A094015 -id:A094015 - cp's OEIS frontend

A096252 Array read by rows, starting with n=0: row n lists A057077(n+1)8^(n+1)/2, A057077(n+2)8^(n+1)/2, A057077(n+1)*8^(n+1).

4, -4, 8, -32, -32, -64, -256, 256, -512, 2048, 2048, 4096, 16384, -16384, 32768, -131072, -131072, -262144, -1048576, 1048576, -2097152, 8388608, 8388608, 16777216, 67108864, -67108864, 134217728, -536870912, -536870912, -1073741824

Offset: 0

Creighton Dement, Jul 31 2004

a(n) = ves( ('i + 'ii' + 'ij' + 'ik')^n ) a(n) = ves( ('j + 'jj' + 'ji' + 'jk')^n ) a(n) = ves( ('k + 'kk' + 'ki' + 'kj')^n ).
The elements x = 'i + 'ii' + 'ij' + 'ik'; y = 'j + 'jj' + 'ji' + 'jk'; and z = 'k + 'kk' + 'ki' + 'kj' are elements of the ring generated from the quaternion factor space Q X Q / {(1,1), (-1,-1)}. Each is represented by a gray shaded area of "Floret's cube". The elements x/2, y/2, z/2 are members of a group, itself a subset of the real algebra generated from Q X Q / {(1,1), (-1,-1)}, which is isomorphic to Q X C_3 (order 24).
This sequence is the term-wise sum of three sequences: a(n) = ves(x^n) = jes(x^n) + les(x^n) + tes(x^n), where jes(x^n)=(1, -6, 8, -24, 16, 0, -64, 384, -512, 1536, -1024, 0, 4096, -24576, 32768, -98304, ...), les(x^n)=(3, 0, 0, 0, -48, 0 -192, 0, 0, 0, 3072, 0, 12288, 0, 0, 0, ...), tes(x^n)=(0, 2, 0, -8, 0, -64, 0, -128, 0, 512, 0, 4096, 0, 8192, 0, -32768, ...). Concerning "les"- notice that if (..., s, 0, 0, 0, t, ...), then t = -16s and if (..., s, 0, t, ...), then t = 4s.

Danny Rorabaugh, Table of n, a(n) for n = 0..1000
C. Dement, The Math Forum.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-16).

Cf. A048473, A094015.

CoefficientList[Series[4(1-x-2x^2-4x^3)/(1-4x^2+16x^4),{x,0,40}],x] (* or *) LinearRecurrence[ {0,4,0,-16},{4,-4,8,-32},40] (* Harvey P. Dale, Feb 15 2024 *)

[(-1)^(floor((floor(n/3)+((n%3)%2)+1)/2)) * 8^(floor(n/3)+1) / 2^(((n+1)^2)%3) for n in range(30)]
# Danny Rorabaugh, May 13 2016

a(n)= 4*a(n-2)-16*a(n-4). G.f.: 4*(1-x-2*x^2-4*x^3)/(1-4*x^2+16*x^4). - R. J. Mathar, Nov 26 2008
a(n) = (-1)^(floor((floor(n/3)+((n mod 3) mod 2)+1)/2)) * 8^(floor(n/3)+1) / 2^(((n+1)^2) mod 3). - Danny Rorabaugh, May 13 2016
a(n) = 4*(-1)^floor((n+1)/2)*A138230(n). - R. J. Mathar, May 21 2019

Edited with clearer definition by Omar E. Pol, Dec 29 2008

A113835 a(n) = a(n-1) + 2^(A007494(n-1)).

Original entry on oeis.org

1, 5, 13, 45, 109, 365, 877, 2925, 7021, 23405, 56173, 187245, 449389, 1497965, 3595117, 11983725, 28760941, 95869805, 230087533, 766958445, 1840700269, 6135667565, 14725602157, 49085340525, 117804817261, 392682724205

Offset: 1

Artur Jasinski, Jan 27 2006

nonn

Cf. A099974, A112627, A080355, A080567, A099969, A099970, A099971.

Partial sums of A094015.

Empirical g.f.: x*(4*x+1) / ((x-1)*(8*x^2-1)). - Colin Barker, Sep 01 2013

Edited with better definition and offset corrected by Omar E. Pol, Jan 08 2009

A096980 Expansion of (1+3x)/(1-2x-7x^2).

Original entry on oeis.org

1, 5, 17, 69, 257, 997, 3793, 14565, 55681, 213317, 816401, 3126021, 11966849, 45815845, 175399633, 671510181, 2570817793, 9842206853, 37680138257, 144255724485, 552272416769, 2114334904933, 8094576727249, 30989497789029

Offset: 0

Paul Barry, Jul 17 2004

Second binomial transform is A002315 (NSW numbers). Binomial transform of A094015.

Binomial transform is A108051 (shifted left, without leading zero). - R. J. Mathar, Jul 11 2012

Index entries for linear recurrences with constant coefficients, signature (2,7).

Cf. A002315, A094015, A108051, A015519.

x='x + O('x^24); Vec((1 + 3*x)/(1 - 2*x - 7*x^2)) \\ Indranil Ghosh, Apr 11 2017

a(n) = (1+sqrt(2))*(1+2*sqrt(2))^n/2 + (1-sqrt(2))*(1-2*sqrt(2))^n/2.
a(n) = 3*Sum_{k=0..floor((n-1)/2)} binomial(n-k-1, k)*(7/2)^k*2^(n-k-1) + Sum_{k=0..floor(n/2)} binomial(n-k, k)*(7/2)^k*2^(n-k).
a(n) = A015519(n+1) + 3*A015519(n). - R. J. Mathar, Jul 11 2012
Satisfies recurrence relation system a(n) = 3*a(n-1) + 2*b(n-1), b(n) = 2*a(n-1) - b(n-1), a(0)=1, b(0)=1. - Ilya Gutkovskiy, Apr 11 2017

A152842 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 3, 1, 4, 3, 1, 7, 15, 9, 1, 8, 22, 24, 9, 1, 11, 46, 90, 81, 27, 1, 12, 57, 136, 171, 108, 27, 1, 15, 93, 307, 579, 621, 351, 81, 1, 16, 108, 400, 886, 1200, 972, 432, 81, 1, 19, 156, 724, 2086, 3858, 4572, 3348, 1377, 243, 1, 20, 175, 880, 2810, 5944, 8430, 7920

Offset: 0

Philippe Deléham, Dec 14 2008

			The triangle T(n,k) begins:
n\k  0   1    2     3     4      5      6      7      8      9     10    11   12
0:   1
1:   1   3
2:   1   4    3
3:   1   7   15     9
4:   1   8   22    24     9
5:   1  11   46    90    81     27
6:   1  12   57   136   171    108     27
7:   1  15   93   307   579    621    351     81
8:   1  16  108   400   886   1200    972    432     81
9:   1  19  156   724  2086   3858   4572   3348   1377    243
10:  1  20  175   880  2810   5944   8430   7920   4725   1620    243
11:  1  23  235  1405  5450  14374  26262  33210  28485  15795   5103   729
12:  1  24  258  1640  6855  19824  40636  59472  61695  44280  20898  5832  729
... reformatted and extended. - _Franck Maminirina Ramaharo_, Feb 28 2018

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

Cf. A152815, A007318, A064861.

a152842 n k = a152842_tabl !! n !! k
a152842_row n = a152842_tabl !! n
a152842_tabl = map fst $ iterate f ([1], 3) where
   f (xs, z) = (zipWith (+) ([0] ++ map (* z) xs) (xs ++ [0]), 4 - z)
-- Reinhard Zumkeller, May 01 2014

T(n,k) = T(n-1,k) + (2-(-1)^n)*T(n-1,k-1).
Sum_{k=0..n} T(n,k) = A094015(n).
T(n,n) = A108411(n+1).
T(2n,n) = A069835(n).
G.f.: (1+x+x*y)/(1-x^2-4*x^2*y-3*x^2*y^2). - Philippe Deléham , Nov 09 2013
T(n,k) = T(n-2,k) + 4*T(n-2,k-1) + 3*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = 3, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Nov 09 2013

A135838 Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 4, 12, 12, 4, 4, 16, 24, 16, 4, 8, 40, 80, 80, 40, 8, 8, 48, 120, 160, 120, 48, 8, 16, 112, 336, 560, 560, 336, 112, 16, 16, 128, 448, 896, 1120, 896, 448, 128, 16, 32, 288, 1152, 2688, 4032, 4032, 2688, 1152, 288, 32

Offset: 1

Gary W. Adamson, Dec 01 2007

			First few rows of the triangle are:
  1;
  2,  2;
  2,  4,  2;
  4, 12, 12,  4;
  4, 16, 24, 16,  4;
  8, 40, 80, 80, 40, 8;
  ...

Gheorghe Coserea, Rows n = 1..100, flattened

Cf. A016116, A059304, A069720, A093968, A094015 (row sums), A135837.

A135838 := proc(n,k)
    2^floor(n/2)*binomial(n-1,k-1) ;
end proc:
seq(seq( A135838(n,k),k=1..n),n=1..10) ; # R. J. Mathar, Aug 15 2022

T[n_, k_]:= 2^Floor[n/2]*Binomial[n-1, k-1];
Table[T[n, k], {n,12}, {k,n}] //Flatten (* G. C. Greubel, Feb 07 2022 *)

A(n,k) = 2^(n\2)*binomial(n-1,k-1);
concat(vector(10, n, vector(n, k, A(n,k))))  \\ Gheorghe Coserea, May 18 2016

flatten([[2^(n//2)*binomial(n-1, k-1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 07 2022

M * Pascal's triangle as infinite lower triangular matrices, where M = a triangle with (1, 2, 2, 4, 4, 8, 8, 16, 16, ...) in the main diagonal and the rest zeros.
Sum_{k=1..n} T(n, k) = A094015(n-1).
From G. C. Greubel, Feb 07 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 1) = A016116(n).
T(n, 2) = 2*A093968(n-1).
T(2*n-1, n) = A059304(n-1).
T(2*n, n) = 2*A069720(n). (End)

A164588 a(n) = ((3 + sqrt(18))(5 + sqrt(8))^n + (3 - sqrt(18))(5 - sqrt(8))^n)/6.

Original entry on oeis.org

1, 9, 73, 577, 4529, 35481, 277817, 2174993, 17027041, 133295529, 1043495593, 8168931937, 63949894289, 500627099961, 3919122796697, 30680567267633, 240180585132481, 1880236207775049, 14719292130498313, 115228905772807297, 902061091509601649

Offset: 0

Al Hakanson (hawkuu(AT)gmail.com), Aug 17 2009

nonn

Binomial transform of A057084. Second binomial transform of A002315. Third binomial transform of A108051 without initial 0. Fourth binomial transform of A096980. Fifth binomial transform of A094015.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-17).

Cf. A057084, A002315, A108051, A096980, A094015.

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+3*r)*(5+2*r)^n+(3-3*r)*(5-2*r)^n)/6: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 24 2009

LinearRecurrence[{10,-17},{1,9},30] (* Harvey P. Dale, Sep 11 2016 *)

x='x+O('x^50); Vec((1-x)/(1-10*x+17*x^2)) \\ G. C. Greubel, Aug 12 2017

a(n) = 10*a(n-1) - 17*a(n-2) for n > 1; a(0) = 1, a(1) = 9.
G.f.: (1-x)/(1-10*x+17*x^2).
E.g.f.: (1/3)*exp(5*x)*(3*cosh(2*sqrt(2)*x) + 3*sqrt(2)*sinh(2*sqrt(2)*x)). - G. C. Greubel, Aug 12 2017

Extended by Klaus Brockhaus and R. J. Mathar Aug 24 2009

A107904 Expansion of (1+6x)/(1-12x^2).

Original entry on oeis.org

1, 6, 12, 72, 144, 864, 1728, 10368, 20736, 124416, 248832, 1492992, 2985984, 17915904, 35831808, 214990848, 429981696, 2579890176, 5159780352, 30958682112, 61917364224, 371504185344, 743008370688, 4458050224128, 8916100448256, 53496602689536, 106993205379072

Offset: 0

Paul Barry, May 27 2005

Fourth binomial transform is A107903.

Index entries for linear recurrences with constant coefficients, signature (0,12).

Cf. A094015, A107903, A108411.

LinearRecurrence[{0,12},{1,6},30] (* Harvey P. Dale, Sep 22 2014 *)

a(n) = ((1+sqrt(3))*(2*sqrt(3))^n + (1-sqrt(3))*(-2*sqrt(3))^n)/2.
a(2n) = 12^n, a(2n+1) = 6*12^n.
a(n) = 2^n*A108411(n+1). - R. J. Mathar, Aug 15 2023
From Amiram Eldar, Dec 06 2024: (Start)
Sum_{n>=0} 1/a(n) = 14/11.
Sum_{n>=0} (-1)^n/a(n) = 10/11. (End)

cp's OEIS Frontend

A096252 Array read by rows, starting with n=0: row n lists A057077(n+1)*8^(n+1)/2, A057077(n+2)*8^(n+1)/2, A057077(n+1)*8^(n+1).

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A113835 a(n) = a(n-1) + 2^(A007494(n-1)).

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A096980 Expansion of (1+3x)/(1-2x-7x^2).

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A152842 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A135838 Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).

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A164588 a(n) = ((3 + sqrt(18))*(5 + sqrt(8))^n + (3 - sqrt(18))*(5 - sqrt(8))^n)/6.

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Magma

Mathematica

PARI

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Extensions

A107904 Expansion of (1+6x)/(1-12x^2).

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Programs

Mathematica

Formula

A096252 Array read by rows, starting with n=0: row n lists A057077(n+1)8^(n+1)/2, A057077(n+2)8^(n+1)/2, A057077(n+1)*8^(n+1).

A164588 a(n) = ((3 + sqrt(18))(5 + sqrt(8))^n + (3 - sqrt(18))(5 - sqrt(8))^n)/6.