A129096 -id:A129096 - cp's OEIS frontend

A129095 Semi-Pell numbers: a(n) = a(n/2) (n even), a(n) = 2*a(n-1) + a(n-2) (n odd >1), with a(1) = 1.

1, 1, 3, 1, 5, 3, 11, 1, 13, 5, 23, 3, 29, 11, 51, 1, 53, 13, 79, 5, 89, 23, 135, 3, 141, 29, 199, 11, 221, 51, 323, 1, 325, 53, 431, 13, 457, 79, 615, 5, 625, 89, 803, 23, 849, 135, 1119, 3, 1125, 141, 1407, 29, 1465, 199, 1863, 11, 1885, 221, 2327, 51, 2429, 323

Offset: 1

Paul D. Hanna, Apr 11 2007

Bisection A129096 is monotonically increasing.

			Terms may be arranged into an irregular-shaped triangle:
  1;
  1, 3;
  1, 5, 3, 11;
  1, 13, 5, 23, 3, 29, 11, 51;
  1, 53, 13, 79, 5, 89, 23, 135, 3, 141, 29, 199, 11, 221, 51, 323;
  ...
where final terms of rows form A129097,
central terms are given by A129098,
and row sums are equal to A129099.

Paul D. Hanna, Table of n, a(n) for n = 1..1024
Cristina Ballantine and George Beck, Partitions enumerated by self-similar sequences, arXiv:2303.11493 [math.CO], 2023.
William J. Keith and Augustine O. Munagi, Binary compositions and semi-Pell compositions, arXiv:1912.11148 [math.CO], 2019.
William J. Keith and Augustine O. Munagi, Power compositions and semi-Pell compositions, Univ. Rochester, Online J. Analytic Comb. (2023) Issue 18, Art No. 2. See p. 2.

Cf. A129096, A129097, A129098, A129099.

Cf. A030067 (semi-Fibonacci), A074364 (semi-tribonacci).

Nest[Append[#1, If[EvenQ[#2], #1[[#2/2]], 2 #1[[-1]] + #1[[-2]] ] ] & @@ {#, Length@ # + 1} &, {1}, 61] (* Michael De Vlieger, Mar 10 2020 *)

a(n)=if(n==1 || n==0,1,if(n%2==0,a(n/2),2*a(n-1)+a(n-2)))

A129097 a(n) = A129095(2^n - 1) for n>=1.

Original entry on oeis.org

1, 3, 11, 51, 323, 3075, 47427, 1230787, 54772163, 4247022531, 582413946819, 143061847179203, 63604391931367363, 51621812365091401667, 77028054123935294320579, 212592144046864728487817155

Offset: 1

Paul D. Hanna, Apr 11 2007

b(n)=A129095(n) obeys the recurrence: b(n) = b(n/2) (n even), b(n) = 2*b(n-1) + b(n-2) (n odd >1), with b(1) = 1.

Cf. A129095, A129096, A129098, A129099.

Block[{e = 16, s}, s = Nest[Append[#1, If[EvenQ[#2], #1[[#2/2]], 2 #1[[-1]] + #1[[-2]] ] ] & @@ {#, Length@ # + 1} &, {1}, 2^e]; Array[s[[2^# - 1]] &, e]] (* Michael De Vlieger, Mar 10 2020 *)

PARI

A129098 a(n) = A129095(2^n + 2^(n-1) - 1) for n>=1.

Original entry on oeis.org

1, 5, 23, 135, 1119, 14319, 300015, 10636463, 652217135, 70382845743, 13551477257519, 4706105734658351, 2973879284783561007, 3444999327807280048431, 7362415635261959807011119, 29188908702092573515760044335

Offset: 1

Paul D. Hanna, Apr 11 2007

b(n)=A129095(n) obeys the recurrence: b(n) = b(n/2) (n even), b(n) = 2*b(n-1) + b(n-2) (n odd >1), with b(1) = 1.

Cf. A129095, A129096, A129097, A129099.

Block[{e = 18, s}, s = Nest[Append[#1, If[EvenQ[#2], #1[[#2/2]], 2 #1[[-1]] + #1[[-2]] ] ] & @@ {#, Length@ # + 1} &, {1}, 2^e]; Array[s[[2^# + 2^(# - 1) - 1]] &, e - 1]] (* Michael De Vlieger, Mar 10 2020 *)

PARI

a(16) from Michael De Vlieger, Mar 10 2020.

A129099 a(n) = Sum_{k=2^(n-1)..2^n-1} A129095(k) for n>=1.

Original entry on oeis.org

1, 4, 20, 136, 1376, 22176, 591680, 26770688, 2096125184, 289083462144, 71239716616192, 31730665042094080, 25779103986580017152, 38488216155785101459456, 106257557996370396596748288, 545336631331873524033714683904

Offset: 1

Paul D. Hanna, Apr 11 2007

b(n)=A129095(n) obeys the recurrence: b(n) = b(n/2) (n even), b(n) = 2*b(n-1) + b(n-2) (n odd >1), with b(1) = 1.

Cf. A129095, A129096, A129097, A129098.

Block[{e = 16, s}, s = Nest[Append[#1, If[EvenQ[#2], #1[[#2/2]], 2 #1[[-1]] + #1[[-2]] ] ] & @@ {#, Length@ # + 1} &, {1}, 2^e]; Array[Total@ s[[2^# ;; 2^(# + 1) - 1]] &, e, 0] ] (* Michael De Vlieger, Mar 10 2020 *)

PARI

a(n) = ( A129097(n+1) - A129097(n) )/2.

a(16) from Michael De Vlieger, Mar 10 2020

cp's OEIS Frontend

A129095 Semi-Pell numbers: a(n) = a(n/2) (n even), a(n) = 2*a(n-1) + a(n-2) (n odd >1), with a(1) = 1.

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A129097 a(n) = A129095(2^n - 1) for n>=1.

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A129098 a(n) = A129095(2^n + 2^(n-1) - 1) for n>=1.

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A129099 a(n) = Sum_{k=2^(n-1)..2^n-1} A129095(k) for n>=1.

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