A001045 -id:A001045 - cp's OEIS frontend

A073377 Seventh convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

1, 8, 52, 264, 1194, 4872, 18516, 66264, 226083, 740608, 2344232, 7202416, 21562164, 63090288, 180884088, 509245776, 1410356133, 3848340312, 10359516684, 27544099704, 72406891326, 188356187448

Offset: 0

Wolfdieter Lang, Aug 02 2002

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (8,-12,-56,154,168,-700,-328,1791,656,-2800,-1344, 2464,1792,-768,-1024,-256).

Eighth (m=7) column of triangle A073370.

Cf. A001045, A073376.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1-2*x))^8 )); // G. C. Greubel, Sep 29 2022

Table[(2^(n+8)*(9539600 +17240268*n +11555460*n^2 +3849489*n^3 +703080*n^4 +71442*n^5 +3780*n^6 +81*n^7) +(-1)^n*(236325040 +225702732*n +87290028*n^2 +17880849*n^3 +2109240*n^4 +144018*n^5 +5292*n^6 +81*n^7))/(7!*3^12), {n,0,60}] (* G. C. Greubel, Sep 29 2022 *)

def A073377(n): return (2^(n+8)*(9539600 +17240268*n +11555460*n^2 +3849489*n^3 +703080*n^4 +71442*n^5 +3780*n^6 +81*n^7) +(-1)^n*(236325040 +225702732*n +87290028*n^2 +17880849*n^3 +2109240*n^4 +144018*n^5 +5292*n^6 +81*n^7))/(factorial(7)*3^12)
[A073377(n) for n in range(40)] # G. C. Greubel, Sep 29 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073376(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+7, 7) * binomial(n-k, k) * 2^k.
a(n) = ((328247920 +332102604*n +131833680*n^2 +26450901*n^3 +2844099*n^4 + 156087*n^5 +3429*n^6)*(n+1)*U(n+1) + 2(141143240 +150941694*n +62335731*n^2 + 12873492*n^3 +1414314*n^4 +78894*n^5 +1755*n^6)*(n+2)*U(n))/(7!*3^11) with U(n) = A001045(n+1), n>=0.
G.f.: 1/(1-(1+2*x)*x)^8 = 1/((1+x)*(1-2*x))^8.
E.g.f.: (1/(7!*3^12))*( 4096*(596225 +4177950*x +7304850*x^2 +5109300*x^3 +1691550*x^4 +278964*x^5 +21924*x^6 +648*x^7)*exp(2*x) + (236325040 -333132240*x +158026680*x^2 -34637400*x^3 +3921750*x^4 -234738*x^5 +6993*x^6 -81*x^7)*exp(-x) ). - G. C. Greubel, Sep 29 2022

A073378 Eighth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

Original entry on oeis.org

1, 9, 63, 345, 1665, 7227, 29073, 109791, 394020, 1354210, 4486482, 14397318, 44932446, 136817370, 407566350, 1190446866, 3415935699, 9645169743, 26836557825, 73670997015, 199751003991, 535449185469

Offset: 0

Wolfdieter Lang, Aug 02 2002

For a(n) in terms of U(n+1) and U(n) with U(n) = A001045(n+1) see A073370 and the row polynomials of triangles A073399 and A073400.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-18,-60,234,126,-1176,36,3519,-479,-7038,144, 9408,2016,-7488,-3840,2304,2304,512).

Ninth (m=8) column of triangle A073370.

Cf. A001045, A073377, A073399, A073400, A073401.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x)*(1-2*x))^9 )); // G. C. Greubel, Oct 01 2022

CoefficientList[Series[1/((1+x)*(1-2*x))^9, {x,0,40}], x] (* G. C. Greubel, Oct 01 2022 *)

def A073378_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*(1-2*x))^9 ).list()
A073378_list(40) # G. C. Greubel, Oct 01 2022

a(n) = Sum_{k=0..n} b(k)*c(n-k), with b(k) = A001045(k+1) and c(k) = A073377(k).
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k+8, 8) * binomial(n-k, k) * 2^k.
G.f.: 1/(1-(1+2*x)*x)^9 = 1/((1+x)*(1-2*x))^9.

A073402 Coefficient triangle of polynomials (rising powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073401.

Original entry on oeis.org

2, 33, 9, 831, 396, 45, 28236, 18297, 3744, 243, 1210140, 968679, 273483, 32481, 1377, 62686440, 58920534, 20681811, 3418767, 268029, 8019, 3810867480, 4075425738, 1683064737, 347584284, 38186478, 2130138, 47385

Offset: 0

Wolfdieter Lang, Aug 02 2002

The row polynomials are q(k,x) := sum(a(k,m)*x^m,m=0..k), k=0,1,2,...

The k-th convolution of U0(n) := A001045(n+1), n>= 0, ((1,2) Fibonacci numbers starting with U0(0)=1) with itself is Uk(n) := A073370(n+k,k) = (p(k-1,n)*(n+1)*U0(n+1) + q(k-1,n)*(n+2)*2*U0(n))/(k!*9^k)), k=1,2,..., where the companion polynomials p(k,n) := sum(b(k,m)*n^m,m=0..k), k >= 0, are the row polynomials of triangle b(k,m)= A073401(k,m).

			k=2: U2(n)=((30+9*n)*(n+1)*U0(n+1)+(33+9*n)*(n+2)*2*U0(n))/(2*9^2), cf. A073372.
1; 33,9; 831,396,45; ... (lower triangular matrix a(k,m), k >= m >= 0, else 0).

Sean A. Irvine, Table of n, a(n) for n = 0..989
Wolfdieter Lang, First 7 rows.

Cf. A001045, A073370, A073399, A073372, A073400.

Recursion for row polynomials defined in the comments: p(k, n)= (n+2)*p(k-1, n+1)+4*(n+2*(k+1))*p(k-1, n)+2*(n+3)*q(k-1, n+1); q(k, n)= (n+1)*p(k-1, n+1)+4*(n+2*(k+1))*q(k-1, n), k >= 1. [Corrected by Sean A. Irvine, Nov 25 2024]

A081857 Jacobsthal sequence (A001045) as represented in base 4.

Original entry on oeis.org

0, 1, 1, 3, 11, 23, 111, 223, 1111, 2223, 11111, 22223, 111111, 222223, 1111111, 2222223, 11111111, 22222223, 111111111, 222222223, 1111111111, 2222222223, 11111111111, 22222222223, 111111111111, 222222222223, 1111111111111

Offset: 0

Matthew Vandermast, Apr 11 2003

			a(8)= 1111 because A001045(8)= 85 (in base 10) = 64 + 16 + 4 + 1 = 1 * (4^3) + 1 * (4^2) + 1 * (4^1) + 1.

Eric Weisstein's World of Mathematics, Repunit.

Cf. A002450.

from gmpy2 import digits
def A081857(n): return int(digits(((1<Chai Wah Wu, Apr 18 2025

For n > 0, a(2n) is represented as a string of n 1's and a(2n+1) as a string of (n-1) 2's followed by a 3.

A091084 a(n) = A001045(n) mod 10.

Original entry on oeis.org

0, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5, 1, 1, 3, 5

Offset: 0

Paul Barry, Dec 18 2003

A001045(0), followed by A001045(1), A001045(2), A001045(3), A001045(4) repeating.

Decimal expansion of 227/19998. - Elmo R. Oliveira, May 11 2024

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001045.

CoefficientList[Series[x (1+x+3x^2+5x^3)/(1-x^4), {x,0,150}], x]  (* Harvey P. Dale, Mar 26 2011 *)

G.f.: x*(1+x+3x^2+5x^3)/(1-x^4); e.g.f.: 2*cos(x) - sin(x) + exp(-x)/2 + 5*exp(x)/2 - 5; a(n) = 2*cos(Pi*n/2) - sin(Pi*n/2) + (-1)^n/2 + 5/2 - 5*0^n.

a(n) = a(n-4) for n > 4. - Elmo R. Oliveira, May 11 2024

A102563 a(n) = A000120(A001045(n)) - A001045(A000120(n)).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 1, 3, 4, 4, 3, 5, 4, 4, 3, 7, 8, 8, 7, 9, 8, 8, 7, 11, 10, 10, 9, 11, 10, 10, 5, 15, 16, 16, 15, 17, 16, 16, 15, 19, 18, 18, 17, 19, 18, 18, 13, 23, 22, 22, 21, 23, 22, 22, 17, 25, 24, 24, 19, 25, 20, 20, 11, 31, 32, 32, 31, 33, 32, 32, 31, 35, 34, 34, 33, 35, 34

Offset: 0

Paul Barry, Jan 14 2005

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A000079, A000120, A000225, A001045, A102564, 102565.

j[n_] := (2^n - (-1)^n)/3; s[n_] := DigitCount[n, 2, 1]; a[n_] := s[j[n]] - j[s[n]]; Array[a, 100, 0] (* Amiram Eldar, Jul 22 2023 *)

It would appear that a(2^(n+1)) = A000225(n).
a(2^(n+1)-1) = A001045(n).
a(2^(n+1)+1) = A000079(n).
a(2n+1) = n; a(4n+1) - a(2n+1) = n; a(8n+1) - a(4n+1) = 2n; a(8n+1) - a(2n+1) = 3n.

A108924 J(n)^2+J(n+1)^2, with J(n) the Jacobsthal number A001045(n).

Original entry on oeis.org

1, 2, 10, 34, 146, 562, 2290, 9074, 36466, 145522, 582770, 2329714, 9321586, 37280882, 149134450, 596515954, 2386107506, 9544342642, 38177545330, 152709831794, 610840026226, 2443358706802, 9773437623410, 39093744901234

Offset: 0

Paul Barry, Jul 17 2005

Index entries for linear recurrences with constant coefficients, signature (3,6,-8).

Total/@Partition[LinearRecurrence[{1,2},{0,1},40]^2,2,1] (* or *) LinearRecurrence[ {3,6,-8},{1,2,10},40] (* Harvey P. Dale, Apr 11 2013 *)

def A108924(n): return (((1<Chai Wah Wu, Apr 18 2025

G.f.: (1-x-2x^2)/((1-x)(1-4x)(1+2x)); a(n)=3a(n-1)+6a(n-2)-8a(n-3); a(n)=(5/9)4^n+(2/9)(-2)^n+2/9.

a(0)=1, a(1)=2, a(2)=10, a(n)=3*a(n-1)+6*a(n-2)-8*a(n-3). - Harvey P. Dale, Apr 11 2013

A114283 Sequence array for binomial transform of Jacobsthal numbers A001045(n+1).

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 18, 6, 2, 1, 54, 18, 6, 2, 1, 162, 54, 18, 6, 2, 1, 486, 162, 54, 18, 6, 2, 1, 1458, 486, 162, 54, 18, 6, 2, 1, 4374, 1458, 486, 162, 54, 18, 6, 2, 1, 13122, 4374, 1458, 486, 162, 54, 18, 6, 2, 1, 39366, 13122, 4374, 1458, 486, 162, 54, 18, 6, 2, 1

Offset: 0

Paul Barry, Nov 20 2005

Sequence array for A025192. Row sums are 3^n, A000244. Diagonal sums are A015518(n+1). Inverse is A114284.

			Triangle begins
1;
2,1;
6,2,1;
18,6,2,1;
54,18,6,2,1;
162,54,18,6,2,1;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

a114283 n k = a114283_tabl !! n !! k
a114283_row n = a114283_tabl !! n
a114283_tabl = iterate
   (\row -> (sum $ zipWith (+) row $ reverse row) : row) [1]
-- Reinhard Zumkeller, Nov 27 2012

Riordan array ((1-x)/(1-3x), x).

A123638 Consider the 2^n compositions of n and count only those ending in an odd part with row sum A001045.

Original entry on oeis.org

1, 1, 3, 8, 25, 83, 299, 1158, 4813, 21373, 100955, 504916, 2662761, 14754311, 85643459, 519493938, 3285790317, 21628225041, 147887079907, 1048634836288, 7698589399833, 58432476430139, 457901993065915, 3700291495531166

Offset: 1

Alford Arnold, Oct 04 2006

nonn

Compositions ending in an even part yield sequence 0 1 2 6 18 ... A123639. and a(n)+A123639(n) = A047970(n). Ending parity of compositions can be detected using mod(A065120,2)

			4
31 32 33
211 221 222
1111
Consider the above multisets: permute and note the parity of the ending part of each of the 14 compositions.
4
31 13 32 23 33
211 121 112 221 212 122 222
1111
4 is even
31 13 23 and 33 are odd
32 is even
etc
there are 0 + 4 + 3 + 1 = 8 odd compositions therefore a(4)=8.

Cf. A001045 A047970 A065120 A123639 A123640 A123641.

g:= proc(b,t,l,m) option remember; if t=0 then b*l else add (g(b, t-1, irem(k, 2), m), k=1..m-1) +g(1, t-1, irem(m, 2), m) fi end: a:= n-> add (g(0, k, 0, n+1-k), k=1..n): seq (a(n), n=1..30);

g[b_, t_, l_, m_] := g[b, t, l, m] = If[t == 0 , b*l , Sum[g[b, t-1, Mod[k, 2], m], {k, 1, m-1}] + g[1, t-1, Mod[m, 2], m]]; a[n_] := Sum[g[0, k, 0, n+1-k], {k, 1, n}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 04 2013, translated from Alois P. Heinz's Maple program *)

Offset corrected, Maple program and more terms added by Alois P. Heinz, Nov 06 2009

A129362 a(n) = Sum_{k=floor((n+1)/2)..n} J(k+1), J(k) = A001045(k).

Original entry on oeis.org

1, 1, 4, 8, 19, 37, 80, 160, 331, 661, 1344, 2688, 5419, 10837, 21760, 43520, 87211, 174421, 349184, 698368, 1397419, 2794837, 5591040, 11182080, 22366891, 44733781, 89473024, 178946048, 357903019, 715806037

Offset: 0

Paul Barry, Apr 11 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,0,-2,-4).

Cf. A001045, A129361.

A001045:= func< n | (2^n - (-1)^n)/3 >;
[(&+[A001045(n-j+1): j in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jan 31 2024

LinearRecurrence[{1,3,-1,0,-2,-4},{1,1,4,8,19,37},30] (* Harvey P. Dale, Oct 22 2011 *)

def A001045(n): return (2^n - (-1)^n)/3
def A129362(n): return sum(A001045(n-j+1) for j in range(1+(n//2)))
[A129362(n) for n in range(31)] # G. C. Greubel, Jan 31 2024

G.f.: (1+2*x^3)/((1-x-2*x^2)*(1-x^2-2*x^4)).
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - 2*a(n-5) - 4*a(n-6).
a(n) = Sum_{k=0..n} ( J(k+1) - J((k+1)/2)*(1-(-1)^k)/2 ).
a(n) = Sum_{j=0..floor(n/2)} A001045(n-j+1). - G. C. Greubel, Jan 31 2024

cp's OEIS Frontend

A073377 Seventh convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

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A073378 Eighth convolution of A001045(n+1) (generalized (1,2)-Fibonacci), n>=0, with itself.

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A073402 Coefficient triangle of polynomials (rising powers) related to convolutions of A001045(n+1), n >= 0, (generalized (1,2)-Fibonacci). Companion triangle is A073401.

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A081857 Jacobsthal sequence (A001045) as represented in base 4.

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A091084 a(n) = A001045(n) mod 10.

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A102563 a(n) = A000120(A001045(n)) - A001045(A000120(n)).

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A108924 J(n)^2+J(n+1)^2, with J(n) the Jacobsthal number A001045(n).

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A114283 Sequence array for binomial transform of Jacobsthal numbers A001045(n+1).

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