A115451 -id:A115451 - cp's OEIS frontend

A139760 First quadrisection of A115451.

1, 9, 137, 2185, 34953, 559241, 8947849, 143165577, 2290649225, 36650387593, 586406201481, 9382499223689, 150119987579017, 2401919801264265, 38430716820228233, 614891469123651721, 9838263505978427529, 157412216095654840457, 2518595457530477447305

Offset: 0

Paul Curtz, May 20 2008

Vincenzo Librandi, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (17,-16)

[(7+8*16^n)/15: n in [0..20]]; // Vincenzo Librandi, Aug 09 2011

LinearRecurrence[{17,-16},{1,9},20] (* Harvey P. Dale, Apr 02 2018 *)

a(n) = 16*a(n-1) - 7.

G.f.: ( 1-8*x ) / ( (16*x-1)*(x-1) ). - R. J. Mathar, Feb 06 2011

A131865 Partial sums of powers of 16.

Original entry on oeis.org

1, 17, 273, 4369, 69905, 1118481, 17895697, 286331153, 4581298449, 73300775185, 1172812402961, 18764998447377, 300239975158033, 4803839602528529, 76861433640456465, 1229782938247303441, 19676527011956855057, 314824432191309680913, 5037190915060954894609

Offset: 0

Reinhard Zumkeller, Jul 22 2007

16 = 2^4 is the growth measure for the Jacobsthal spiral (compare with phi^4 for the Fibonacci spiral). - Paul Barry, Mar 07 2008
Second quadrisection of A115451. - Paul Curtz, May 21 2008
Let A be the Hessenberg matrix of order n, defined by: A[1,j]=1, A[i,i]:=16, (i>1), A[i,i-1]=-1, and A[i,j]=0 otherwise. Then, for n >= 1, a(n-1) = det(A). - Milan Janjic, Feb 21 2010
Partial sums are in A014899. Also, the sequence is related to A014931 by A014931(n+1) = (n+1)*a(n) - Sum_{i=0..n-1} a(i) for n>0. - Bruno Berselli, Nov 07 2012
a(n) is the total number of holes in a certain box fractal (start with 16 boxes, 1 hole) after n iterations. See illustration in links. - Kival Ngaokrajang, Jan 28 2015
Except for 1 and 17, all terms are Brazilian repunits numbers in base 16, and so belong to A125134. All terms >= 273 are composite because a(n) = ((4^(n+1) + 1) * (4^(n+1) - 1))/15. - Bernard Schott, Jun 06 2017
The sequence in binary is 1, 10001, 100010001, 1000100010001, 10001000100010001, ... cf. Plouffe link, A330135. - Frank Ellermann, Mar 05 2020

			a(3) = 1 + 16 + 256 + 4096 = 4369 = in binary: 1000100010001.
a(4) = (16^5 - 1)/15 = (4^5 + 1) * (4^5 - 1)/15 = 1025 * 1023/15 = 205 * 341 = 69905 = 11111_16. - _Bernard Schott_, Jun 06 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..800
A. Abdurrahman, CM Method and Expansion of Numbers, arXiv:1909.10889 [math.NT], 2019.
Kival Ngaokrajang, Illustration of initial terms
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.
Simon Plouffe, Identities and approximations inspired from Ramanujan notebooks, III, 2009.
Index entries related to partial sums.
Index entries related to q-numbers.
Index entries for linear recurrences with constant coefficients, signature (17,-16).

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A016123, A016125, A091030, A135519, A135518, A091045, A218721, A218722, A064108, A218724-A218734, A132469, A218736-A218753, A133853, A094028, A218723. - M. F. Hasler, Nov 05 2012

[(16^(n+1)-1)/15: n in [0..20]]; // Vincenzo Librandi, Sep 17 2011

A131865:=n->(16^(n+1)-1)/15: seq(A131865(n), n=0..30); # Wesley Ivan Hurt, Apr 29 2017

Table[(2^(4 n) - 1)/15, {n, 16}] (* Robert G. Wilson v, Aug 22 2007 *)
Accumulate[16^Range[0,20]] (* or *) LinearRecurrence[{17,-16},{1,17},20] (* Harvey P. Dale, Jul 19 2019 *)

a[0]:0$
a[n]:=16*a[n-1]+1$
A131865(n):=a[n]$
makelist(A131865(n),n,1,30); /* Martin Ettl, Nov 05 2012 */

A131865(n)=16^n\15  \\ M. F. Hasler, Nov 05 2012

def A131865(n): return (1<<(n+1<<2))//15 # Chai Wah Wu, Nov 10 2022

[gaussian_binomial(n,1,16) for n in range(1,18)] # Zerinvary Lajos, May 28 2009

a(n) = if n=0 then 1 else a(n-1) + A001025(n).
for n > 0: A131851(a(n)) = n and abs(A131851(m)) < n for m < a(n).
a(n) = A098704(n+2)/2.
a(n) = (16^(n+1) - 1)/15. - Bernard Schott, Jun 06 2017
a(n) = (A001025(n+1) - 1)/15.
a(n) = 16*a(n-1) + 1. - Paul Curtz, May 20 2008
G.f.: 1 / ( (16*x-1)*(x-1) ). - R. J. Mathar, Feb 06 2011
E.g.f.: exp(x)*(16*exp(15*x) - 1)/15. - Stefano Spezia, Mar 06 2020

A098704 Decimal form of the binary numbers 10, 100010, 1000100010, 10001000100010, 100010001000100010,...

Original entry on oeis.org

2, 34, 546, 8738, 139810, 2236962, 35791394, 572662306, 9162596898, 146601550370, 2345624805922, 37529996894754, 600479950316066, 9607679205057058, 153722867280912930, 2459565876494606882

Offset: 2

Simone Severini, Oct 26 2004

Decimal form of the hexadecimal numbers 2, 22, 222, 2222, 22222, 222222, ...; e.g., 2*16^0 + 2*16^1 = 2 + 32 = 34. - Zerinvary Lajos, Feb 01 2007
For n>0: A131852(a(n+1))=n and ABS(A131852(m))A131865(n-2). - Reinhard Zumkeller, Jul 22 2007
Third quadrisection of A115451. - Klaus Purath, Mar 14 2021

Harvey P. Dale, Table of n, a(n) for n = 2..832
Index entries for linear recurrences with constant coefficients, signature (17,-16).

s=0;lst={};Do[s+=2^n;AppendTo[lst, s], {n, 1, 2*5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 07 2008 *)
FromDigits[#,2]&/@Table[Join[PadRight[{},4n,{1,0,0,0}],{1,0}],{n,0,20}] (* Harvey P. Dale, Apr 06 2020 *)

for(n=0,20,print(2*sum(k=0,n,2^(4*k))))
for(k=0,20,print(2*(1-16^(k+1))/-15))

lim_{n -> infinity} a(n)/a(n-k) = 2^(4*(n-k)).
2*Sum_{k=0..n} 16^k = 2*(16^(n+1) - 1)/15.
From Klaus Purath, Mar 14 2021: (Start)
a(n) = (2^(4*n-3)-2)/15.
a(n) = 17*a(n-1) - 16*a(n-2).
a(n) = 16*a(n-1) + 2.
a(n) = 2*16^(n-2) + a(n-1).
a(n) = 2*A131865(n-2). (End)

More terms from Ray Chandler, Nov 02 2004

More terms from Vladimir Joseph Stephan Orlovsky, Nov 07 2008

A349839 Triangle T(n,k) built by placing all ones on the left edge, [1,0,0,0] repeated on the right edge, and filling the body using the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 1, 1, 4, 6, 4, 2, 0, 1, 5, 10, 10, 6, 2, 0, 1, 6, 15, 20, 16, 8, 2, 0, 1, 7, 21, 35, 36, 24, 10, 2, 1, 1, 8, 28, 56, 71, 60, 34, 12, 3, 0, 1, 9, 36, 84, 127, 131, 94, 46, 15, 3, 0, 1, 10, 45, 120, 211, 258, 225, 140, 61, 18, 3, 0, 1, 11, 55, 165, 331, 469, 483, 365, 201, 79, 21, 3, 1

Offset: 0

Michael A. Allen, Dec 01 2021

This is the m=4 member in the sequence of triangles A007318, A059259, A118923, A349839, A349841 which have all ones on the left side, ones separated by m-1 zeros on the other side, and whose interiors obey Pascal's recurrence.
T(n,k) is the (n,n-k)-th entry of the (1/(1-x^4),x/(1-x)) Riordan array.
For n>0, T(n,n-1) = A008621(n-1).
For n>1, T(n,n-2) = A001972(n-2).
For n>2, T(n,n-3) = A122046(n).
Sums of rows give A115451.
Sums of antidiagonals give A349840.

			Triangle begins:
  1;
  1,   0;
  1,   1,   0;
  1,   2,   1,   0;
  1,   3,   3,   1,   1;
  1,   4,   6,   4,   2,   0;
  1,   5,  10,  10,   6,   2,   0;
  1,   6,  15,  20,  16,   8,   2,   0;
  1,   7,  21,  35,  36,  24,  10,   2,   1;
  1,   8,  28,  56,  71,  60,  34,  12,   3,   0;
  1,   9,  36,  84, 127, 131,  94,  46,  15,   3,   0;
  1,  10,  45, 120, 211, 258, 225, 140,  61,  18,   3,   0;
  1,  11,  55, 165, 331, 469, 483, 365, 201,  79,  21,   3,   1;

Michael A. Allen and Kenneth Edwards, On Two Families of Generalizations of Pascal's Triangle, J. Int. Seq. 25 (2022) Article 22.7.1.

Other members of sequence of triangles: A007318, A059259, A118923, A349841.
Columns: A000012, A001477, A000217, A000292, A145126, A051745.
Diagonals: A121262, A008621, A001972, A122046.

Flatten[Table[CoefficientList[Series[(1-x*y)/((1-(x*y)^4)(1 - x - x*y)), {x, 0, 24}, {y, 0, 12}], {x, y}][[n+1,k+1]],{n,0,12},{k,0,n}]]

G.f.: (1-x*y)/((1-(x*y)^4)(1-x-x*y)) in the sense that T(n,k) is the coefficient of x^n*y^k in the series expansion of the g.f.
T(n,0) = 1.
T(n,n) = delta(n mod 4,0).
T(n,1) = n-1 for n>0.
T(n,2) = (n-1)*(n-2)/2 for n>1.
T(n,3) = (n-1)*(n-2)*(n-3)/6 for n>2.
T(n,4) = C(n-1,4) + 1 for n>3.
T(n,5) = C(n-1,5) + n - 5 for n>4.
For 0 <= k < n, T(n,k) = (n-k)*Sum_{j=0..floor(k/4)} binomial(n-4*j,n-k)/(n-4*j).
The g.f. of the n-th subdiagonal is 1/((1-x^4)(1-x)^n).

A115450 Number triangle (1/((1-x)(1-2x)),-x)-(x/((1-x)(1-2x)),-x^2) (expressed in the notation of Riordan arrays).

Original entry on oeis.org

1, 2, -1, 4, -3, 1, 8, -6, 3, -1, 16, -12, 7, -3, 1, 32, -24, 14, -7, 3, -1, 64, -48, 28, -15, 7, -3, 1, 128, -96, 56, -30, 15, -7, 3, -1, 256, -192, 112, -60, 31, -15, 7, -3, 1, 512, -384, 224, -120, 62, -31, 15, -7, 3, -1, 1024, -768, 448, -240, 124, -63, 31, -15, 7, -3, 1, 2048, -1536, 896, -480, 248, -126, 63, -31, 15, -7, 3

Offset: 0

Paul Barry, Jan 22 2006

Row sums are A115451. Inverse is A115452. Row sums of inverse matrix are the Fredholm-Rueppel sequence A036987.

			Triangle begins
1,
2, -1,
4, -3, 1,
8, -6, 3, -1,
16, -12, 7, -3, 1,
32, -24, 14, -7, 3, -1,
64, -48, 28, -15, 7, -3, 1,
128, -96, 56, -30, 15, -7, 3, -1,
256, -192, 112, -60, 31, -15, 7, -3, 1,
512, -384, 224, -120, 62, -31, 15, -7, 3, -1,
1024, -768, 448, -240, 124, -63, 31, -15, 7, -3, 1,
2048, -1536, 896, -480, 248, -126, 63, -31, 15, -7, 3, -1,
4096, -3072, 1792, -960, 496, -252, 127, -63, 31, -15, 7, -3, 1

Column k has g.f. 1/((1-x)(1-2x))*(-x)^k*(1-x^(k+1)); Number triangle T(n, k)=(if(k<=n, 2^(n-k+1)-1, 0)-if(k<=floor(n/2), 2^(n-2k)-1, 0))(-1)^k.

A349842 Expansion of 1/((1 - 2x)(1 + x + x^2 + x^3 + x^4)).

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 33, 66, 132, 264, 529, 1057, 2114, 4228, 8456, 16913, 33825, 67650, 135300, 270600, 541201, 1082401, 2164802, 4329604, 8659208, 17318417, 34636833, 69273666, 138547332, 277094664, 554189329, 1108378657, 2216757314, 4433514628, 8867029256, 17734058513

Offset: 0

Michael A. Allen, Dec 13 2021

Number of ways to tile an n-board (an n X 1 array of 1 X 1 cells) using squares, dominoes, trominoes, tetrominoes, black pentominoes, and white pentominoes.

Row sums of A349841.

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

Row sums of triangles in the same family as A349841: A000079, A001045, A077947, A115451.

CoefficientList[Series[(1 - x)/((1 - x^5)(1 - 2x)), {x, 0, 35}], x]

a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + 2*a(n-5) + delta(n,0), a(n<0)=0.
a(n) = 2*a(n-1) + a(n-5) - 2*a(n-6) + delta(n,0) - delta(n,1), a(n<0)=0.
G.f.: 1/(1-x-x^2-x^3-x^4-2*x^5).

A181586 a(0)=0; a(n+1) = 2*a(n) + period 4:repeat 0,1,-2,1.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 5, 8, 17, 34, 69, 136, 273, 546, 1093, 2184, 4369, 8738, 17477, 34952, 69905, 139810, 279621, 559240, 1118481, 2236962, 4473925, 8947848, 17895697, 35791394, 71582789, 143165576, 286331153, 572662306, 1145324613, 2290649224

Offset: 0

Paul Curtz, Jan 30 2011

(**) See A115451, A139763, A139800, A139806, A139814.

a(n) + a(n+1) + a(n+2) + a(n+3) = 2^n.

			a(1)=2*a(0)+0=0, a(2)=2*a(1)+1=0+1=1, a(3)=2*a(2)-2=2-2=0, a(4)=2*a(3)+1=0+1=1, a(5)=2*a(4)+0=2+0=2, a(6)=2*a(5)+1=4+1=5.

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

Cf. A180343.

a:= proc(n) option remember;
      `if`(n=0, 0, 2*a(n-1) +[0, 1, -2, 1][irem(n-1, 4)+1])
    end:
seq(a(n), n=0..40); # Alois P. Heinz, Jan 30 2011

LinearRecurrence[{1, 1, 1, 2}, {0, 0, 1, 0}, 40] (* Jean-François Alcover, May 18 2018 *)

a(n) = a(n-4) + 2^(n-4).
a(n) = -a(n-2) + A078008(n).
a(n) = a(n-2) + A118405(n-2) unsigned.
a(n) = a(n-1) + a(n-2) + a(n-3) + 2*a(n-4) (**).
G.f.: x^2*(-1+x) / ( (2*x-1)*(1+x)*(x^2+1) ). - R. J. Mathar, Feb 06 2011

A268349 Expansion of (1 + x - x^2 - 6x^3)/(1 - x - 2x^2 - 3x^3 - 4x^4).

Original entry on oeis.org

1, 2, 3, 4, 20, 45, 109, 275, 708, 1765, 4442, 11196, 28207, 70985, 178755, 450130, 1133423, 2853888, 7186144, 18094709, 45562353, 114725755, 288879164, 727396569, 1831581574, 4611915224, 11612784735, 29240946181, 73628587619, 185396495082

Offset: 0

Ilya Gutkovskiy, Feb 02 2016

In general, the ordinary generating function for the recurrence relation b(n) = b(n - 1) + 2*b(n - 2) + 3*b(n - 3) + 4*b(n - 4) + ... + k*b(n - k), with n > k - 1 and initial values b(i-1) = i for i = 1..k, is (Sum_{m = 0..(k - 1)} (-m^3 - 3*m^2 + 4*m + 6)*x^m/6)/(1 - Sum_{m = 1..k} m*x^m).

Index entries for linear recurrences with constant coefficients, signature (1,2,3,4).

Cf. A115451, A111586.

[n le 4 select n else Self(n-1)+2*Self(n-2)+3*Self(n-3)+4*Self(n-4): n in [1..35]]; // Vincenzo Librandi, Feb 04 2016

LinearRecurrence[{1, 2, 3, 4}, {1, 2, 3, 4}, 30]
CoefficientList[Series[(1 + x - x^2 - 6 x^3) / (1 - x - 2 x^2 - 3 x^3 - 4 x^4), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 04 2016 *)

Vec((1+x-x^2-6*x^3)/(1-x-2*x^2-3*x^3-4*x^4) + O(x^40)) \\ Michel Marcus, Feb 02 2016

G.f.: (1 + x - x^2 - 6*x^3)/(1 - x - 2*x^2 - 3*x^3 - 4*x^4).

cp's OEIS Frontend

A139760 First quadrisection of A115451.

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A131865 Partial sums of powers of 16.

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A098704 Decimal form of the binary numbers 10, 100010, 1000100010, 10001000100010, 100010001000100010,...

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A349839 Triangle T(n,k) built by placing all ones on the left edge, [1,0,0,0] repeated on the right edge, and filling the body using the Pascal recurrence T(n,k) = T(n-1,k) + T(n-1,k-1).

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A115450 Number triangle (1/((1-x)(1-2x)),-x)-(x/((1-x)(1-2x)),-x^2) (expressed in the notation of Riordan arrays).

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A349842 Expansion of 1/((1 - 2*x)*(1 + x + x^2 + x^3 + x^4)).

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A181586 a(0)=0; a(n+1) = 2*a(n) + period 4:repeat 0,1,-2,1.

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A268349 Expansion of (1 + x - x^2 - 6*x^3)/(1 - x - 2*x^2 - 3*x^3 - 4*x^4).

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A349842 Expansion of 1/((1 - 2x)(1 + x + x^2 + x^3 + x^4)).

A268349 Expansion of (1 + x - x^2 - 6x^3)/(1 - x - 2x^2 - 3x^3 - 4x^4).