n - 1) + 2^n - 1, {n, 0, 70}]

a(n) = 2*n*(n-1) + 2^n - 1; \\ Michel Marcus, Nov 14 2017

G.f.: (x + 2*x^2 - 7*x^3)/((1 - x)^3*(1 - 2*x)).
a(0)=0, a(1)=1, a(2)=7, a(3)=19; for n>3, a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4).
a(n) = 2*A131924(n-1) - 1 for n>0, a(0)=0.
a(n) = a(n-1) + A000079(n-1) + A008586(n-1) for n>0, a(0)=0.
a(n) = A126646(n-1) + A046092(n-1) for n>0, a(0)=0.
a(n+1) - 2*a(n) + a(n-1) = A140504(n-1) for n>0, a(0)=0.
E.g.f.: exp(2*x) - (1 - 2*x^2)*exp(x). - G. C. Greubel, Oct 17 2018

A146528 a(0) = 4; for n >= 1, a(n) = 2^n + 4.

Original entry on oeis.org

4, 6, 8, 12, 20, 36, 68, 132, 260, 516, 1028, 2052, 4100, 8196, 16388, 32772, 65540, 131076, 262148, 524292, 1048580, 2097156, 4194308, 8388612, 16777220, 33554436, 67108868, 134217732, 268435460, 536870916, 1073741828

Offset: 0

Roger L. Bagula, Oct 30 2008

This is one of many possible ways to extend the Platonic solids sequence A053016.

Cf. A053016, A342759.

Essentially the same as A140504.

v[n_] := 2*(If[n == 0, 0, 2^(n - 1)] + 2); Table[v[n], {n, 0, 30}]

a(n)=A140504(n), n>0. [From R. J. Mathar, Nov 05 2008]

Edited by N. J. A. Sloane, Mar 21 2021

A267615 a(n) = 2^n + 11.

Original entry on oeis.org

12, 13, 15, 19, 27, 43, 75, 139, 267, 523, 1035, 2059, 4107, 8203, 16395, 32779, 65547, 131083, 262155, 524299, 1048587, 2097163, 4194315, 8388619, 16777227, 33554443, 67108875, 134217739, 268435467, 536870923, 1073741835, 2147483659, 4294967307, 8589934603, 17179869195, 34359738379

Offset: 0

Ilya Gutkovskiy, Jan 18 2016

Recurrence relation b(n) = 3*b(n - 1) - 2*b(n - 2) for n>1, b(0) = k, b(1) = k + 1, gives the closed form b(n) = 2^n + k - 1.

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

Cf. sequences with closed form 2^n + k - 1: A168616 (k=-4), A028399 (k=-3), A036563 (k=-2), A000918 (k=-1), A000225 (k=0), A000079 (k=1), A000051 (k=2), A052548 (k=3), A062709 (k=4), A140504 (k=5), A168614 (k=6), A153972 (k=7), A168415 (k=8), A242475 (k=9), A188165 (k=10), A246139 (k=11), this sequence (k=12).

Cf. A156940.

[2^n+11: n in [0..30]]; // Vincenzo Librandi, Jan 19 2016

Table[2^n + 11, {n, 0, 35}]
LinearRecurrence[{3, -2}, {12, 13}, 40] (* Vincenzo Librandi, Jan 19 2016 *)

a(n) = 2^n + 11; \\ Altug Alkan, Jan 18 2016

G.f.: (12 - 23*x)/(1 - 3*x + 2*x^2).
a(n) = 3*a(n - 1) - 2*a(n - 2) for n>1, a(0)=12, a(1)=13.
a(n) = A000079(n) + A010850(n).
Sum_{n>=0} 1/a(n) = 0.367971714327125...
Lim_{n->oo} a(n + 1)/a(n) = 2.
E.g.f.: exp(2*x) + 11*exp(x). - Elmo R. Oliveira, Nov 08 2023

cp's OEIS Frontend

A254367 a(n) = 5*2^(n+2) + 2^(2n+2) + 10*3^n + 5^n + 35.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A242475 a(n) = 2^n + 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A246139 a(n) = 2^n + 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A140505 Second differences of Jacobsthal sequence A001045, pairs with even and odd indices swapped.

Views

Author

Keywords

Comments

Formula

Extensions

A254465 a(n) = 35*2^n + 10*4^n + 20*3^n + 4*5^n + 6^n + 56.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A254466 a(n) = 56*2^n + 20*4^n + 35*3^n + 4*6^n + 10*5^n + 7^n + 84.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A295077 a(n) = 2*n*(n-1) + 2^n - 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A146528 a(0) = 4; for n >= 1, a(n) = 2^n + 4.

A254367 a(n) = 52^(n+2) + 2^(2n+2) + 103^n + 5^n + 35.

A254465 a(n) = 352^n + 104^n + 203^n + 45^n + 6^n + 56.

A254466 a(n) = 562^n + 204^n + 353^n + 46^n + 10*5^n + 7^n + 84.

A295077 a(n) = 2n(n-1) + 2^n - 1.