A337277 -id:A337277 - cp's OEIS frontend

A334627 T(n,k) is the number of k's in the n-th row of Stern's triangle (A337277); triangle T(n,k), n >= 0, 1 <= k <= A000045(n+1), read by rows.

1, 3, 5, 2, 7, 4, 4, 9, 6, 8, 4, 4, 11, 8, 12, 8, 12, 0, 8, 4, 13, 10, 16, 12, 20, 4, 16, 8, 8, 4, 8, 4, 4, 15, 12, 20, 16, 28, 8, 28, 12, 16, 8, 24, 8, 16, 8, 4, 4, 8, 8, 8, 0, 4, 17, 14, 24, 20, 36, 12, 40, 20, 24, 12, 40, 12, 36, 16, 8, 16, 28, 16, 24, 4, 8, 8, 16, 4, 12, 8, 8, 0, 12, 4, 8, 0, 0, 4

Offset: 0

Alois P. Heinz, Sep 09 2020

All terms in the first column are odd, all other terms are even.

			T(0,1) = 1 because Stern's triangle has one 1 in row n=0.
T(2,2) = 2 because Stern's triangle has two 2's in row n=2.
T(4,3) = 8 because Stern's triangle has eight 3's in row n=4.
Triangle T(n,k) begins:
   1;
   3;
   5,  2;
   7,  4,  4;
   9,  6,  8,  4,  4;
  11,  8, 12,  8, 12, 0,  8,  4;
  13, 10, 16, 12, 20, 4, 16,  8,  8, 4,  8, 4,  4;
  15, 12, 20, 16, 28, 8, 28, 12, 16, 8, 24, 8, 16, 8, 4, 4, 8, 8, 8, 0, 4;
  ...

Alois P. Heinz, Rows n = 0..21, flattened

Column k=1 gives A005408.
Row sums give A126646.
Row lengths give A000045(n+1).
Cf. A000244, A337277.

b:= proc(n) option remember; `if`(n=0, [1], (l-> [1, l[1],
      seq([l[i-1]+l[i], l[i]][], i=2..nops(l)), 1])(b(n-1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(add(x^j, j=b(n))):
seq(T(n), n=0..8);

Sum_{k=1..A000045(n+1)} k * T(n,k) = A000244(n).

A339025 Sum of n-th powers of entries in the n-th row of Stern's triangle (A337277).

Original entry on oeis.org

1, 3, 13, 147, 4277, 314403, 58215317, 27104094867, 31830051961045, 94398513955640643, 709919097675516974293, 13569078873978509433342387, 661668739571948876787281152277, 82526665791586458931717457637364323, 26412772665617176235336349304356162390677

Offset: 0

Alois P. Heinz, Nov 19 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..28

Cf. A337277.

b:= proc(n) option remember; `if`(n=0, 1, (h-> [1, h[1], seq(
      [h[i-1]+h[i], h[i]][], i=2..nops(h)), 1][])([b(n-1)]))
    end:
a:= proc(n) option remember; add(i^n, i=[b(n)]) end:
seq(a(n), n=0..15);

nmax = 15;
T = Nest[Append[#, Flatten@Join[{1}, If[Length@# > 1, Map[{#1, #1 + #2}& @@ #&, Partition[#[[-1]], 2, 1]], {}], {#[[-1, -1]]}, {1}]]&, {{1}}, nmax];
a[n_] := T[[n+1]]^n // Total;
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, May 30 2022, after Michael De Vlieger in A337277 *)

A338717 a(n) = sum of 4th powers of entries in row n of Stern's triangle A337277.

Original entry on oeis.org

1, 3, 37, 395, 4277, 46251, 500213, 5409835, 58507765, 632765739, 6843407605, 74011952171, 800444658677, 8656867341099, 93624651434741, 1012557431099947, 10950882439229941, 118434591969329451, 1280878746784164085, 13852797030687146027, 149819009843990278133

Offset: 0

N. J. A. Sloane, Nov 19 2020

Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019. Also American Mathematical Monthly 127.2 (2020): 99-111.
Index entries for linear recurrences with constant coefficients, signature (10,9,-2).

Cf. A337277.

For 2nd and 3rd powers see A052984, A169634.

LinearRecurrence[{10,9,-2},{1,3,37},30] (* Harvey P. Dale, Apr 07 2022 *)

G.f.: -(2*x^2+7*x-1)/((x+1)*(2*x^2-11*x+1)). - Alois P. Heinz, Nov 19 2020

A106709 Expansion of g.f. -2x/(1 - 5x + 2*x^2).

Original entry on oeis.org

0, -2, -10, -46, -210, -958, -4370, -19934, -90930, -414782, -1892050, -8630686, -39369330, -179585278, -819187730, -3736768094, -17045465010, -77753788862, -354678014290, -1617882493726, -7380056440050, -33664517212798, -153562473183890, -700483331493854

Offset: 0

Roger L. Bagula, May 30 2005

Let T(n,k) denote the k-th element of row n of Stern's triangle (see A337277). Then b(n) = Sum_k T(n,k)*T(n,k+1) gives the present sequence (without the signs). - N. J. A. Sloane, Nov 19 2020

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019. Also American Mathematical Monthly 127.2 (2020): 99-111.
Index entries for linear recurrences with constant coefficients, signature (5,-2).

Cf. A107839, A337277.

I:=[0,-2]; [n le 2 select I[n] else 5*Self(n-1) - 2*Self(n-2): n in [1..41]]; // G. C. Greubel, Sep 10 2021

a:= n-> (<<0|-2>, <1|5>>^n)[1,2]:
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 19 2020

LinearRecurrence[{5,-2}, {0,-2}, 41] (* G. C. Greubel, Sep 10 2021 *)

[-round(sqrt(2)^(n+1)*chebyshev_U(n-1, 5/(2*sqrt(2)))) for n in (0..40)] # G. C. Greubel, Sep 10 2021

a(n) = -2*A107839(n-1), n>0.
a(n) = first entry of v(n), where v(n) = M*v(n-1), M is the 2 X 2 matrix ({0, -2}, {1, 5}) and v(0) is the column vector (0, 1).
G.f.: -2*x/(1-5*x+2*x^2). - Alois P. Heinz, Nov 26 2020
a(n) = -sqrt(2)^(n+1)*ChebyshevU(n-1, 5/(2*sqrt(2))). - G. C. Greubel, Sep 10 2021

Edited by N. J. A. Sloane, Apr 30 2006

New name by G. C. Greubel, Sep 10 2021

cp's OEIS Frontend

A334627 T(n,k) is the number of k's in the n-th row of Stern's triangle (A337277); triangle T(n,k), n >= 0, 1 <= k <= A000045(n+1), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A339025 Sum of n-th powers of entries in the n-th row of Stern's triangle (A337277).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A338717 a(n) = sum of 4th powers of entries in row n of Stern's triangle A337277.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A106709 Expansion of g.f. -2*x/(1 - 5*x + 2*x^2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Sage

Formula

Extensions

A106709 Expansion of g.f. -2x/(1 - 5x + 2*x^2).