A052984 -id:A052984 - cp's OEIS frontend

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

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

A385082 Sum of squared coefficients of Product_{i=0..n-1} 1+x^(2^i+1)+x^(2^(i+1)+1).

Original entry on oeis.org

1, 3, 13, 55, 249, 1121, 5025, 22607, 101931, 460877, 2088687, 9482763, 43109307, 196163983, 893222041, 4069162197, 18543631161, 84525140297, 385343891847, 1756959373157, 8011450183181, 36533108258455, 166602342944307, 759783053580809, 3465042771956289, 15802856371611411

Offset: 0

Michel Marcus, Jun 16 2025

nonn

Shalosh B. Ekhad and Doron Zeilberger, Automated Generation of Generating Functions Related to Generalized Stern's Diatomic Arrays in the footsteps of Richard Stanley, arXiv:2103.12855 [math.CO], 2021-2024.
Jinlong Tang and Guoce Xin, Meeting a Challenge raised by Ekhad and Zeilberger related to Stern's Triangle, arXiv:2506.13375 [math.CO], 2025.

Cf. A052984 (with 1+x^(2^i)+x^(2^(i+1)) instead).

b:= proc(n) option remember; expand(`if`(n<0, 1,
       b(n-1)*(1+x^(2^n+1)+x^(2^(n+1)+1))))
    end:
a:= n-> add(i^2, i=[coeffs(b(n-1))]):
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 17 2025

a[n_]:=Total[CoefficientList[Product[ 1+x^(2^i+1)+x^(2^(i+1)+1),{i,0,n-1}],x]^2]; Array[a,20,0] (* Stefano Spezia, Jun 17 2025 *)

a(n) = norml2(Vec(prod(i=0, n-1, 1+x^(2^i+1)+x^(2^(i+1)+1))));

from collections import Counter
from itertools import count, islice
def A385082_gen(): # generator of terms
    c = Counter({0:1})
    for n in count(0):
        yield sum(i**2 for i in c.values())
        c = sum((Counter({i:j,(m:=1<A385082_list = list(islice(A385082_gen(),10)) # Chai Wah Wu, Jun 18 2025

cp's OEIS Frontend

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

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