A053088 -id:A053088 - cp's OEIS frontend

A084219 Inverse binomial transform of A053088.

1, -1, 4, -8, 20, -44, 100, -220, 484, -1052, 2276, -4892, 10468, -22300, 47332, -100124, 211172, -444188, 932068, -1951516, 4077796, -8505116, 17709284, -36816668, 76429540, -158451484, 328087780, -678545180, 1401829604

Offset: 0

Paul Barry, May 20 2003

Contribution from Gary W. Adamson, Jan 05 2009: (Start)
Unsigned, starting with offset 1: generated from iterates of M * [1,1,1,...]
where M = a tridiagonal matrix with [0,1,1,1,...] as the main diagonal,
[1,1,1,...] as the uperdiagonal and [2,0,0,0,...] as the subdiagonal. (End)
Define a triangle via T(n,0) = T(n,n) = A001045(n) and T(r,c) = T(r-1,c-1) + T(r-1,c). The row sums of the triangle are s(n) = 0, 2, 4, 12, ... = 2*A059570(n), and their first differences are s(n+1)-s(n) = 2*|a(n)|. J. M. Bergot, May 15 2013

Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015.
Index entries for linear recurrences with constant coefficients, signature (-3,0,4).

LinearRecurrence[{-3,0,4},{1,-1,4},30] (* Harvey P. Dale, Dec 16 2016 *)

a(n) = (4 - 3*n*(-2)^(n-1) + 5*(-2)^n)/9.
a(n) = (1/4) + Sum_{k=0..n} (-2)^k*(k+3)/4.
G.f.: (1+x)^2/((1-x)(1+2x)^2).

A232603 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=2, q=-1/2.

Original entry on oeis.org

0, -1, 2, -5, 6, -13, 10, -29, 6, -69, -38, -197, -250, -669, -1142, -2509, -4762, -9813, -19302, -38965, -77530, -155501, -310518, -621565, -1242554, -2485733, -4970790, -9942309, -19883834, -39768509, -79536118

Offset: 0

Stanislav Sykora, Nov 27 2013

The factor 2^n (i.e., |1/q|^n) is present to keep the values integer.

See also A232600 and references therein for integer values of q.

			a(3) = 2^3 * [0^2/2^0 - 1^2/2^1 + 2^2/2^2 - 3^2/2^3] = -5.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-1,3,5,2).

Cf. A001045 (p=0,q=-1/2), A053088 (p=1,q=-1/2), A232604 (p=3,q=-1/2), A000225 (p=0,q=1/2), A000295 and A125128 (p=1,q=1/2), A047520 (p=2,q=1/2), A213575 (p=3,q=1/2), A232599 (p=3,q=-1), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2).

[((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27: n in [0..30]]; // G. C. Greubel, Mar 31 2021

A232603:= n-> ((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27; seq(A232603(n), n=0..35); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-1,3,5,2}, {0,-1,2,-5}, 35] (* G. C. Greubel, Mar 31 2021 *)

a(n)=((-1)^n*(9*n^2+12*n+2)-2^(n+1))/27;

[((-1)^n*(2+12*n+9*n^2) -2^(n+1))/27 for n in (0..30)] # G. C. Greubel, Mar 31 2021

a(n) = ((-1)^n*(9*n^2+12*n+2) - 2^(n+1))/27.
G.f.: x*(-1+x)/( (1-2*x)*(1+x)^3 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (1/27)*(-2*exp(2*x) + (2 -21*x +9*x^2)*exp(-x)). - G. C. Greubel, Mar 31 2021
a(n) = - a(n-1) + 3*a(n-2) + 5*a(n-3) + 2*a(n-4). - Wesley Ivan Hurt, Mar 31 2021

A232604 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=3, q=-1/2.

Original entry on oeis.org

0, -1, 6, -15, 34, -57, 102, -139, 234, -261, 478, -375, 978, -241, 2262, 1149, 6394, 7875, 21582, 36305, 80610, 151959, 314566, 616965, 1247754, 2479883, 4977342, 9935001, 19891954, 39759519, 79546038, 159062285

Offset: 0

Stanislav Sykora, Nov 27 2013

The factor 2^n (i.e., |1/q|^n) is present to make the values integers.
See also A232600 and references therein for integer values of q.
The same values with different signs are produced by a(n) = n^3 - 2*a(n). The signs are all positive until n = 15, with negative signs on values for all subsequent odd indices. - Richard R. Forberg, Feb 17 2014.

			a(3) = 2^3 * (0^3/2^0 - 1^3/2^1 + 2^3/2^2 - 3^3/2^3) = 0-4+16-27 = -15.

Stanislav Sykora, Table of n, a(n) for n = 0..1000
S. Sykora, Finite and Infinite Sums of the Power Series (k^p)(x^k), DOI 10.3247/SL1Math06.002, Section V.
Index entries for linear recurrences with constant coefficients, signature (-2,2,8,7,2).

Cf. A001045 (p=0,q=-1/2), A053088 (p=1,q=-1/2), A232603 (p=2,q=-1/2), A000225 (p=0,q=1/2), A000295 and A125128 (p=1,q=1/2), A047520 (p=2,q=1/2), A213575 (p=3,q=1/2), A232599 (p=3,q=-1), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2).

[(2^(n+1) + (-1)^n*(9*n^3 +18*n^2 +6*n -2))/27: n in [0..35]]; // G. C. Greubel, Mar 31 2021

A232604:= n-> (2^(n+1) +(-1)^n*(9*n^3 +18*n^2 +6*n -2))/27; seq(A232604(n), n=0..30); # G. C. Greubel, Mar 31 2021

LinearRecurrence[{-2,2,8,7,2}, {0,-1,6,-15,34}, 35] (* G. C. Greubel, Mar 31 2021 *)

a(n)=(2^(n+1)+(-1)^n*(9*n^3+18*n^2+6*n-2))/27;

[(2^(n+1) + (-1)^n*(9*n^3 +18*n^2 +6*n -2))/27 for n in (0..35)] # G. C. Greubel, Mar 31 2021

a(n) = (2^(n+1) + (-1)^n*(9*n^3+18*n^2+6*n-2))/27.
G.f.: x*(1-4*x+x^2) / ( (2*x-1)*(1+x)^4 ). - R. J. Mathar, Nov 23 2014
E.g.f.: (1/27)*(2*exp(2*x) - (2 +33*x -45*x^2 +9*x^3)*exp(-x)). - G. C. Greubel, Mar 31 2021
a(n) = - 2*a(n-1) + 2*a(n-2) + 8*a(n-3) + 7*a(n-4) + 2*a(n-5). - Wesley Ivan Hurt, Mar 31 2021

A077898 Expansion of (1 - x)^(-1)/(1 + x - 2*x^2).

Original entry on oeis.org

1, 0, 3, -2, 9, -12, 31, -54, 117, -224, 459, -906, 1825, -3636, 7287, -14558, 29133, -58248, 116515, -233010, 466041, -932060, 1864143, -3728262, 7456549, -14913072, 29826171, -59652314, 119304657, -238609284, 477218599, -954437166, 1908874365, -3817748696, 7635497427

Offset: 0

N. J. A. Sloane, Nov 17 2002

Partial sums of A077925 (signed Jacobsthal numbers). - Paul Barry, Aug 26 2003

The generalized (3,-2)-Padovan sequence p(3,-2;n). See the W. Lang link under A000931 with (A,B)=(3,-2). - Wolfdieter Lang, Jun 28 2010

			(3,-2)-Padovan combinatorics from the (3,2)-Morse code with weights -2 and 3 for 3-lines -- and 2-lines -, respectively (see the W. Lang link under A000931). n=5: two codes - -- and -- - with the weights (3^1)*(-2)^1 and (-2)^1*3^1, respectively, adding up to 2*(3)(-2) = -12 = a(5). - _Wolfdieter Lang_, Jun 28 2010

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

Cf. A000931, A053088, A077925.

[(5+(-1)^n*2^(2+n)+3*n)/9 : n in [0..50]]; // Wesley Ivan Hurt, Apr 21 2016

A077898:=n->(5+(-1)^n*2^(2+n)+3*n)/9: seq(A077898(n), n=0..50); # Wesley Ivan Hurt, Apr 21 2016

CoefficientList[Series[(1 - x)^(-1)/(1 + x - 2*x^2), {x, 0, 40}], x] (* Wesley Ivan Hurt, Apr 21 2016 *)

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

G.f.: (1-x)^(-1)/(1+x-2*x^2).
a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} binomial(j, i)*(-3)^i. - Paul Barry, Aug 26 2003
a(n) = (-1)^n * A053088(n). - R. J. Mathar, Aug 30 2008
From Colin Barker, Apr 21 2016: (Start)
a(n) = 3*a(n-2) - 2*a(n-3) for n>2.
a(n) = (5+(-1)^n*2^(2+n)+3*n)/9. (End)
E.g.f.: (4*exp(-2*x) + (5 + 3*x)*exp(x))/9. - Ilya Gutkovskiy, Apr 21 2016
a(n) = Sum_{k=0..n} (n+1-k)*(-2)^k. - Bruno Berselli, May 15 2018

A103196 a(n) = (1/9)(2^(n+3)-(-1)^n(3n-1)).

Original entry on oeis.org

1, 2, 3, 8, 13, 30, 55, 116, 225, 458, 907, 1824, 3637, 7286, 14559, 29132, 58249, 116514, 233011, 466040, 932061, 1864142, 3728263, 7456548, 14913073, 29826170, 59652315, 119304656, 238609285, 477218598, 954437167

Offset: 0

Creighton Dement, Mar 18 2005

A floretion-generated sequence relating to the Jacobsthal sequence A001045 as well as to A095342 (Number of elements in n-th string generated by a Kolakoski(5,1) rule starting with a(1)=1). (a(n)) may be seen as the result of a certain transform of the natural numbers (see program code).

Floretion Algebra Multiplication Program, FAMP Code: 4jesleftforseq[A*B] with A = + 'i + 'j + i' + j' + 'ii' + 'jj' + 'ij' + 'ji' + e and B = - .25'i + .25'j + .25'k + .25i' - .25j' + .25k' - .25'ii' + .25'jj' + .25'kk' + .25'ij' + .25'ik' + .25'ji' + .25'jk' - .25'ki' - .25'kj' - .25e; 1vesforseq[A*B](n) = n, ForType: 1A.

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

Cf. A001045, A095342, A053088, A034299, A052953, A026644.

Table[(2^(n+3)-(-1)^n (3n-1))/9,{n,0,30}] (* or *) LinearRecurrence[ {0,3,2},{1,2,3},40] (* Harvey P. Dale, Jul 09 2018 *)

G.f. (2x+1)/((1-2x)(x+1)^2); Superseeker results: a(n) + a(n+1) = A001045(n+3); a(n+1) - a(n) = A095342(n+1); a(n+2) - a(n+1) - a(n) = A053088(n+1) = A034299(n+1) - A034299(n); a(n) + 2a(n+1) + a(n+2) = 2^(n+3); a(n+2) - 2a(n+1) + a(n) = A053088(n+1) - A053088(n); a(n+2) - a(n) = A001045(n+4) - A001045(n+3) = A052953(n+3) - A052953(n+2) = A026644(n+2) - A026644(n+1);

a(n)=sum{k=0..n+2, (-1)^(n-k)*C(n+2, k)phi(phi(3^k))}; a(n)=sum{k=0..n+2, (-1)^(n-k)*C(n+2, k)(2*3^k/9+C(1, k)/3+4*C(0, k)/9)}; a(n)=sum{k=0..n+2, J(n-k+3)((-1)^(k+1)-2C(1, k)+4C(0, k))} where J(n)=A001045(n); a(n)=A113954(n+2). - Paul Barry, Nov 09 2005

A172285 a(n) = (52^n - 5(-1)^n - 3n(-1)^n) / 9.

Original entry on oeis.org

0, 2, 1, 6, 7, 20, 33, 74, 139, 288, 565, 1142, 2271, 4556, 9097, 18210, 36403, 72824, 145629, 291278, 582535, 1165092, 2330161, 4660346, 9320667, 18641360, 37282693, 74565414, 149130799, 298261628, 596523225, 1193046482, 2386092931, 4772185896

Offset: 0

Paul Curtz, Jan 30 2010

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

Cf. A000079, A001045, A053088, A141291, A164044.

[(5*2^n - 5*(-1)^n - 3*n*(-1)^n) / 9: n in [0..40]]; // Vincenzo Librandi, Aug 05 2011

A172295 := proc(n) (5*2^n - 5*(-1)^n - 3*n*(-1)^n) / 9 ; end proc: seq(A172295(n), n=0..100) ; # R. J. Mathar, Feb 02 2010

Table[(5*2^n - 5*(-1)^n - 3*n*(-1)^n)/9, {n, 0, 40}] (* Wesley Ivan Hurt, Aug 27 2015 *)

first(m)=vector(m,i,i--;(5*2^i -5*(-1)^i - 3*i*(-1)^i ) / 9) \\ Anders Hellström, Aug 27 2015

a(n) = 3*a(n-2) + 2*a(n-3), n>2.
a(n+1) = 2*a(n) + (-1)^n * (2+n).
a(n) = A053088(n-1) + A001045(n), n>0.
a(n) = A000079(n) - A053088(n).
a(2n) = A141291(n). a(2n+1) = 2*A164044(n).
G.f.: x*(2+x)/( (1-2*x)*(1+x)^2 ).

Definition replaced by explicit formula; g.f. added - R. J. Mathar, Feb 02 2010

A176737 Expansion of 1 / (1-4x^2-3x^3). (4,3)-Padovan sequence.

Original entry on oeis.org

1, 0, 4, 3, 16, 24, 73, 144, 364, 795, 1888, 4272, 9937, 22752, 52564, 120819, 278512, 640968, 1476505, 3399408, 7828924, 18027147, 41513920, 95595360, 220137121, 506923200, 1167334564, 2688104163, 6190107856, 14254420344, 32824743913, 75588004944, 174062236684

Offset: 0

Wolfdieter Lang, Jun 26 2010

See A000931 (Padovan), and the W. Lang link given there.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,4,3).

Cf. A053088 ((3,2)-Padovan).

CoefficientList[Series[1/(1-4*x^2-3*x^3),{x,0,40}],x] (* or *) LinearRecurrence[ {0,4,3},{1,0,4},40] (* Harvey P. Dale, Jan 21 2013 *)

Vec(1 / ((1 + x)*(1 - x - 3*x^2)) + O(x^40)) \\ Colin Barker, Dec 25 2017

O.g.f.: 1/((1-x-3*x^2)*(1+x)) = (2-3*x)/(1-x-3*x^2) -1/(1+x).
a(n) = 2*b(n) - 3*b(n-1) - (-1)^n, n>=0, with b(n):=A006130(n) ((1,3)-Fibonacci), b(-1):=0.
From Wolfdieter Lang, Aug 26 2010: (Start)
a(n) = a(n-1) + 3*a(n-2) + (-1)^n, n>=2, a(0)=1, a(1)=0.
Due to the identity for the o.g.f. A(x): A(x)= x*(1 + 3*x)*A(x) + 1/(1+x).
(This recurrence was observed by Gary Detlefs in an Aug 24 2010 email to the author.)
(End)
a(n) = 4*a(n-2) + 3*a(n-3) for n>2. - Harvey P. Dale, Jan 21 2013
a(n) = (-1)^(n+1)*A140165(n+2)-(-1)^n. - R. J. Mathar, Apr 22 2013
a(n) = ((-1)^(1+n) + (2^(-n)*((-2+sqrt(13))*(1+sqrt(13))^n + (1-sqrt(13))^n*(2+sqrt(13)))) / sqrt(13)). - Colin Barker, Dec 25 2017

A113678 Sequence array for A078008.

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 2, 2, 0, 1, 6, 2, 2, 0, 1, 10, 6, 2, 2, 0, 1, 22, 10, 6, 2, 2, 0, 1, 42, 22, 10, 6, 2, 2, 0, 1, 86, 42, 22, 10, 6, 2, 2, 0, 1, 170, 86, 42, 22, 10, 6, 2, 2, 0, 1, 342, 170, 86, 42, 22, 10, 6, 2, 2, 0, 1, 682, 342, 170, 86, 42, 22, 10, 6, 2, 2, 0, 1, 1366, 682, 342, 170, 86

Offset: 0

Paul Barry, Nov 04 2005

Row sums are A001045(n+1). Diagonal sums are A053088. Inverse is A113680.

			Triangle begins
1;
0, 1;
2, 0, 1;
2, 2, 0, 1;
6, 2, 2, 0, 1;
10, 6, 2, 2, 0, 1;
22, 10, 6, 2, 2, 0, 1;

Riordan array ((1-x)/(1-x-2x^2), x); Number triangle T(n, k)=if(k<=n, (2^(n-k)+2(-1)^(n-k))/3, 0); T(n, k)=sum{i=0..n, C(n-i, k)C(k, n-i)(2^i+2(-1)^i)/3}.

A366987 Triangle read by rows: T(n, k) = -(2^(n - k)*(-1)^n + 2^k + (-1)^k)/3.

Original entry on oeis.org

-1, 0, 0, -2, -1, -2, 2, 1, -1, -2, -6, -3, -3, -3, -6, 10, 5, 1, -1, -5, -10, -22, -11, -7, -5, -7, -11, -22, 42, 21, 9, 3, -3, -9, -21, -42, -86, -43, -23, -13, -11, -13, -23, -43, -86, 170, 85, 41, 19, 5, -5, -19, -41, -85, -170, -342, -171, -87, -45, -27, -21, -27, -45, -87, -171, -342

Offset: 0

Paul Curtz and Thomas Scheuerle, Oct 31 2023

			Triangle T(n, k) starts:
   -1
    0   0
   -2  -1  -2
    2   1  -1  -2
   -6  -3  -3  -3  -6
   10   5   1  -1  -5 -10
  -22 -11  -7  -5  -7 -11 -22
   42  21   9   3  -3  -9 -21 -42
   ...
Note the symmetrical distribution of the absolute values of the terms in each row.

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened)

Rows sums: -A282577(n+2), if the conjectures from Colin Barker in A282577 are true.
First column: -(-1)^n * A078008(n).
Second column: (-1)^n * A001045(n).
Third column: -A140966(n).
Fourth column: (-1)^n * A155980(n+2).
Cf. A000079, A020989, A048573, A053088, A059570, A077898, A140966.

T := (n, k) -> -(2^(n-k)*(-1)^n + 2^k + (-1)^k)/3:
seq(seq(T(n, k), k = 0..n), n = 0..10);  # Peter Luschny, Nov 02 2023

A366987row[n_]:=Table[-(2^(n-k)(-1)^n+2^k+(-1)^k)/3,{k,0,n}];Array[A366987row,15,0] (* Paolo Xausa, Nov 07 2023 *)

T(n, k) = (-2^(k+1) + 2*(-1)^(k+1) + (-1)^(n+1)*2^(1+n-k))/6 \\ Thomas Scheuerle, Nov 01 2023

T(n, 0) = -((-2)^n + 2)/3.
T(n, k+1) - T(n, k) = T(n-1, k) + (-1)^k.
T(2*n+1, n) = A001045(n).
T(2*n+1, n+1) = -A001045(n).
T(2*n, n+1) = -A048573(n-1), for n > 0.
Note that the definition of T extends to negative parameters:
T(2*n-2, n-1) = -A001045(n).
-2^n*Sum_{k=0..n} (-1)^k*T(-n, -k) = A059570(n+1).
-4^n*Sum_{k=0..2*n} T(-2*n, -k) = A020989(n).
-Sum_{k=0..n} (-1)^k*T(n, k) = A077898(n). See also A053088.
Sum_{k = 0..2*n} |T(2*n, k)| = (4^(n+1) - 1)/3.
Sum_{k = 0..2*n+1} |T(2*n+1, k)| = (1 + (-1)^n - 2^(2 + n) + 2^(1 + 2*n))/3.
G.f.: (-1 - x + x*y)/((1 - x)*(1 + 2*x)*(1 + x*y)*(1 - 2*x*y)). - Stefano Spezia, Nov 03 2023

a(42) corrected by Paolo Xausa, Nov 07 2023

A099095 Riordan array (1,3+2x).

Original entry on oeis.org

1, 0, 3, 0, 2, 9, 0, 0, 12, 27, 0, 0, 4, 54, 81, 0, 0, 0, 36, 216, 243, 0, 0, 0, 8, 216, 810, 729, 0, 0, 0, 0, 96, 1080, 2916, 2187, 0, 0, 0, 0, 16, 720, 4860, 10206, 6561, 0, 0, 0, 0, 0, 240, 4320, 20412, 34992, 19683, 0, 0, 0, 0, 0, 32, 2160, 22680, 81648, 118098, 59049, 0, 0

Offset: 0

Paul Barry, Sep 25 2004

Row sums are A007482. Diagonal sums are A053088. The Riordan array (1,s+tx) defines T(n,k)=binomial(k,n-k)s^k(t/s)^(n-k). The row sums satisfy a(n)=s*a(n-1)+t*a(n-2) and the diagonal sums satisfy a(n)=s*a(n-2)+t*a(n-3).

			Rows begin {1}, {0,3}, {0,2,9}, {0,0,12,27}, {0,0,4,54,81},...

Eric W. Weisstein, Fermat Polynomial

Cf. A038220.

Number triangle T(n, k)=binomial(k, n-k)3^k*(2/3)^(n-k); Columns have g.f. (3x+2x^2)^k.

cp's OEIS Frontend

A084219 Inverse binomial transform of A053088.

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A232603 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=2, q=-1/2.

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A232604 a(n) = 2^n * Sum_{k=0..n} k^p*q^k, where p=3, q=-1/2.

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A077898 Expansion of (1 - x)^(-1)/(1 + x - 2*x^2).

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A103196 a(n) = (1/9)(2^(n+3)-(-1)^n(3n-1)).

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A172285 a(n) = (5*2^n - 5*(-1)^n - 3*n*(-1)^n) / 9.

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A176737 Expansion of 1 / (1-4*x^2-3*x^3). (4,3)-Padovan sequence.

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A172285 a(n) = (52^n - 5(-1)^n - 3n(-1)^n) / 9.

A176737 Expansion of 1 / (1-4x^2-3x^3). (4,3)-Padovan sequence.